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Existence and non-existence of ergodic Hilbert transform for
admissible processesDogan Comez

Sketch of the proof eHt along moving averages sequences

Set up

Dirichlet problem in the upper half plane:

Given f (x), find u(x , y), harmonic in the upper half plane and agrees with f (x) on the real line; that is,

∂2u ∂x2 + ∂2u

Solution: u(x , y) = y π

∫∞ −∞

(x−s)2+y2 ds.

Harmonic conjugate of u(x , y) in the upper half plane is

v(x , y) = 1

∫∞ −∞

s ds.

Besicovitch (early 1930’s): Hf exists a.e. for all integrable f on R.

Dogan Comez Existence and non-existence of ergodic Hilbert transform for admissible processes

Set up History.

Sketch of the proof eHt along moving averages sequences

Set up

Dirichlet problem in the upper half plane: Given f (x), find u(x , y), harmonic in the upper half plane and agrees with f (x) on the real line; that is,

∂2u ∂x2 + ∂2u

Solution: u(x , y) = y π

∫∞ −∞

(x−s)2+y2 ds.

Harmonic conjugate of u(x , y) in the upper half plane is

v(x , y) = 1

∫∞ −∞

s ds.

Besicovitch (early 1930’s): Hf exists a.e. for all integrable f on R.

Dogan Comez Existence and non-existence of ergodic Hilbert transform for admissible processes

Set up History.

Sketch of the proof eHt along moving averages sequences

Set up

Dirichlet problem in the upper half plane: Given f (x), find u(x , y), harmonic in the upper half plane and agrees with f (x) on the real line; that is,

∂2u ∂x2 + ∂2u

Solution: u(x , y) = y π

∫∞ −∞

(x−s)2+y2 ds.

Harmonic conjugate of u(x , y) in the upper half plane is

v(x , y) = 1

∫∞ −∞

s ds.

Besicovitch (early 1930’s): Hf exists a.e. for all integrable f on R.

Dogan Comez Existence and non-existence of ergodic Hilbert transform for admissible processes

Set up History.

Sketch of the proof eHt along moving averages sequences

Set up

Dirichlet problem in the upper half plane: Given f (x), find u(x , y), harmonic in the upper half plane and agrees with f (x) on the real line; that is,

∂2u ∂x2 + ∂2u

Solution: u(x , y) = y π

∫∞ −∞

(x−s)2+y2 ds.

Harmonic conjugate of u(x , y) in the upper half plane is

v(x , y) = 1

∫∞ −∞

s ds.

Besicovitch (early 1930’s): Hf exists a.e. for all integrable f on R.

Dogan Comez Existence and non-existence of ergodic Hilbert transform for admissible processes

Set up History.

Sketch of the proof eHt along moving averages sequences

Set up

Dirichlet problem in the upper half plane: Given f (x), find u(x , y), harmonic in the upper half plane and agrees with f (x) on the real line; that is,

∂2u ∂x2 + ∂2u

Solution: u(x , y) = y π

∫∞ −∞

(x−s)2+y2 ds.

Harmonic conjugate of u(x , y) in the upper half plane is

v(x , y) = 1

∫∞ −∞

s ds.

Besicovitch (early 1930’s): Hf exists a.e. for all integrable f on R.

Dogan Comez Existence and non-existence of ergodic Hilbert transform for admissible processes

Set up History.

Sketch of the proof eHt along moving averages sequences

Set up

Dirichlet problem in the upper half plane: Given f (x), find u(x , y), harmonic in the upper half plane and agrees with f (x) on the real line; that is,

∂2u ∂x2 + ∂2u

Solution: u(x , y) = y π

∫∞ −∞

(x−s)2+y2 ds.

Harmonic conjugate of u(x , y) in the upper half plane is

v(x , y) = 1

∫∞ −∞

s ds.

Besicovitch (early 1930’s): Hf exists a.e. for all integrable f on R.

Dogan Comez Existence and non-existence of ergodic Hilbert transform for admissible processes

Set up History.

Sketch of the proof eHt along moving averages sequences

Ergodic setting

Observe: Lebesgue measure is translation invariant; hence, the map Us(x) = x − s

preserves Lebesgue measure, namely, m(U−1 s E) = m(E).

Hilbert transform of f can be rewritten as:

Hf (x) = lim ε→0

∫ ε<|s|<1/ε

f (x − s)

s ds = lim

f (Usx)

s dt.

More generally, if τ = {Tt}t∈R is a group of invertible m.p.t’s on (X ,Σ, µ), then the ergodic Hilbert transform (eHt) of f is defined as

Hτ f (x) = lim ε→0

∫ ε<|s|<1/ε

f (Tsx)

s ds.

Discrete version: If T : X → X is an i.m.p.t. (i.e., τ = {T n}n∈Z), the (discrete) ergodic Hilbert transform of f is

HT f (x) = lim n→∞

∑ 0<|k|≤n

f (T kx)

Dogan Comez Existence and non-existence of ergodic Hilbert transform for admissible processes

Set up History.

Sketch of the proof eHt along moving averages sequences

Ergodic setting

Observe: Lebesgue measure is translation invariant; hence, the map Us(x) = x − s

preserves Lebesgue measure, namely, m(U−1 s E) = m(E).

Hilbert transform of f can be rewritten as:

Hf (x) = lim ε→0

∫ ε<|s|<1/ε

f (x − s)

s ds = lim

f (Usx)

s dt.

More generally, if τ = {Tt}t∈R is a group of invertible m.p.t’s on (X ,Σ, µ), then the ergodic Hilbert transform (eHt) of f is defined as

Hτ f (x) = lim ε→0

∫ ε<|s|<1/ε

f (Tsx)

s ds.

Discrete version: If T : X → X is an i.m.p.t. (i.e., τ = {T n}n∈Z), the (discrete) ergodic Hilbert transform of f is

HT f (x) = lim n→∞

∑ 0<|k|≤n

f (T kx)

Dogan Comez Existence and non-existence of ergodic Hilbert transform for admissible processes

Set up History.

Sketch of the proof eHt along moving averages sequences

Ergodic setting

Observe: Lebesgue measure is translation invariant; hence, the map Us(x) = x − s

preserves Lebesgue measure, namely, m(U−1 s E) = m(E).

Hilbert transform of f can be rewritten as:

Hf (x) = lim ε→0

∫ ε<|s|<1/ε

f (x − s)

s ds = lim

f (Usx)

s dt.

More generally, if τ = {Tt}t∈R is a group of invertible m.p.t’s on (X ,Σ, µ), then the ergodic Hilbert transform (eHt) of f is defined as

Hτ f (x) = lim ε→0

∫ ε<|s|<1/ε

f (Tsx)

s ds.

Discrete version: If T : X → X is an i.m.p.t. (i.e., τ = {T n}n∈Z), the (discrete) ergodic Hilbert transform of f is

HT f (x) = lim n→∞

∑ 0<|k|≤n

f (T kx)

Dogan Comez Existence and non-existence of ergodic Hilbert transform for admissible processes

Set up History.

Sketch of the proof eHt along moving averages sequences

Ergodic setting

Observe: Lebesgue measure is translation invariant; hence, the map Us(x) = x − s

preserves Lebesgue measure, namely, m(U−1 s E) = m(E).

Hilbert transform of f can be rewritten as:

Hf (x) = lim ε→0

∫ ε<|s|<1/ε

f (x − s)

s ds = lim

f (Usx)

s dt.

More generally, if τ = {Tt}t∈R is a group of invertible m.p.t’s on (X ,Σ, µ), then the ergodic Hilbert transform (eHt) of f is defined as

Hτ f (x) = lim ε→0

∫ ε<|s|<1/ε

f (Tsx)

s ds.

Discrete version: If T : X → X is an i.m.p.t. (i.e., τ = {T n}n∈Z), the (discrete) ergodic Hilbert transform of f is

HT f (x) = lim n→∞

∑ 0<|k|≤n

f (T kx)

Dogan Comez Existence and non-existence of ergodic Hilbert transform for admissible processes

Set up History.

Sketch of the proof eHt along moving averages sequences

Ergodic theorems

Let (X ,Σ, µ) be a probability space. A m.pt. T : X → X is called ergodic if T−1E = E implies µ(E) = 0 or µ(E) = 1.

Crown jewel of ergodic theory:

Birkhoff’s Ergodic Theorem. (1931)

If T : X → X is m.p.t., then, limn→∞ 1 n

∑n−1 k=0 f (T kx) = f ∗(x) exists a.e. for all

f ∈ L1. If, furthermore, T is ergodic, then f ∗ = ∫ f .

Continuous parameter version:

Local Ergodic Theorem. (Wiener, 1939)

If τ is a m.p. flow and f ∈ L1, then limt→0+ 1 t

∫ t 0 f (Tsx)ds = f (x) a.e.

Question: Analogous results for the (discrete or local) eHt?

Dogan Comez Existence and non-existence of ergodic Hilbert transform for admissible processes

Set up History.

Sketch of the proof eHt along moving averages sequences

Ergodic theorems

Let (X ,Σ, µ) be a probability space. A m.pt. T : X → X is called ergodic if T−1E = E implies µ(E) = 0 or µ(E) = 1.

Crown jewel of ergodic theory:

Birkhoff’s Ergodic Theorem. (1931)

If T : X → X is m.p.t., then, limn→∞ 1 n

∑n−1 k=0 f (T kx) = f ∗(x) exists a.e. for all

f ∈ L1. If, furthermore, T is ergodic, then f ∗ = ∫ f .

Continuous parameter version:

Local Ergodic Theorem. (Wiener, 1939)

If τ is a m.p. flow and f ∈ L1, then limt→0+ 1 t

∫ t 0 f (Tsx)ds = f (x) a.e.

Question: Analogous results for the (discrete or local) eHt?

Dogan Comez Existence and non-existence of ergodic Hilbert transform for admissible processes

Set up History.

Sketch of the proof eHt along moving averages sequences

Ergodic theorems

Let (X ,Σ, µ) be a probability space. A m.pt. T : X → X is called ergodic if T−1E = E implies µ(E) = 0 or µ(E) = 1.

Crown jewel of ergodic theory:

Birkhoff’s Ergodic Theorem. (1931)

If T : X → X is m.p.t., then, limn→∞ 1 n

∑n−1 k=0 f (T kx) = f ∗(x) exists a.e. for all

f ∈ L1. If, furthermore, T is ergodic, then f ∗ = ∫ f .

Continuous parameter version:

Local Ergodic Theorem. (Wiener, 1939)

If τ is a m.p. flow and f ∈ L1, then limt→0+ 1 t

∫ t 0 f (Tsx)ds = f (x) a.e.

Question: Analogous results for the (discrete or local) eHt?

Dogan Comez Existence and non-existence of ergodic Hilbert transform for admissible processes

Set up History.

Sketch of the proof eHt along moving averages sequences

Ergodic theorems

Let (X ,Σ, µ) be a probability space. A m.pt. T : X → X is called ergodic if T−1E = E implies µ(E) = 0 or µ(E) = 1.

Crown jewel of ergodic theory:

Birkhoff’s Ergodic Theorem. (1931)

If T : X → X is m.p.t., then, limn→∞ 1 n

∑n−1 k=0 f (T kx) = f ∗(x) exists a.e. for all

f ∈ L1. If, furthermore, T is ergodic, then f ∗ = ∫ f .

Continuous parameter version:

Local Ergodic Theorem. (Wiener, 1939)

If τ is a m.p. flow and f ∈ L1, then limt→0+ 1 t

∫ t 0 f (Tsx)ds = f (x) a.e.

Question: Analogous results for the (discrete or local) eHt?

Dogan Comez Existence and non-existence of ergodic Hilbert transform for admissible processes

Set up History.

Sketch of the proof eHt along moving averages sequences

Existence of eHt

HT f (x) = lim n→∞

∑ 0<|k|≤n

f (T kx)

ε→0

M. Cotlar (1955)

If τ is an i.m.p. flow (or T is an i.m.p.t.) on X , then Hτ f (x) exists a.e. for all f ∈ L1.

K. Petersen (1983, 1985) - Another proof of eHt (both local and discrete).

R. Sato (1989) - eHt in the operator setting (discrete).

L.M. Fernadez-Cabrera, F. Martin-Reyes and J.L. Torrea (1995) - connection between ergodic everages and eHt (discrete).

D. Comez (2006) - eHt for admissible processes (discrete).

Dogan Comez Existence and non-existence of ergodic Hilbert transform for admissible processes

Set up History.

Sketch of the proof eHt along moving averages sequences

Existence of eHt

HT f (x) = lim n→∞

∑ 0<|k|≤n

f (T kx)

ε→0

M. Cotlar (1955)

If τ is an i.m.p. flow (or T is an i.m.p.t.) on X , then Hτ f (x) exists a.e. for all f ∈ L1.

K. Petersen (1983, 1985) - Another proof of eHt (both local and discrete).

R. Sato (1989) - eHt in the operator setting (discrete).

L.M. Fernadez-Cabrera, F. Martin-Reyes and J.L. Torrea (1995) - connection between ergodic everages and eHt (discrete).

D. Comez (2006) - eHt for admissible processes (discrete).

Dogan Comez Existence and non-existence of ergodic Hilbert transform for admissible processes

Set up History.

Sketch of the proof eHt along moving averages sequences

Existence of eHt

HT f (x) = lim n→∞

∑ 0<|k|≤n

f (T kx)

ε→0

M. Cotlar (1955)

If τ is an i.m.p. flow (or T is an i.m.p.t.) on X , then Hτ f (x) exists a.e. for all f ∈ L1.

K. Petersen (1983, 1985) - Another proof of eHt (both local and discrete).

R. Sato (1989) - eHt in the operator setting (discrete).

L.M. Fernadez-Cabrera, F. Martin-Reyes and J.L. Torrea (1995) - connection between ergodic everages and eHt (discrete).

D. Comez (2006) - eHt for admissible processes (discrete).

Dogan Comez Existence and non-existence of ergodic Hilbert transform for admissible processes

Set up History.

Sketch of the proof eHt along moving averages sequences

Existence of eHt

HT f (x) = lim n→∞

∑ 0<|k|≤n

f (T kx)

ε→0

M. Cotlar (1955)

If τ is an i.m.p. flow (or T is an i.m.p.t.) on X , then Hτ f (x) exists a.e. for all f ∈ L1.

K. Petersen (1983, 1985) - Another proof of eHt (both local and discrete).

R. Sato (1989) - eHt in the operator setting (discrete).

L.M. Fernadez-Cabrera, F. Martin-Reyes and J.L. Torrea (1995) - connection between ergodic everages and eHt (discrete).

D. Comez (2006) - eHt for admissible processes (discrete).

Dogan Comez Existence and non-existence of ergodic Hilbert transform for admissible processes

Set up History.

Sketch of the proof eHt along moving averages sequences

Existence of eHt

HT f (x) = lim n→∞

∑ 0<|k|≤n

f (T kx)

ε→0

M. Cotlar (1955)

If τ is an i.m.p. flow (or T is an i.m.p.t.) on X , then Hτ f (x) exists a.e. for all f ∈ L1.

K. Petersen (1983, 1985) - Another proof of eHt (both local and discrete).

R. Sato (1989) - eHt in the operator setting (discrete).

L.M. Fernadez-Cabrera, F. Martin-Reyes and J.L. Torrea (1995) - connection between ergodic everages and eHt (discrete).

D. Comez (2006) - eHt for admissible processes (discrete).

Dogan Comez Existence and non-existence of ergodic Hilbert transform for admissible processes

Set up History.

Sketch of the proof eHt along moving averages sequences

Existence of eHt

HT f (x) = lim n→∞

∑ 0<|k|≤n

f (T kx)

ε→0

M. Cotlar (1955)

If τ is an i.m.p. flow (or T is an i.m.p.t.) on X , then Hτ f (x) exists a.e. for all f ∈ L1.

K. Petersen (1983, 1985) - Another proof of eHt (both local and discrete).

R. Sato (1989) - eHt in the operator setting (discrete).

L.M. Fernadez-Cabrera, F. Martin-Reyes and J.L. Torrea (1995) - connection between ergodic everages and eHt (discrete).

D. Comez (2006) - eHt for admissible processes (discrete).

Dogan Comez Existence and non-existence of ergodic Hilbert transform for admissible processes

Set up History.

Sketch of the proof eHt along moving averages sequences

Existence of eHt

HT f (x) = lim n→∞

∑ 0<|k|≤n

f (T kx)

ε→0

M. Cotlar (1955)

If τ is an i.m.p. flow (or T is an i.m.p.t.) on X , then Hτ f (x) exists a.e. for all f ∈ L1.

K. Petersen (1983, 1985) - Another proof of eHt (both local and discrete).

R. Sato (1989) - eHt in the operator setting (discrete).

L.M. Fernadez-Cabrera, F. Martin-Reyes and J.L. Torrea (1995) - connection between ergodic everages and eHt (discrete).

D. Comez (2006) - eHt for admissible processes (discrete).

Dogan Comez Existence and non-existence of ergodic Hilbert transform for admissible processes

Set up History.

Sketch of the proof eHt along moving averages sequences

Results on the modulated eHt

Some other eHt results:

C. Demeter, M. Lacey, T. Tao & C. Thiele (2007)- Return time theorem for the eHt (discrete).

M. Lacey & E. Terwilleger (2008)- Wiener-Wintner type theorem for the eHt (local and discrete).

Corollary: limN→∞ ∑

induced by a trigonometric polynomial.

Comez (2006) - There is a bounded Besicovitch sequence {ak} such that the limit fails to exist.

A. Akhmedov & D. Comez, (2014)- Let Mα = {a : ∑n −n |ak | = O( nα−1

logα n )},

1 < α ≤ 2. If a ∈ Mα, then it is universally good for the eHt in L1.

There is a subclass Bα of bounded Besicovitch sequences s.t. if a ∈ Bα, then it is universally good for the eHt in L2.

Question. Does eHt exist along moving averages sequences? We will return!

Dogan Comez Existence and non-existence of ergodic Hilbert transform for admissible processes

Set up History.

Sketch of the proof eHt along moving averages sequences

Results on the modulated eHt

Some other eHt results:

C. Demeter, M. Lacey, T. Tao & C. Thiele (2007)- Return time theorem for the eHt (discrete).

M. Lacey & E. Terwilleger (2008)- Wiener-Wintner type theorem for the eHt (local and discrete).

Corollary: limN→∞ ∑

induced by a trigonometric polynomial.

Comez (2006) - There is a bounded Besicovitch sequence {ak} such that the limit fails to exist.

A. Akhmedov & D. Comez, (2014)- Let Mα = {a : ∑n −n |ak | = O( nα−1

logα n )},

1 < α ≤ 2. If a ∈ Mα, then it is universally good for the eHt in L1.

There is a subclass Bα of bounded Besicovitch sequences s.t. if a ∈ Bα, then it is universally good for the eHt in L2.

Question. Does eHt exist along moving averages sequences? We will return!

Dogan Comez Existence and non-existence of ergodic Hilbert transform for admissible processes

Set up History.

Sketch of the proof eHt along moving averages sequences

Results on the modulated eHt

Some other eHt results:

C. Demeter, M. Lacey, T. Tao & C. Thiele (2007)- Return time theorem for the eHt (discrete).

M. Lacey & E. Terwilleger (2008)- Wiener-Wintner type theorem for the eHt (local and discrete).

Corollary: limN→∞ ∑

induced by a trigonometric polynomial.

Comez (2006) - There is a bounded Besicovitch sequence {ak} such that the limit fails to exist.

A. Akhmedov & D. Comez, (2014)- Let Mα = {a : ∑n −n |ak | = O( nα−1

logα n )},

1 < α ≤ 2. If a ∈ Mα, then it is universally good for the eHt in L1.

There is a subclass Bα of bounded Besicovitch sequences s.t. if a ∈ Bα, then it is universally good for the eHt in L2.

Question. Does eHt exist along moving averages sequences? We will return!

Dogan Comez Existence and non-existence of ergodic Hilbert transform for admissible processes

Set up History.

Sketch of the proof eHt along moving averages sequences

Results on the modulated eHt

Some other eHt results:

C. Demeter, M. Lacey, T. Tao & C. Thiele (2007)- Return time theorem for the eHt (discrete).

M. Lacey & E. Terwilleger (2008)- Wiener-Wintner type theorem for the eHt (local and discrete).

Corollary: limN→∞ ∑

induced by a trigonometric polynomial.

Comez (2006) - There is a bounded Besicovitch sequence {ak} such that the limit fails to exist.

A. Akhmedov & D. Comez, (2014)- Let Mα = {a : ∑n −n |ak | = O( nα−1

logα n )},

1 < α ≤ 2. If a ∈ Mα, then it is universally good for the eHt in L1.

There is a subclass Bα of bounded Besicovitch sequences s.t. if a ∈ Bα, then it is universally good for the eHt in L2.

Question. Does eHt exist along moving averages sequences? We will return!

Dogan Comez Existence and non-existence of ergodic Hilbert transform for admissible processes

Set up History.

Sketch of the proof eHt along moving averages sequences

Results on the modulated eHt

Some other eHt results:

C. Demeter, M. Lacey, T. Tao & C. Thiele (2007)- Return time theorem for the eHt (discrete).

M. Lacey & E. Terwilleger (2008)- Wiener-Wintner type theorem for the eHt (local and discrete).

Corollary: limN→∞ ∑

induced by a trigonometric polynomial.

Comez (2006) - There is a bounded Besicovitch sequence {ak} such that the limit fails to exist.

A. Akhmedov & D. Comez, (2014)- Let Mα = {a : ∑n −n |ak | = O( nα−1

logα n )},

1 < α ≤ 2. If a ∈ Mα, then it is universally good for the eHt in L1.

There is a subclass Bα of bounded Besicovitch sequences s.t. if a ∈ Bα, then it is universally good for the eHt in L2.

Question. Does eHt exist along moving averages sequences? We will return!

Dogan Comez Existence and non-existence of ergodic Hilbert transform for admissible processes

Set up History.

Sketch of the proof eHt along moving averages sequences

Results on the modulated eHt

Some other eHt results:

C. Demeter, M. Lacey, T. Tao & C. Thiele (2007)- Return time theorem for the eHt (discrete).

M. Lacey & E. Terwilleger (2008)- Wiener-Wintner type theorem for the eHt (local and discrete).

Corollary: limN→∞ ∑

induced by a trigonometric polynomial.

Comez (2006) - There is a bounded Besicovitch sequence {ak} such that the limit fails to exist.

A. Akhmedov & D. Comez, (2014)- Let Mα = {a : ∑n −n |ak | = O( nα−1

logα n )},

1 < α ≤ 2. If a ∈ Mα, then it is universally good for the eHt in L1.

There is a subclass Bα of bounded Besicovitch sequences s.t. if a ∈ Bα, then it is universally good for the eHt in L2.

Question. Does eHt exist along moving averages sequences? We will return!

Dogan Comez Existence and non-existence of ergodic Hilbert transform for admissible processes

Set up History.

Sketch of the proof eHt along moving averages sequences

Results on the modulated eHt

Some other eHt results:

C. Demeter, M. Lacey, T. Tao & C. Thiele (2007)- Return time theorem for the eHt (discrete).

M. Lacey & E. Terwilleger (2008)- Wiener-Wintner type theorem for the eHt (local and discrete).

Corollary: limN→∞ ∑

induced by a trigonometric polynomial.

Comez (2006) - There is a bounded Besicovitch sequence {ak} such that the limit fails to exist.

A. Akhmedov & D. Comez, (2014)- Let Mα = {a : ∑n −n |ak | = O( nα−1

logα n )},

1 < α ≤ 2. If a ∈ Mα, then it is universally good for the eHt in L1.

There is a subclass Bα of bounded Besicovitch sequences s.t. if a ∈ Bα, then it is universally good for the eHt in L2.

Question. Does eHt exist along moving averages sequences? We will return!

Dogan Comez Existence and non-existence of ergodic Hilbert transform for admissible processes

Set up History.

Sketch of the proof eHt along moving averages sequences

Results on the modulated eHt

Some other eHt results:

C. Demeter, M. Lacey, T. Tao & C. Thiele (2007)- Return time theorem for the eHt (discrete).

M. Lacey & E. Terwilleger (2008)- Wiener-Wintner type theorem for the eHt (local and discrete).

Corollary: limN→∞ ∑

induced by a trigonometric polynomial.

Comez (2006) - There is a bounded Besicovitch sequence {ak} such that the limit fails to exist.

A. Akhmedov & D. Comez, (2014)- Let Mα = {a : ∑n −n |ak | = O( nα−1

logα n )},

1 < α ≤ 2. If a ∈ Mα, then it is universally good for the eHt in L1.

There is a subclass Bα of bounded Besicovitch sequences s.t. if a ∈ Bα, then it is universally good for the eHt in L2.

Question. Does eHt exist along moving averages sequences?

We will return!

Dogan Comez Existence and non-existence of ergodic Hilbert transform for admissible processes

Set up History.

Sketch of the proof eHt along moving averages sequences

Results on the modulated eHt

Some other eHt results:

C. Demeter, M. Lacey, T. Tao & C. Thiele (2007)- Return time theorem for the eHt (discrete).

M. Lacey & E. Terwilleger (2008)- Wiener-Wintner type theorem for the eHt (local and discrete).

Corollary: limN→∞ ∑

induced by a trigonometric polynomial.

Comez (2006) - There is a bounded Besicovitch sequence {ak} such that the limit fails to exist.

A. Akhmedov & D. Comez, (2014)- Let Mα = {a : ∑n −n |ak | = O( nα−1

logα n )},

1 < α ≤ 2. If a ∈ Mα, then it is universally good for the eHt in L1.

There is a subclass Bα of bounded Besicovitch sequences s.t. if a ∈ Bα, then it is universally good for the eHt in L2.

Question. Does eHt exist along moving averages sequences? We will return!

Dogan Comez Existence and non-existence of ergodic Hilbert transform for admissible processes

Set up History.

Sketch of the proof eHt along moving averages sequences

Admissible processes

F = {ft}t∈R ⊂ Lp is called a τ -admissible process on R if Ts ft ≤ fs+t and T−s f−t ≤ f−s−t for t, s ≥ 0.

Discrete version: F = {fn}n∈Z is T -admissible on Z if Tfk ≤ fk+1 and T−1f−k ≤ f−k−1 for k ∈ Z+.

An admissible process F = {ft}t∈R is called

positive if ft ≥ 0 for all t ∈ R, strongly bounded if γF := supt∈R ftp <∞, symmetric if T2t f−t = ft for all t ∈ R+.

D. Comez (2006) - If T : X → X is an i.m.p.t and F = {fn}n∈Z ⊂ L1 is a strongly

bounded symmetric T -admissible process, then limn ∑

1≤|k|≤n fk (x) k

exists a.e.

Question. How about extending the local eHt result to the admissible processes?

Dogan Comez Existence and non-existence of ergodic Hilbert transform for admissible processes

Set up History.

Sketch of the proof eHt along moving averages sequences

Admissible processes

F = {ft}t∈R ⊂ Lp is called a τ -admissible process on R if Ts ft ≤ fs+t and T−s f−t ≤ f−s−t for t, s ≥ 0.

Discrete version: F = {fn}n∈Z is T -admissible on Z if Tfk ≤ fk+1 and T−1f−k ≤ f−k−1 for k ∈ Z+.

An admissible process F = {ft}t∈R is called

positive if ft ≥ 0 for all t ∈ R, strongly bounded if γF := supt∈R ftp <∞, symmetric if T2t f−t = ft for all t ∈ R+.

D. Comez (2006) - If T : X → X is an i.m.p.t and F = {fn}n∈Z ⊂ L1 is a strongly

bounded symmetric T -admissible process, then limn ∑

1≤|k|≤n fk (x) k

exists a.e.

Question. How about extending the local eHt result to the admissible processes?

Dogan Comez Existence and non-existence of ergodic Hilbert transform for admissible processes

Set up History.

Sketch of the proof eHt along moving averages sequences

Admissible processes

F = {ft}t∈R ⊂ Lp is called a τ -admissible process on R if Ts ft ≤ fs+t and T−s f−t ≤ f−s−t for t, s ≥ 0.

Discrete version: F = {fn}n∈Z is T -admissible on Z if Tfk ≤ fk+1 and T−1f−k ≤ f−k−1 for k ∈ Z+.

An admissible process F = {ft}t∈R is called

positive if ft ≥ 0 for all t ∈ R, strongly bounded if γF := supt∈R ftp <∞, symmetric if T2t f−t = ft for all t ∈ R+.

D. Comez (2006) - If T : X → X is an i.m.p.t and F = {fn}n∈Z ⊂ L1 is a strongly

bounded symmetric T -admissible process, then limn ∑

1≤|k|≤n fk (x) k

exists a.e.

Question. How about extending the local eHt result to the admissible processes?

Dogan Comez Existence and non-existence of ergodic Hilbert transform for admissible processes

Set up History.

Sketch of the proof eHt along moving averages sequences

Admissible processes

F = {ft}t∈R ⊂ Lp is called a τ -admissible process on R if Ts ft ≤ fs+t and T−s f−t ≤ f−s−t for t, s ≥ 0.

Discrete version: F = {fn}n∈Z is T -admissible on Z if Tfk ≤ fk+1 and T−1f−k ≤ f−k−1 for k ∈ Z+.

An admissible process F = {ft}t∈R is called

positive if ft ≥ 0 for all t ∈ R, strongly bounded if γF := supt∈R ftp <∞, symmetric if T2t f−t = ft for all t ∈ R+.

D. Comez (2006) - If T : X → X is an i.m.p.t and F = {fn}n∈Z ⊂ L1 is a strongly

bounded symmetric T -admissible process, then limn ∑

1≤|k|≤n fk (x) k

exists a.e.

Question. How about extending the local eHt result to the admissible processes?

Dogan Comez Existence and non-existence of ergodic Hilbert transform for admissible processes

Set up History.

Sketch of the proof eHt along moving averages sequences

Admissible processes

F = {ft}t∈R ⊂ Lp is called a τ -admissible process on R if Ts ft ≤ fs+t and T−s f−t ≤ f−s−t for t, s ≥ 0.

Discrete version: F = {fn}n∈Z is T -admissible on Z if Tfk ≤ fk+1 and T−1f−k ≤ f−k−1 for k ∈ Z+.

An admissible process F = {ft}t∈R is called

positive if ft ≥ 0 for all t ∈ R, strongly bounded if γF := supt∈R ftp <∞, symmetric if T2t f−t = ft for all t ∈ R+.

D. Comez (2006) - If T : X → X is an i.m.p.t and F = {fn}n∈Z ⊂ L1 is a strongly

bounded symmetric T -admissible process, then limn ∑

1≤|k|≤n fk (x) k

exists a.e.

Question. How about extending the local eHt result to the admissible processes?

Dogan Comez Existence and non-existence of ergodic Hilbert transform for admissible processes

Set up History.

Sketch of the proof eHt along moving averages sequences

LeHt for admissible processes-set up

Recall: Local eHt of f is Hf (x) = limε→0

∫ ε≤|t|≤1/ε

ds.

Local ergodic Hilbert transform of a symmetric admissible process F = {ft}t∈R :

HF (x) = limε→0

ds.

Let F = {ft} be a positive, symmetric, strongly bounded τ -admissible process. Then:

M.A. Akcoglu & U. Krengel (1981) - ∃ a maximal τ -additive process {g Tt} such that g Tt(x) ≤ ft(x) a.e. for all t ∈ R. (Hence, we can assume a given admissible F a positive process.)

∃ a family of non-negative integrable functions {ut} s.t. ft = Ttu|t| for all t ∈ R ∃ δ ∈ L1 s.t. ft ≤ Ttδ, ∀ t ∈ R, and δ1 = supt∈R ft1 (exact dominant of F ).

Dogan Comez Existence and non-existence of ergodic Hilbert transform for admissible processes

Set up History.

Sketch of the proof eHt along moving averages sequences

LeHt for admissible processes-set up

Recall: Local eHt of f is Hf (x) = limε→0

∫ ε≤|t|≤1/ε

ds.

Local ergodic Hilbert transform of a symmetric admissible process F = {ft}t∈R :

HF (x) = limε→0

ds.

Let F = {ft} be a positive, symmetric, strongly bounded τ -admissible process. Then:

M.A. Akcoglu & U. Krengel (1981) - ∃ a maximal τ -additive process {g Tt} such that g Tt(x) ≤ ft(x) a.e. for all t ∈ R. (Hence, we can assume a given admissible F a positive process.)

∃ a family of non-negative integrable functions {ut} s.t. ft = Ttu|t| for all t ∈ R ∃ δ ∈ L1 s.t. ft ≤ Ttδ, ∀ t ∈ R, and δ1 = supt∈R ft1 (exact dominant of F ).

Dogan Comez Existence and non-existence of ergodic Hilbert transform for admissible processes

Set up History.

Sketch of the proof eHt along moving averages sequences

LeHt for admissible processes-set up

Recall: Local eHt of f is Hf (x) = limε→0

∫ ε≤|t|≤1/ε

ds.

Local ergodic Hilbert transform of a symmetric admissible process F = {ft}t∈R :

HF (x) = limε→0

ds.

Let F = {ft} be a positive, symmetric, strongly bounded τ -admissible process. Then:

M.A. Akcoglu & U. Krengel (1981) - ∃ a maximal τ -additive process {g Tt} such that g Tt(x) ≤ ft(x) a.e. for all t ∈ R. (Hence, we can assume a given admissible F a positive process.)

∃ a family of non-negative integrable functions {ut} s.t. ft = Ttu|t| for all t ∈ R ∃ δ ∈ L1 s.t. ft ≤ Ttδ, ∀ t ∈ R, and δ1 = supt∈R ft1 (exact dominant of F ).

Dogan Comez Existence and non-existence of ergodic Hilbert transform for admissible processes

Set up History.

Sketch of the proof eHt along moving averages sequences

LeHt for admissible processes-set up

Recall: Local eHt of f is Hf (x) = limε→0

∫ ε≤|t|≤1/ε

ds.

Local ergodic Hilbert transform of a symmetric admissible process F = {ft}t∈R :

HF (x) = limε→0

ds.

Let F = {ft} be a positive, symmetric, strongly bounded τ -admissible process. Then:

M.A. Akcoglu & U. Krengel (1981) - ∃ a maximal τ -additive process {g Tt} such that g Tt(x) ≤ ft(x) a.e. for all t ∈ R. (Hence, we can assume a given admissible F a positive process.)

∃ a family of non-negative integrable functions {ut} s.t. ft = Ttu|t| for all t ∈ R

∃ δ ∈ L1 s.t. ft ≤ Ttδ, ∀ t ∈ R, and δ1 = supt∈R ft1 (exact dominant of F ).

Dogan Comez Existence and non-existence of ergodic Hilbert transform for admissible processes

Set up History.

Sketch of the proof eHt along moving averages sequences

LeHt for admissible processes-set up

Recall: Local eHt of f is Hf (x) = limε→0

∫ ε≤|t|≤1/ε

ds.

Local ergodic Hilbert transform of a symmetric admissible process F = {ft}t∈R :

HF (x) = limε→0

ds.

Let F = {ft} be a positive, symmetric, strongly bounded τ -admissible process. Then:

M.A. Akcoglu & U. Krengel (1981) - ∃ a maximal τ -additive process {g Tt} such that g Tt(x) ≤ ft(x) a.e. for all t ∈ R. (Hence, we can assume a given admissible F a positive process.)

∃ a family of non-negative integrable functions {ut} s.t. ft = Ttu|t| for all t ∈ R ∃ δ ∈ L1 s.t. ft ≤ Ttδ, ∀ t ∈ R, and δ1 = supt∈R ft1 (exact dominant of F ).

Dogan Comez Existence and non-existence of ergodic Hilbert transform for admissible processes

Set up History.

Sketch of the proof eHt along moving averages sequences

LeHt for admissible processes-results

Let τ = {Tt}t∈R be an i.m.p. flow and F = {ft} ⊂ L1 be a symmetric, strongly bounded τ -admissible process with exact dominant δ.

Theorem.1 (Maximal Inequality)

For any λ > 0, there exists a constant C such that,

µ{x : sup q>0 | ∫ q≤|s|≤1/q

fs(x)

HF (x) = lim q→0

∫ q≤|s|≤1/q

Dogan Comez Existence and non-existence of ergodic Hilbert transform for admissible processes

Set up History.

Sketch of the proof eHt along moving averages sequences

LeHt for admissible processes-results

Let τ = {Tt}t∈R be an i.m.p. flow and F = {ft} ⊂ L1 be a symmetric, strongly bounded τ -admissible process with exact dominant δ.

Theorem.1 (Maximal Inequality)

For any λ > 0, there exists a constant C such that,

µ{x : sup q>0 | ∫ q≤|s|≤1/q

fs(x)

HF (x) = lim q→0

∫ q≤|s|≤1/q

Dogan Comez Existence and non-existence of ergodic Hilbert transform for admissible processes

Set up History.

Sketch of the proof eHt along moving averages sequences

LeHt for admissible processes-results

Let τ = {Tt}t∈R be an i.m.p. flow and F = {ft} ⊂ L1 be a symmetric, strongly bounded τ -admissible process with exact dominant δ.

Theorem.1 (Maximal Inequality)

For any λ > 0, there exists a constant C such that,

µ{x : sup q>0 | ∫ q≤|s|≤1/q

fs(x)

HF (x) = lim q→0

∫ q≤|s|≤1/q

Dogan Comez Existence and non-existence of ergodic Hilbert transform for admissible processes

Set up History.

Sketch of the proof eHt along moving averages sequences

One-sided leHt

∫ q<|s|<1/q

One-sided variant: limq→0

Ergodic version:

∫ q<|s|≤a

s ds exists a.e. for all f ∈ L1, a > 0.

Hence: existence of the local eHt and (*) implies that

(∗∗) H l f (x) := lim q→0

∫ a≤|s|<1/q

s ds exists a.e. for all f ∈ L1, a > 0.

Dogan Comez Existence and non-existence of ergodic Hilbert transform for admissible processes

Set up History.

Sketch of the proof eHt along moving averages sequences

One-sided leHt

∫ q<|s|<1/q

One-sided variant: limq→0

Ergodic version:

∫ q<|s|≤a

s ds exists a.e. for all f ∈ L1, a > 0.

Hence: existence of the local eHt and (*) implies that

(∗∗) H l f (x) := lim q→0

∫ a≤|s|<1/q

s ds exists a.e. for all f ∈ L1, a > 0.

Dogan Comez Existence and non-existence of ergodic Hilbert transform for admissible processes

Set up History.

Sketch of the proof eHt along moving averages sequences

One-sided leHt

∫ q<|s|<1/q

One-sided variant: limq→0

Ergodic version:

∫ q<|s|≤a

s ds exists a.e. for all f ∈ L1, a > 0.

Hence: existence of the local eHt and (*) implies that

(∗∗) H l f (x) := lim q→0

∫ a≤|s|<1/q

s ds exists a.e. for all f ∈ L1, a > 0.

Dogan Comez Existence and non-existence of ergodic Hilbert transform for admissible processes

Set up History.

Sketch of the proof eHt along moving averages sequences

One-sided leHt

∫ q<|s|<1/q

One-sided variant: limq→0

Ergodic version:

∫ q<|s|≤a

s ds exists a.e. for all f ∈ L1, a > 0.

Hence: existence of the local eHt and (*) implies that

(∗∗) H l f (x) := lim q→0

∫ a≤|s|<1/q

s ds exists a.e. for all f ∈ L1, a > 0.

Dogan Comez Existence and non-existence of ergodic Hilbert transform for admissible processes

Set up History.

Sketch of the proof eHt along moving averages sequences

One-sided leHt

∫ q<|s|<1/q

One-sided variant: limq→0

Ergodic version:

∫ q<|s|≤a

s ds exists a.e. for all f ∈ L1, a > 0.

Hence: existence of the local eHt and (*) implies that

(∗∗) H l f (x) := lim q→0

∫ a≤|s|<1/q

s ds exists a.e. for all f ∈ L1, a > 0.

Dogan Comez Existence and non-existence of ergodic Hilbert transform for admissible processes

Set up History.

Sketch of the proof eHt along moving averages sequences

Sketch of proof

Sketch of proof of Theorem.2 (assuming Theorem.1):

We can assume that ft ≥ 0 for each t. Recall ft = Ttu|t|, and ut ≥ 0, for all t ∈ R.

Let G1 = {Ttu0}t∈R. Then G1 t =

∫ t 0 Tsu0ds ≤ Ft for all t ∈ R.

Fix ε > 0, and pick t0 > 0 such that ut01 > δ1 − ε, and let G2 = {Tt ft0}. Then G2

t = ∫ t

0 Tsut0ds ≤ Ft for all t ≥ t0 and γG2 = ut01 > γF − ε.

Define the process G = {gt} by

gt =

Ttut0 when t0 ≤ t.

Then: G is τ -admissible, Gt ≤ Ft for all t ∈ R, and γG > γF − ε.

Dogan Comez Existence and non-existence of ergodic Hilbert transform for admissible processes

Set up History.

Sketch of the proof eHt along moving averages sequences

Sketch of proof

Sketch of proof of Theorem.2 (assuming Theorem.1):

We can assume that ft ≥ 0 for each t. Recall ft = Ttu|t|, and ut ≥ 0, for all t ∈ R.

Let G1 = {Ttu0}t∈R. Then G1 t =

∫ t 0 Tsu0ds ≤ Ft for all t ∈ R.

Fix ε > 0, and pick t0 > 0 such that ut01 > δ1 − ε, and let G2 = {Tt ft0}. Then G2

t = ∫ t

0 Tsut0ds ≤ Ft for all t ≥ t0 and γG2 = ut01 > γF − ε.

Define the process G = {gt} by

gt =

Ttut0 when t0 ≤ t.

Then: G is τ -admissible, Gt ≤ Ft for all t ∈ R, and γG > γF − ε.

Dogan Comez Existence and non-existence of ergodic Hilbert transform for admissible processes

Set up History.

Sketch of the proof eHt along moving averages sequences

Sketch of proof

Sketch of proof of Theorem.2 (assuming Theorem.1):

We can assume that ft ≥ 0 for each t. Recall ft = Ttu|t|, and ut ≥ 0, for all t ∈ R.

Let G1 = {Ttu0}t∈R. Then G1 t =

∫ t 0 Tsu0ds ≤ Ft for all t ∈ R.

Fix ε > 0, and pick t0 > 0 such that ut01 > δ1 − ε, and let G2 = {Tt ft0}. Then G2

t = ∫ t

0 Tsut0ds ≤ Ft for all t ≥ t0 and γG2 = ut01 > γF − ε.

Define the process G = {gt} by

gt =

Ttut0 when t0 ≤ t.

Then: G is τ -admissible, Gt ≤ Ft for all t ∈ R, and γG > γF − ε.

Dogan Comez Existence and non-existence of ergodic Hilbert transform for admissible processes

Set up History.

Sketch of the proof eHt along moving averages sequences

Sketch of proof

Sketch of proof of Theorem.2 (assuming Theorem.1):

We can assume that ft ≥ 0 for each t. Recall ft = Ttu|t|, and ut ≥ 0, for all t ∈ R.

Let G1 = {Ttu0}t∈R. Then G1 t =

∫ t 0 Tsu0ds ≤ Ft for all t ∈ R.

Fix ε > 0, and pick t0 > 0 such that ut01 > δ1 − ε, and let G2 = {Tt ft0}. Then G2

t = ∫ t

γG2 = ut01 > γF − ε.

Define the process G = {gt} by

gt =

Ttut0 when t0 ≤ t.

Then: G is τ -admissible, Gt ≤ Ft for all t ∈ R, and γG > γF − ε.

Dogan Comez Existence and non-existence of ergodic Hilbert transform for admissible processes

Set up History.

Sketch of the proof eHt along moving averages sequences

Sketch of proof

Sketch of proof of Theorem.2 (assuming Theorem.1):

We can assume that ft ≥ 0 for each t. Recall ft = Ttu|t|, and ut ≥ 0, for all t ∈ R.

Let G1 = {Ttu0}t∈R. Then G1 t =

∫ t 0 Tsu0ds ≤ Ft for all t ∈ R.

Fix ε > 0, and pick t0 > 0 such that ut01 > δ1 − ε, and let G2 = {Tt ft0}. Then G2

t = ∫ t

0 Tsut0ds ≤ Ft for all t ≥ t0 and γG2 = ut01 > γF − ε.

Define the process G = {gt} by

gt =

Ttut0 when t0 ≤ t.

Then: G is τ -admissible, Gt ≤ Ft for all t ∈ R, and γG > γF − ε.

Dogan Comez Existence and non-existence of ergodic Hilbert transform for admissible processes

Set up History.

Sketch of the proof eHt along moving averages sequences

Sketch of proof

Sketch of proof of Theorem.2 (assuming Theorem.1):

We can assume that ft ≥ 0 for each t. Recall ft = Ttu|t|, and ut ≥ 0, for all t ∈ R.

Let G1 = {Ttu0}t∈R. Then G1 t =

∫ t 0 Tsu0ds ≤ Ft for all t ∈ R.

Fix ε > 0, and pick t0 > 0 such that ut01 > δ1 − ε, and let G2 = {Tt ft0}. Then G2

t = ∫ t

0 Tsut0ds ≤ Ft for all t ≥ t0 and γG2 = ut01 > γF − ε.

Define the process G = {gt} by

gt =

Ttut0 when t0 ≤ t.

Then: G is τ -admissible, Gt ≤ Ft for all t ∈ R, and γG > γF − ε.

Dogan Comez Existence and non-existence of ergodic Hilbert transform for admissible processes

Set up History.

Sketch of the proof eHt along moving averages sequences

Sketch of proof

Sketch of proof of Theorem.2 (assuming Theorem.1):

We can assume that ft ≥ 0 for each t. Recall ft = Ttu|t|, and ut ≥ 0, for all t ∈ R.

Let G1 = {Ttu0}t∈R. Then G1 t =

∫ t 0 Tsu0ds ≤ Ft for all t ∈ R.

Fix ε > 0, and pick t0 > 0 such that ut01 > δ1 − ε, and let G2 = {Tt ft0}. Then G2

t = ∫ t

0 Tsut0ds ≤ Ft for all t ≥ t0 and γG2 = ut01 > γF − ε.

Define the process G = {gt} by

gt =

Ttut0 when t0 ≤ t.

Then: G is τ -admissible, Gt ≤ Ft for all t ∈ R, and γG > γF − ε.

Dogan Comez Existence and non-existence of ergodic Hilbert transform for admissible processes

Set up History.

Sketch of the proof eHt along moving averages sequences

Sketch of proof

Define a new process Z = {zt} by zt(x) = ft(x)− Ttg(x) for all t ∈ R.

Then: 0 < zt(x) ≤ Tt(δ − g)(x), and δ − g1 < ε.

Furthermore, Z = {zt} is a strongly bounded, symmetric τ -admissible process with exact dominant δ − g such that γZ = δ − g < ε.

Also, HqF (x) = HqZ(x) + ∫ q≤|s|≤t0

u0(Ts x) s

s ds.

Since the last two limits exist a.e. by (*) and (**), it remains to prove that limq→0 HqZ(x) exists a.e.

Dogan Comez Existence and non-existence of ergodic Hilbert transform for admissible processes

Set up History.

Sketch of the proof eHt along moving averages sequences

Sketch of proof

Define a new process Z = {zt} by zt(x) = ft(x)− Ttg(x) for all t ∈ R. Then: 0 < zt(x) ≤ Tt(δ − g)(x), and δ − g1 < ε.

Furthermore, Z = {zt} is a strongly bounded, symmetric τ -admissible process with exact dominant δ − g such that γZ = δ − g < ε.

Also, HqF (x) = HqZ(x) + ∫ q≤|s|≤t0

u0(Ts x) s

s ds.

Since the last two limits exist a.e. by (*) and (**), it remains to prove that limq→0 HqZ(x) exists a.e.

Dogan Comez Existence and non-existence of ergodic Hilbert transform for admissible processes

Set up History.

Sketch of the proof eHt along moving averages sequences

Sketch of proof

Define a new process Z = {zt} by zt(x) = ft(x)− Ttg(x) for all t ∈ R. Then: 0 < zt(x) ≤ Tt(δ − g)(x), and δ − g1 < ε.

Furthermore, Z = {zt} is a strongly bounded, symmetric τ -admissible process with exact dominant δ − g such that γZ = δ − g < ε.

Also, HqF (x) = HqZ(x) + ∫ q≤|s|≤t0

u0(Ts x) s

s ds.

Since the last two limits exist a.e. by (*) and (**), it remains to prove that limq→0 HqZ(x) exists a.e.

Dogan Comez Existence and non-existence of ergodic Hilbert transform for admissible processes

Set up History.

Sketch of the proof eHt along moving averages sequences

Sketch of proof

Define a new process Z = {zt} by zt(x) = ft(x)− Ttg(x) for all t ∈ R. Then: 0 < zt(x) ≤ Tt(δ − g)(x), and δ − g1 < ε.

Furthermore, Z = {zt} is a strongly bounded, symmetric τ -admissible process with exact dominant δ − g such that γZ = δ − g < ε.

Also, HqF (x) = HqZ(x) + ∫ q≤|s|≤t0

u0(Ts x) s

s ds.

Since the last two limits exist a.e. by (*) and (**), it remains to prove that limq→0 HqZ(x) exists a.e.

Dogan Comez Existence and non-existence of ergodic Hilbert transform for admissible processes

Set up History.

Sketch of the proof eHt along moving averages sequences

Sketch of proof

Define a new process Z = {zt} by zt(x) = ft(x)− Ttg(x) for all t ∈ R. Then: 0 < zt(x) ≤ Tt(δ − g)(x), and δ − g1 < ε.

Furthermore, Z = {zt} is a strongly bounded, symmetric τ -admissible process with exact dominant δ − g such that γZ = δ − g < ε.

Also, HqF (x) = HqZ(x) + ∫ q≤|s|≤t0

u0(Ts x) s

s ds.

Since the last two limits exist a.e. by (*) and (**), it remains to prove that limq→0 HqZ(x) exists a.e.

Dogan Comez Existence and non-existence of ergodic Hilbert transform for admissible processes

Set up History.

Sketch of the proof eHt along moving averages sequences

Sketch of proof

Let E = {x : lim supq HqF (x)− lim infq HqF (x) > λ}, then

E ⊂ {x : sup q |HqZ(x)| >

λ

2 }.

Since ε > 0 is arbitrary, so µ(E) = 0.

Thus HF (x) = limq→0 HqF (x) exists a.e.

Dogan Comez Existence and non-existence of ergodic Hilbert transform for admissible processes

Set up History.

Sketch of the proof eHt along moving averages sequences

Sketch of proof

Let E = {x : lim supq HqF (x)− lim infq HqF (x) > λ}, then

E ⊂ {x : sup q |HqZ(x)| >

λ

2 }.

Since ε > 0 is arbitrary, so µ(E) = 0.

Thus HF (x) = limq→0 HqF (x) exists a.e.

Dogan Comez Existence and non-existence of ergodic Hilbert transform for admissible processes

Set up History.

Sketch of the proof eHt along moving averages sequences

Sketch of proof

Let E = {x : lim supq HqF (x)− lim infq HqF (x) > λ}, then

E ⊂ {x : sup q |HqZ(x)| >

λ

2 }.

Since ε > 0 is arbitrary, so µ(E) = 0.

Thus HF (x) = limq→0 HqF (x) exists a.e.

Dogan Comez Existence and non-existence of ergodic Hilbert transform for admissible processes

Set up History.

Sketch of the proof eHt along moving averages sequences

Moving averages

Moving averages sequences.

A sequence w = {(vn, rn)} in Z× Z+, where rn > 0, satisfies the cone condition (CC) if ∃ C such that ∀ s ∈ R+, |α(s)| ≤ Cs, where

α = {(t, s) : ∃n, |t − vn| ≤ α(s − rn)}, and α(s) = {x ∈ R : (x , s) ∈ α}. Examples: {(n, n)} and {(22n ,

√ 22n )} satisfy CC, but {(n,

√ n)} does not.

M.A. Akcoglu and A. delJunco (1975): limn 1√ n

∑√n k=0 f (T n+kx) fails to exist a.e.

Notice: {(n, √ n)} does not satisfy cone condition.

A. Bellow, R. Jones and J. Rosenblatt (1990): {(vn, rn)} satisfy CC is necessary and sufficient for the moving average 1

rn

∑rn k=0 f (T vn+kx) to converge a.e.

S. Ferrando (1995): extended it to superadditive processes setting.

Question. How about the analogous results for the eHt?

Dogan Comez Existence and non-existence of ergodic Hilbert transform for admissible processes

Set up History.

Sketch of the proof eHt along moving averages sequences

Moving averages

Moving averages sequences.

A sequence w = {(vn, rn)} in Z× Z+, where rn > 0, satisfies the cone condition (CC) if ∃ C such that ∀ s ∈ R+, |α(s)| ≤ Cs, where

α = {(t, s) : ∃n, |t − vn| ≤ α(s − rn)}, and α(s) = {x ∈ R : (x , s) ∈ α}.

Examples: {(n, n)} and {(22n , √

22n )} satisfy CC, but {(n, √ n)} does not.

M.A. Akcoglu and A. delJunco (1975): limn 1√ n

∑√n k=0 f (T n+kx) fails to exist a.e.

Notice: {(n, √ n)} does not satisfy cone condition.

A. Bellow, R. Jones and J. Rosenblatt (1990): {(vn, rn)} satisfy CC is necessary and sufficient for the moving average 1

rn

∑rn k=0 f (T vn+kx) to converge a.e.

S. Ferrando (1995): extended it to superadditive processes setting.

Question. How about the analogous results for the eHt?

Dogan Comez Existence and non-existence of ergodic Hilbert transform for admissible processes

Set up History.

Sketch of the proof eHt along moving averages sequences

Moving averages

Moving averages sequences.

A sequence w = {(vn, rn)} in Z× Z+, where rn > 0, satisfies the cone condition (CC) if ∃ C such that ∀ s ∈ R+, |α(s)| ≤ Cs, where

α = {(t, s) : ∃n, |t − vn| ≤ α(s − rn)}, and α(s) = {x ∈ R : (x , s) ∈ α}. Examples: {(n, n)} and {(22n ,

√ 22n )} satisfy CC, but {(n,

√ n)} does not.

M.A. Akcoglu and A. delJunco (1975): limn 1√ n

∑√n k=0 f (T n+kx) fails to exist a.e.

Notice: {(n, √ n)} does not satisfy cone condition.

A. Bellow, R. Jones and J. Rosenblatt (1990): {(vn, rn)} satisfy CC is necessary and sufficient for the moving average 1

rn

∑rn k=0 f (T vn+kx) to converge a.e.

S. Ferrando (1995): extended it to superadditive processes setting.

Question. How about the analogous results for the eHt?

Dogan Comez Existence and non-existence of ergodic Hilbert transform for admissible processes

Set up History.

Sketch of the proof eHt along moving averages sequences

Moving averages

Moving averages sequences.

A sequence w = {(vn, rn)} in Z× Z+, where rn > 0, satisfies the cone condition (CC) if ∃ C such that ∀ s ∈ R+, |α(s)| ≤ Cs, where

α = {(t, s) : ∃n, |t − vn| ≤ α(s − rn)}, and α(s) = {x ∈ R : (x , s) ∈ α}. Examples: {(n, n)} and {(22n ,

√ 22n )} satisfy CC, but {(n,

√ n)} does not.

M.A. Akcoglu and A. delJunco (1975): limn 1√ n

∑√n k=0 f (T n+kx) fails to exist a.e.

Notice: {(n, √ n)} does not satisfy cone condition.

A. Bellow, R. Jones and J. Rosenblatt (1990): {(vn, rn)} satisfy CC is necessary and sufficient for the moving average 1

rn

∑rn k=0 f (T vn+kx) to converge a.e.

S. Ferrando (1995): extended it to superadditive processes setting.

Question. How about the analogous results for the eHt?

Dogan Comez Existence and non-existence of ergodic Hilbert transform for admissible processes

Set up History.

Sketch of the proof eHt along moving averages sequences

Moving averages

Moving averages sequences.

A sequence w = {(vn, rn)} in Z× Z+, where rn > 0, satisfies the cone condition (CC) if ∃ C such that ∀ s ∈ R+, |α(s)| ≤ Cs, where

α = {(t, s) : ∃n, |t − vn| ≤ α(s − rn)}, and α(s) = {x ∈ R : (x , s) ∈ α}. Examples: {(n, n)} and {(22n ,

√ 22n )} satisfy CC, but {(n,

√ n)} does not.

M.A. Akcoglu and A. delJunco (1975): limn 1√ n

∑√n k=0 f (T n+kx) fails to exist a.e.

Notice: {(n, √ n)} does not satisfy cone condition.

A. Bellow, R. Jones and J. Rosenblatt (1990): {(vn, rn)} satisfy CC is necessary and sufficient for the moving average 1

rn

∑rn k=0 f (T vn+kx) to converge a.e.

S. Ferrando (1995): extended it to superadditive processes setting.

Question. How about the analogous results for the eHt?

Dogan Comez Existence and non-existence of ergodic Hilbert transform for admissible processes

Set up History.

Sketch of the proof eHt along moving averages sequences

Moving averages

Moving averages sequences.

A sequence w = {(vn, rn)} in Z× Z+, where rn > 0, satisfies the cone condition (CC) if ∃ C such that ∀ s ∈ R+, |α(s)| ≤ Cs, where

α = {(t, s) : ∃n, |t − vn| ≤ α(s − rn)}, and α(s) = {x ∈ R : (x , s) ∈ α}. Examples: {(n, n)} and {(22n ,

√ 22n )} satisfy CC, but {(n,

√ n)} does not.

M.A. Akcoglu and A. delJunco (1975): limn 1√ n

∑√n k=0 f (T n+kx) fails to exist a.e.

Notice: {(n, √ n)} does not satisfy cone condition.

A. Bellow, R. Jones and J. Rosenblatt (1990): {(vn, rn)} satisfy CC is necessary and sufficient for the moving average 1

rn

∑rn k=0 f (T vn+kx) to converge a.e.

S. Ferrando (1995): extended it to superadditive processes setting.

Question. How about the analogous results for the eHt?

Dogan Comez Existence and non-existence of ergodic Hilbert transform for admissible processes

Set up History.

Sketch of the proof eHt along moving averages sequences

Moving averages

Moving averages sequences.

A sequence w = {(vn, rn)} in Z× Z+, where rn > 0, satisfies the cone condition (CC) if ∃ C such that ∀ s ∈ R+, |α(s)| ≤ Cs, where

α = {(t, s) : ∃n, |t − vn| ≤ α(s − rn)}, and α(s) = {x ∈ R : (x , s) ∈ α}. Examples: {(n, n)} and {(22n ,

√ 22n )} satisfy CC, but {(n,

√ n)} does not.

M.A. Akcoglu and A. delJunco (1975): limn 1√ n

∑√n k=0 f (T n+kx) fails to exist a.e.

Notice: {(n, √ n)} does not satisfy cone condition.

A. Bellow, R. Jones and J. Rosenblatt (1990): {(vn, rn)} satisfy CC is necessary and sufficient for the moving average 1

rn

∑rn k=0 f (T vn+kx) to converge a.e.

S. Ferrando (1995): extended it to superadditive processes setting.

Question. How about the analogous results for the eHt?

Dogan Comez Existence and non-existence of ergodic Hilbert transform for admissible processes

Set up History.

Sketch of the proof eHt along moving averages sequences

Moving averages

Moving averages sequences.

A sequence w = {(vn, rn)} in Z× Z+, where rn > 0, satisfies the cone condition (CC) if ∃ C such that ∀ s ∈ R+, |α(s)| ≤ Cs, where

α = {(t, s) : ∃n, |t − vn| ≤ α(s − rn)}, and α(s) = {x ∈ R : (x , s) ∈ α}. Examples: {(n, n)} and {(22n ,

√ 22n )} satisfy CC, but {(n,

√ n)} does not.

M.A. Akcoglu and A. delJunco (1975): limn 1√ n

∑√n k=0 f (T n+kx) fails to exist a.e.

Notice: {(n, √ n)} does not satisfy cone condition.

A. Bellow, R. Jones and J. Rosenblatt (1990): {(vn, rn)} satisfy CC is necessary and sufficient for the moving average 1

rn

∑rn k=0 f (T vn+kx) to converge a.e.

S. Ferrando (1995): extended it to superadditive processes setting.

Question. How about the analogous results for the eHt?

Dogan Comez Existence and non-existence of ergodic Hilbert transform for admissible processes

Set up History.

Sketch of the proof eHt along moving averages sequences

Moving eHt

Let T : X → X be an i.m.p.t. and w = {(vn, rn)}. Define the moving eHt of f by

HT w f = limn

, if exists.

As usual, one needs the maximal inequality and the a.e. existence of HT w f for all f in a

dense subset of Lp .

Maximal inequality is OK: For continuous parameter additive processes (Ferrando, Jones and Reinhold, 1995). For (discrete) admissible processes (Comez, 2015).

Convergence on a dense subset of Lp?

Dogan Comez Existence and non-existence of ergodic Hilbert transform for admissible processes

Set up History.

Sketch of the proof eHt along moving averages sequences

Moving eHt

Let T : X → X be an i.m.p.t. and w = {(vn, rn)}. Define the moving eHt of f by

HT w f = limn

, if exists.

As usual, one needs the maximal inequality and the a.e. existence of HT w f for all f in a

dense subset of Lp .

Maximal inequality is OK: For continuous parameter additive processes (Ferrando, Jones and Reinhold, 1995). For (discrete) admissible processes (Comez, 2015).

Convergence on a dense subset of Lp?

Dogan Comez Existence and non-existence of ergodic Hilbert transform for admissible processes

Set up History.

Sketch of the proof eHt along moving averages sequences

Moving eHt

Let T : X → X be an i.m.p.t. and w = {(vn, rn)}. Define the moving eHt of f by

HT w f = limn

, if exists.

As usual, one needs the maximal inequality and the a.e. existence of HT w f for all f in a

dense subset of Lp .

Maximal inequality is OK: For continuous parameter additive processes (Ferrando, Jones and Reinhold, 1995). For (discrete) admissible processes (Comez, 2015).

Convergence on a dense subset of Lp?

Dogan Comez Existence and non-existence of ergodic Hilbert transform for admissible processes

Set up History.

Sketch of the proof eHt along moving averages sequences

Moving eHt

Let T : X → X be an i.m.p.t. and w = {(vn, rn)}. Define the moving eHt of f by

HT w f = limn

, if exists.

As usual, one needs the maximal inequality and the a.e. existence of HT w f for all f in a

dense subset of Lp .

Maximal inequality is OK: For continuous parameter additive processes (Ferrando, Jones and Reinhold, 1995). For (discrete) admissible processes (Comez, 2015).

Convergence on a dense subset of Lp?

Dogan Comez Existence and non-existence of ergodic Hilbert transform for admissible processes

Set up History.

Sketch of the proof eHt along moving averages sequences

Example

Example. Let (X ,F , µ,T ), where X = {0, 1, 2}, F = P(X ), µ({i}) = 1/3, 0 ≤ i ≤ 2,

and T (0) = 1, T (1) = 2, T (2) = 0.

Let w = {(n, n)}. w satisfies the cone condition and vn = n = rn →∞.

Consider two subsequences: w1 consists of those that are multiples of 3, w2 consists of those that are of the form n = 3k + 1.

Let f : X → R be such that f (0) = f (2) = 0 and f (1) = 1.

Hf (1) = ∑ k 6=0

f (T k1)

f (T k0)

Dogan Comez Existence and non-existence of ergodic Hilbert transform for admissible processes

Set up History.

Sketch of the proof eHt along moving averages sequences

Example

Example. Let (X ,F , µ,T ), where X = {0, 1, 2}, F = P(X ), µ({i}) = 1/3, 0 ≤ i ≤ 2, and T (0) = 1, T (1) = 2, T (2) = 0.

Let w = {(n, n)}. w satisfies the cone condition and vn = n = rn →∞.

Consider two subsequences: w1 consists of those that are multiples of 3, w2 consists of those that are of the form n = 3k + 1.

Let f : X → R be such that f (0) = f (2) = 0 and f (1) = 1.

Hf (1) = ∑ k 6=0

f (T k1)

f (T k0)

Dogan Comez Existence and non-existence of ergodic Hilbert transform for admissible processes

Set up History.

Sketch of the proof eHt along moving averages sequences

Example

Example. Let (X ,F , µ,T ), where X = {0, 1, 2}, F = P(X ), µ({i}) = 1/3, 0 ≤ i ≤ 2, and T (0) = 1, T (1) = 2, T (2) = 0.

Let w = {(n, n)}.

w satisfies the cone condition and vn = n = rn →∞.

Consider two subsequences: w1 consists of those that are multiples of 3, w2 consists of those that are of the form n = 3k + 1.

Let f : X → R be such that f (0) = f (2) = 0 and f (1) = 1.

Hf (1) = ∑ k 6=0

f (T k1)

f (T k0)

Dogan Comez Existence and non-existence of ergodic Hilbert transform for admissible processes

Set up History.

Sketch of the proof eHt along moving averages sequences

Example

Example. Let (X ,F , µ,T ), where X = {0, 1, 2}, F = P(X ), µ({i}) = 1/3, 0 ≤ i ≤ 2, and T (0) = 1, T (1) = 2, T (2) = 0.

Let w = {(n, n)}. w satisfies the cone condition and vn = n = rn →∞.

Consider two subsequences: w1 consists of those that are multiples of 3, w2 consists of those that are of the form n = 3k + 1.

Let f : X → R be such that f (0) = f (2) = 0 and f (1) = 1.

Hf (1) = ∑ k 6=0

f (T k1)

f (T k0)

Dogan Comez Existence and non-existence of ergodic Hilbert transform for admissible processes

Set up History.

Sketch of the proof eHt along moving averages sequences

Example

Example. Let (X ,F , µ,T ), where X = {0, 1, 2}, F = P(X ), µ({i}) = 1/3, 0 ≤ i ≤ 2, and T (0) = 1, T (1) = 2, T (2) = 0.

Let w = {(n, n)}. w satisfies the cone condition and vn = n = rn →∞.

Consider two subsequences:

w1 consists of those that are multiples of 3, w2 consists of those that are of the form n = 3k + 1.

Let f : X → R be such that f (0) = f (2) = 0 and f (1) = 1.

Hf (1) = ∑ k 6=0

f (T k1)

f (T k0)

Dogan Comez Existence and non-existence of ergodic Hilbert transform for admissible processes

Set up History.

Sketch of the proof eHt along moving averages sequences

Example

Example. Let (X ,F , µ,T ), where X = {0, 1, 2}, F = P(X ), µ({i}) = 1/3, 0 ≤ i ≤ 2, and T (0) = 1, T (1) = 2, T (2) = 0.

Let w = {(n, n)}. w satisfies the cone condition and vn = n = rn →∞.

Consider two subsequences: w1 consists of those that are multiples of 3,

w2 consists of those that are of the form n = 3k + 1.

Let f : X → R be such that f (0) = f (2) = 0 and f (1) = 1.

Hf (1) = ∑ k 6=0

f (T k1)

f (T k0)

Dogan Comez Existence and non-existence of ergodic Hilbert transform for admissible processes

Set up History.

Sketch of the proof eHt along moving averages sequences

Example

Example. Let (X ,F , µ,T ), where X = {0, 1, 2}, F = P(X ), µ({i}) = 1/3, 0 ≤ i ≤ 2, and T (0) = 1, T (1) = 2, T (2) = 0.

Let w = {(n, n)}. w satisfies the cone condition and vn = n = rn →∞.

Consider two subsequences: w1 consists of those that are multiples of 3, w2 consists of those that are of the form n = 3k + 1.

Let f : X → R be such that f (0) = f (2) = 0 and f (1) = 1.

Hf (1) = ∑ k 6=0

f (T k1)

f (T k0)

Dogan Comez Existence and non-existence of ergodic Hilbert transform for admissible processes

Set up History.

Sketch of the proof eHt along moving averages sequences

Example

Example. Let (X ,F , µ,T ), where X = {0, 1, 2}, F = P(X ), µ({i}) = 1/3, 0 ≤ i ≤ 2, and T (0) = 1, T (1) = 2, T (2) = 0.

Let w = {(n, n)}. w satisfies the cone condition and vn = n = rn →∞.

Consider two subsequences: w1 consists of those that are multiples of 3, w2 consists of those that are of the form n = 3k + 1.

Let f : X → R be such that f (0) = f (2) = 0 and f (1) = 1.

Hf (1) = ∑ k 6=0

f (T k1)

f (T k0)

Dogan Comez Existence and non-existence of ergodic Hilbert transform for admissible processes

Set up History.

Sketch of the proof eHt along moving averages sequences

Example

Example. Let (X ,F , µ,T ), where X = {0, 1, 2}, F = P(X ), µ({i}) = 1/3, 0 ≤ i ≤ 2, and T (0) = 1, T (1) = 2, T (2) = 0.

Let w = {(n, n)}. w satisfies the cone condition and vn = n = rn →∞.

Consider two subsequences: w1 consists of those that are multiples of 3, w2 consists of those that are of the form n = 3k + 1.

Let f : X → R be such that f (0) = f (2) = 0 and f (1) = 1.

Hf (1) = ∑ k 6=0

f (T k1)

f (T k0)

Dogan Comez Existence and non-existence of ergodic Hilbert transform for admissible processes

Set up History.

Sketch of the proof eHt along moving averages sequences

Example

Along w2,

Hence limn H(vn,rn)f (0) does not exist!

Question. Is it possible to have an affirmative answer under some additional conditions?

Dogan Comez Existence and non-existence of ergodic Hilbert transform for admissible processes

Set up History.

Sketch of the proof eHt along moving averages sequences

Example

Along w2,

Hence limn H(vn,rn)f (0) does not exist!

Question. Is it possible to have an affirmative answer under some additional conditions?

Dogan Comez Existence and non-existence of ergodic Hilbert transform for admissible processes

Set up History.

Sketch of the proof eHt along moving averages sequences

Example

Hence limn H(vn,rn)f (0) does not exist!

Question. Is it possible to have an affirmative answer under some additional conditions?

Dogan Comez Existence and non-existence of ergodic Hilbert transform for admissible processes

Set up History.

Sketch of the proof eHt along moving averages sequences

Example

Along w2,

Hence limn H(vn,rn)f (0) does not exist!

Question. Is it possible to have an affirmative answer under some additional conditions?

Dogan Comez Existence and non-existence of ergodic Hilbert transform for admissible processes

Set up History.

Sketch of the proof eHt along moving averages sequences

Example

Along w2,

Hence limn H(vn,rn)f (0) does not exist!

Question. Is it possible to have an affirmative answer under some additional conditions?

Dogan Comez Existence and non-existence of ergodic Hilbert transform for admissible processes

Set up History.

Sketch of the proof eHt along moving averages sequences

Example

Along w2,

Hence limn H(vn,rn)f (0) does not exist!

Question. Is it possible to have an affirmative answer under some additional conditions?

Dogan Comez Existence and non-existence of ergodic Hilbert transform for admissible processes

Set up History.

Sketch of the proof eHt along moving averages sequences

Example

Along w2,

Hence limn H(vn,rn)f (0) does not exist!

Question. Is it possible to have an affirmative answer under some additional conditions?

Dogan Comez Existence and non-existence of ergodic Hilbert transform for admissible processes

Set up History.

Sketch of the proof eHt along moving averages sequences

Example

Along w2,

Hence limn H(vn,rn)f (0) does not exist!

Question. Is it possible to have an affirmative answer under some additional conditions?

Dogan Comez Existence and non-existence of ergodic Hilbert transform for admissible processes

Set up History.

Sketch of the proof eHt along moving averages sequences

Example

Along w2,

Hence limn H(vn,rn)f (0) does not exist!

Question. Is it possible to have an affirmative answer under some additional conditions?

Dogan Comez Existence and non-existence of ergodic Hilbert transform for admissible processes

Set up History.

Sketch of the proof eHt along moving averages sequences

Example

Along w2,

Hence limn H(vn,rn)f (0) does not exist!

Question. Is it possible to have an affirmative answer under some additional conditions?

Dogan Comez Existence and non-existence of ergodic Hilbert transform for admissible processes

Set up History.

Sketch of the proof eHt along moving averages sequences

Some special dynamical systems

Since, moving eHt of an integrable function fails to exist for arbitrary i.m.p. dynamical system; positive results are possible if one considers:

restricted classes of functions, or

some special classes of dynamical systems, or

further restrictions of the sequence {(vn, rn)}.

T has Lebesgue spectrum if L0 2(X ) = ⊕jHj , where Hj = span{T k fj} for some fj ∈ L0

2

such that < fj ,T k fj >= 0 if k 6= 0.

Theorem (Comez, 2016) Let {(vn, rn)} satisfy CC. If T has Lebesgue spectrum, then, for all f ∈ L0

2, limn H(vn,rn)f (x) exists a.e.

Dogan Comez Existence and non-existence of ergodic Hilbert transform for admissible processes

Set up History.

Sketch of the proof eHt along moving averages sequences

Some special dynamical systems

Since, moving eHt of an integrable function fails to exist for arbitrary i.m.p. dynamical system; positive results are possible if one considers:

restricted classes of functions, or

some special classes of dynamical systems, or

further restrictions of the sequence {(vn, rn)}.

T has Lebesgue spectrum if L0 2(X ) = ⊕jHj , where Hj = span{T k fj} for some fj ∈ L0

2

such that < fj ,T k fj >= 0 if k 6= 0.

Theorem (Comez, 2016) Let {(vn, rn)} satisfy CC. If T has Lebesgue spectrum, then, for all f ∈ L0

2, limn H(vn,rn)f (x) exists a.e.

Dogan Comez Existence and non-existence of ergodic Hilbert transform for admissible processes

Set up History.

Sketch of the proof eHt along moving averages sequences

Some special dynamical systems

Since, moving eHt of an integrable function fails to exist for arbitrary i.m.p. dynamical system; positive results are possible if one considers:

restricted classes of functions, or

some special classes of dynamical systems, or

further restrictions of the sequence {(vn, rn)}.

T has Lebesgue spectrum if L0 2(X ) = ⊕jHj , where Hj = span{T k fj} for some fj ∈ L0

2

such that < fj ,T k fj >= 0 if k 6= 0.

Theorem (Comez, 2016) Let {(vn, rn)} satisfy CC. If T has Lebesgue spectrum, then, for all f ∈ L0

2, limn H(vn,rn)f (x) exists a.e.

Dogan Comez Existence and non-existence of ergodic Hilbert transform for admissible processes

Set up History.

Sketch of the proof eHt along moving averages sequences

Some special dynamical systems

Since, moving eHt of an integrable function fails to exist for arbitrary i.m.p. dynamical system; positive results are possible if one considers:

restricted classes of functions, or

some special classes of dynamical systems, or

further restrictions of the sequence {(vn, rn)}.

T has Lebesgue spectrum if L0 2(X ) = ⊕jHj , where Hj = span{T k fj} for some fj ∈ L0

2

such that < fj ,T k fj >= 0 if k 6= 0.

Theorem (Comez, 2016) Let {(vn, rn)} satisfy CC. If T has Lebesgue spectrum, then, for all f ∈ L0

2, limn H(vn,rn)f (x) exists a.e.

Dogan Comez Existence and non-existence of ergodic Hilbert transform for admissible processes

Set up History.

Sketch of the proof eHt along moving averages sequences

Some special dynamical systems

Since, moving eHt of an integrable function fails to exist for arbitrary i.m.p. dynamical system; positive results are possible if one considers:

restricted classes of functions, or

some special classes of dynamical systems, or

further restrictions of the sequence {(vn, rn)}.

T has Lebesgue spectrum if L0 2(X ) = ⊕jHj , where Hj = span{T k fj} for some fj ∈ L0

2

such that < fj ,T k fj >= 0 if k 6= 0.

Theorem (Comez, 2016) Let {(vn, rn)} satisfy CC. If T has Lebesgue spectrum, then, for all f ∈ L0

2, limn H(vn,rn)f (x) exists a.e.

Dogan Comez Existence and non-existence of ergodic Hilbert transform for admissible processes

Set up History.

Sketch of the proof eHt along moving averages sequences

THANK YOU!

Dogan Comez Existence and non-existence of ergodic Hilbert transform for admissible processes

Set up

Sketch of the proof

Sketch of the proof eHt along moving averages sequences

Set up

Dirichlet problem in the upper half plane:

Given f (x), find u(x , y), harmonic in the upper half plane and agrees with f (x) on the real line; that is,

∂2u ∂x2 + ∂2u

Solution: u(x , y) = y π

∫∞ −∞

(x−s)2+y2 ds.

Harmonic conjugate of u(x , y) in the upper half plane is

v(x , y) = 1

∫∞ −∞

s ds.

Besicovitch (early 1930’s): Hf exists a.e. for all integrable f on R.

Dogan Comez Existence and non-existence of ergodic Hilbert transform for admissible processes

Set up History.

Sketch of the proof eHt along moving averages sequences

Set up

Dirichlet problem in the upper half plane: Given f (x), find u(x , y), harmonic in the upper half plane and agrees with f (x) on the real line; that is,

∂2u ∂x2 + ∂2u

Solution: u(x , y) = y π

∫∞ −∞

(x−s)2+y2 ds.

Harmonic conjugate of u(x , y) in the upper half plane is

v(x , y) = 1

∫∞ −∞

s ds.

Besicovitch (early 1930’s): Hf exists a.e. for all integrable f on R.

Dogan Comez Existence and non-existence of ergodic Hilbert transform for admissible processes

Set up History.

Sketch of the proof eHt along moving averages sequences

Set up

Dirichlet problem in the upper half plane: Given f (x), find u(x , y), harmonic in the upper half plane and agrees with f (x) on the real line; that is,

∂2u ∂x2 + ∂2u

Solution: u(x , y) = y π

∫∞ −∞

(x−s)2+y2 ds.

Harmonic conjugate of u(x , y) in the upper half plane is

v(x , y) = 1

∫∞ −∞

s ds.

Besicovitch (early 1930’s): Hf exists a.e. for all integrable f on R.

Dogan Comez Existence and non-existence of ergodic Hilbert transform for admissible processes

Set up History.

Sketch of the proof eHt along moving averages sequences

Set up

Dirichlet problem in the upper half plane: Given f (x), find u(x , y), harmonic in the upper half plane and agrees with f (x) on the real line; that is,

∂2u ∂x2 + ∂2u

Solution: u(x , y) = y π

∫∞ −∞

(x−s)2+y2 ds.

Harmonic conjugate of u(x , y) in the upper half plane is

v(x , y) = 1

∫∞ −∞

s ds.

Besicovitch (early 1930’s): Hf exists a.e. for all integrable f on R.

Dogan Comez Existence and non-existence of ergodic Hilbert transform for admissible processes

Set up History.

Sketch of the proof eHt along moving averages sequences

Set up

Dirichlet problem in the upper half plane: Given f (x), find u(x , y), harmonic in the upper half plane and agrees with f (x) on the real line; that is,

∂2u ∂x2 + ∂2u

Solution: u(x , y) = y π

∫∞ −∞

(x−s)2+y2 ds.

Harmonic conjugate of u(x , y) in the upper half plane is

v(x , y) = 1

∫∞ −∞

s ds.

Besicovitch (early 1930’s): Hf exists a.e. for all integrable f on R.

Dogan Comez Existence and non-existence of ergodic Hilbert transform for admissible processes

Set up History.

Sketch of the proof eHt along moving averages sequences

Set up

Dirichlet problem in the upper half plane: Given f (x), find u(x , y), harmonic in the upper half plane and agrees with f (x) on the real line; that is,

∂2u ∂x2 + ∂2u

Solution: u(x , y) = y π

∫∞ −∞

(x−s)2+y2 ds.

Harmonic conjugate of u(x , y) in the upper half plane is

v(x , y) = 1

∫∞ −∞

s ds.

Besicovitch (early 1930’s): Hf exists a.e. for all integrable f on R.

Dogan Comez Existence and non-existence of ergodic Hilbert transform for admissible processes

Set up History.

Sketch of the proof eHt along moving averages sequences

Ergodic setting

Observe: Lebesgue measure is translation invariant; hence, the map Us(x) = x − s

preserves Lebesgue measure, namely, m(U−1 s E) = m(E).

Hilbert transform of f can be rewritten as:

Hf (x) = lim ε→0

∫ ε<|s|<1/ε

f (x − s)

s ds = lim

f (Usx)

s dt.

More generally, if τ = {Tt}t∈R is a group of invertible m.p.t’s on (X ,Σ, µ), then the ergodic Hilbert transform (eHt) of f is defined as

Hτ f (x) = lim ε→0

∫ ε<|s|<1/ε

f (Tsx)

s ds.

Discrete version: If T : X → X is an i.m.p.t. (i.e., τ = {T n}n∈Z), the (discrete) ergodic Hilbert transform of f is

HT f (x) = lim n→∞

∑ 0<|k|≤n

f (T kx)

Dogan Comez Existence and non-existence of ergodic Hilbert transform for admissible processes

Set up History.

Sketch of the proof eHt along moving averages sequences

Ergodic setting

Observe: Lebesgue measure is translation invariant; hence, the map Us(x) = x − s

preserves Lebesgue measure, namely, m(U−1 s E) = m(E).

Hilbert transform of f can be rewritten as:

Hf (x) = lim ε→0

∫ ε<|s|<1/ε

f (x − s)

s ds = lim

f (Usx)

s dt.

More generally, if τ = {Tt}t∈R is a group of invertible m.p.t’s on (X ,Σ, µ), then the ergodic Hilbert transform (eHt) of f is defined as

Hτ f (x) = lim ε→0

∫ ε<|s|<1/ε

f (Tsx)

s ds.

Discrete version: If T : X → X is an i.m.p.t. (i.e., τ = {T n}n∈Z), the (discrete) ergodic Hilbert transform of f is

HT f (x) = lim n→∞

∑ 0<|k|≤n

f (T kx)

Dogan Comez Existence and non-existence of ergodic Hilbert transform for admissible processes

Set up History.

Sketch of the proof eHt along moving averages sequences

Ergodic setting

Observe: Lebesgue measure is translation invariant; hence, the map Us(x) = x − s

preserves Lebesgue measure, namely, m(U−1 s E) = m(E).

Hilbert transform of f can be rewritten as:

Hf (x) = lim ε→0

∫ ε<|s|<1/ε

f (x − s)

s ds = lim

f (Usx)

s dt.

More generally, if τ = {Tt}t∈R is a group of invertible m.p.t’s on (X ,Σ, µ), then the ergodic Hilbert transform (eHt) of f is defined as

Hτ f (x) = lim ε→0

∫ ε<|s|<1/ε

f (Tsx)

s ds.

Discrete version: If T : X → X is an i.m.p.t. (i.e., τ = {T n}n∈Z), the (discrete) ergodic Hilbert transform of f is

HT f (x) = lim n→∞

∑ 0<|k|≤n

f (T kx)

Dogan Comez Existence and non-existence of ergodic Hilbert transform for admissible processes

Set up History.

Sketch of the proof eHt along moving averages sequences

Ergodic setting

Observe: Lebesgue measure is translation invariant; hence, the map Us(x) = x − s

preserves Lebesgue measure, namely, m(U−1 s E) = m(E).

Hilbert transform of f can be rewritten as:

Hf (x) = lim ε→0

∫ ε<|s|<1/ε

f (x − s)

s ds = lim

f (Usx)

s dt.

More generally, if τ = {Tt}t∈R is a group of invertible m.p.t’s on (X ,Σ, µ), then the ergodic Hilbert transform (eHt) of f is defined as

Hτ f (x) = lim ε→0

∫ ε<|s|<1/ε

f (Tsx)

s ds.

Discrete version: If T : X → X is an i.m.p.t. (i.e., τ = {T n}n∈Z), the (discrete) ergodic Hilbert transform of f is

HT f (x) = lim n→∞

∑ 0<|k|≤n

f (T kx)

Dogan Comez Existence and non-existence of ergodic Hilbert transform for admissible processes

Set up History.

Sketch of the proof eHt along moving averages sequences

Ergodic theorems

Let (X ,Σ, µ) be a probability space. A m.pt. T : X → X is called ergodic if T−1E = E implies µ(E) = 0 or µ(E) = 1.

Crown jewel of ergodic theory:

Birkhoff’s Ergodic Theorem. (1931)

If T : X → X is m.p.t., then, limn→∞ 1 n

∑n−1 k=0 f (T kx) = f ∗(x) exists a.e. for all

f ∈ L1. If, furthermore, T is ergodic, then f ∗ = ∫ f .

Continuous parameter version:

Local Ergodic Theorem. (Wiener, 1939)

If τ is a m.p. flow and f ∈ L1, then limt→0+ 1 t

∫ t 0 f (Tsx)ds = f (x) a.e.

Question: Analogous results for the (discrete or local) eHt?

Dogan Comez Existence and non-existence of ergodic Hilbert transform for admissible processes

Set up History.

Sketch of the proof eHt along moving averages sequences

Ergodic theorems

Let (X ,Σ, µ) be a probability space. A m.pt. T : X → X is called ergodic if T−1E = E implies µ(E) = 0 or µ(E) = 1.

Crown jewel of ergodic theory:

Birkhoff’s Ergodic Theorem. (1931)

If T : X → X is m.p.t., then, limn→∞ 1 n

∑n−1 k=0 f (T kx) = f ∗(x) exists a.e. for all

f ∈ L1. If, furthermore, T is ergodic, then f ∗ = ∫ f .

Continuous parameter version:

Local Ergodic Theorem. (Wiener, 1939)

If τ is a m.p. flow and f ∈ L1, then limt→0+ 1 t

∫ t 0 f (Tsx)ds = f (x) a.e.

Question: Analogous results for the (discrete or local) eHt?

Dogan Comez Existence and non-existence of ergodic Hilbert transform for admissible processes

Set up History.

Sketch of the proof eHt along moving averages sequences

Ergodic theorems

Let (X ,Σ, µ) be a probability space. A m.pt. T : X → X is called ergodic if T−1E = E implies µ(E) = 0 or µ(E) = 1.

Crown jewel of ergodic theory:

Birkhoff’s Ergodic Theorem. (1931)

If T : X → X is m.p.t., then, limn→∞ 1 n

∑n−1 k=0 f (T kx) = f ∗(x) exists a.e. for all

f ∈ L1. If, furthermore, T is ergodic, then f ∗ = ∫ f .

Continuous parameter version:

Local Ergodic Theorem. (Wiener, 1939)

If τ is a m.p. flow and f ∈ L1, then limt→0+ 1 t

∫ t 0 f (Tsx)ds = f (x) a.e.

Question: Analogous results for the (discrete or local) eHt?

Dogan Comez Existence and non-existence of ergodic Hilbert transform for admissible processes

Set up History.

Sketch of the proof eHt along moving averages sequences

Ergodic theorems

Let (X ,Σ, µ) be a probability space. A m.pt. T : X → X is called ergodic if T−1E = E implies µ(E) = 0 or µ(E) = 1.

Crown jewel of ergodic theory:

Birkhoff’s Ergodic Theorem. (1931)

If T : X → X is m.p.t., then, limn→∞ 1 n

∑n−1 k=0 f (T kx) = f ∗(x) exists a.e. for all

f ∈ L1. If, furthermore, T is ergodic, then f ∗ = ∫ f .

Continuous parameter version:

Local Ergodic Theorem. (Wiener, 1939)

If τ is a m.p. flow and f ∈ L1, then limt→0+ 1 t

∫ t 0 f (Tsx)ds = f (x) a.e.

Question: Analogous results for the (discrete or local) eHt?

Dogan Comez Existence and non-existence of ergodic Hilbert transform for admissible processes

Set up History.

Sketch of the proof eHt along moving averages sequences

Existence of eHt

HT f (x) = lim n→∞

∑ 0<|k|≤n

f (T kx)

ε→0

M. Cotlar (1955)

If τ is an i.m.p. flow (or T is an i.m.p.t.) on X , then Hτ f (x) exists a.e. for all f ∈ L1.

K. Petersen (1983, 1985) - Another proof of eHt (both local and discrete).

R. Sato (1989) - eHt in the operator setting (discrete).

L.M. Fernadez-Cabrera, F. Martin-Reyes and J.L. Torrea (1995) - connection between ergodic everages and eHt (discrete).

D. Comez (2006) - eHt for admissible processes (discrete).

Dogan Comez Existence and non-existence of ergodic Hilbert transform for admissible processes

Set up History.

Sketch of the proof eHt along moving averages sequences

Existence of eHt

HT f (x) = lim n→∞

∑ 0<|k|≤n

f (T kx)

ε→0

M. Cotlar (1955)

If τ is an i.m.p. flow (or T is an i.m.p.t.) on X , then Hτ f (x) exists a.e. for all f ∈ L1.

K. Petersen (1983, 1985) - Another proof of eHt (both local and discrete).

R. Sato (1989) - eHt in the operator setting (discrete).

L.M. Fernadez-Cabrera, F. Martin-Reyes and J.L. Torrea (1995) - connection between ergodic everages and eHt (discrete).

D. Comez (2006) - eHt for admissible processes (discrete).

Dogan Comez Existence and non-existence of ergodic Hilbert transform for admissible processes

Set up History.

Sketch of the proof eHt along moving averages sequences

Existence of eHt

HT f (x) = lim n→∞

∑ 0<|k|≤n

f (T kx)

ε→0

M. Cotlar (1955)

If τ is an i.m.p. flow (or T is an i.m.p.t.) on X , then Hτ f (x) exists a.e. for all f ∈ L1.

K. Petersen (1983, 1985) - Another proof of eHt (both local and discrete).

R. Sato (1989) - eHt in the operator setting (discrete).

L.M. Fernadez-Cabrera, F. Martin-Reyes and J.L. Torrea (1995) - connection between ergodic everages and eHt (discrete).

D. Comez (2006) - eHt for admissible processes (discrete).

Dogan Comez Existence and non-existence of ergodic Hilbert transform for admissible processes

Set up History.

Sketch of the proof eHt along moving averages sequences

Existence of eHt

HT f (x) = lim n→∞

∑ 0<|k|≤n

f (T kx)

ε→0

M. Cotlar (1955)

If τ is an i.m.p. flow (or T is an i.m.p.t.) on X , then Hτ f (x) exists a.e. for all f ∈ L1.

K. Petersen (1983, 1985) - Another proof of eHt (both local and discrete).

R. Sato (1989) - eHt in the operator setting (discrete).

L.M. Fernadez-Cabrera, F. Martin-Reyes and J.L. Torrea (1995) - connection between ergodic everages and eHt (discrete).

D. Comez (2006) - eHt for admissible processes (discrete).

Dogan Comez Existence and non-existence of ergodic Hilbert transform for admissible processes

Set up History.

Sketch of the proof eHt along moving averages sequences

Existence of eHt

HT f (x) = lim n→∞

∑ 0<|k|≤n

f (T kx)

ε→0

M. Cotlar (1955)

If τ is an i.m.p. flow (or T is an i.m.p.t.) on X , then Hτ f (x) exists a.e. for all f ∈ L1.

K. Petersen (1983, 1985) - Another proof of eHt (both local and discrete).

R. Sato (1989) - eHt in the operator setting (discrete).

L.M. Fernadez-Cabrera, F. Martin-Reyes and J.L. Torrea (1995) - connection between ergodic everages and eHt (discrete).

D. Comez (2006) - eHt for admissible processes (discrete).

Dogan Comez Existence and non-existence of ergodic Hilbert transform for admissible processes

Set up History.

Sketch of the proof eHt along moving averages sequences

Existence of eHt

HT f (x) = lim n→∞

∑ 0<|k|≤n

f (T kx)

ε→0

M. Cotlar (1955)

If τ is an i.m.p. flow (or T is an i.m.p.t.) on X , then Hτ f (x) exists a.e. for all f ∈ L1.

K. Petersen (1983, 1985) - Another proof of eHt (both local and discrete).

R. Sato (1989) - eHt in the operator setting (discrete).

L.M. Fernadez-Cabrera, F. Martin-Reyes and J.L. Torrea (1995) - connection between ergodic everages and eHt (discrete).

D. Comez (2006) - eHt for admissible processes (discrete).

Dogan Comez Existence and non-existence of ergodic Hilbert transform for admissible processes

Set up History.

Sketch of the proof eHt along moving averages sequences

Existence of eHt

HT f (x) = lim n→∞

∑ 0<|k|≤n

f (T kx)

ε→0

M. Cotlar (1955)

If τ is an i.m.p. flow (or T is an i.m.p.t.) on X , then Hτ f (x) exists a.e. for all f ∈ L1.

K. Petersen (1983, 1985) - Another proof of eHt (both local and discrete).

R. Sato (1989) - eHt in the operator setting (discrete).

L.M. Fernadez-Cabrera, F. Martin-Reyes and J.L. Torrea (1995) - connection between ergodic everages and eHt (discrete).

D. Comez (2006) - eHt for admissible processes (discrete).

Dogan Comez Existence and non-existence of ergodic Hilbert transform for admissible processes

Set up History.

Sketch of the proof eHt along moving averages sequences

Results on the modulated eHt

Some other eHt results:

C. Demeter, M. Lacey, T. Tao & C. Thiele (2007)- Return time theorem for the eHt (discrete).

M. Lacey & E. Terwilleger (2008)- Wiener-Wintner type theorem for the eHt (local and discrete).

Corollary: limN→∞ ∑

induced by a trigonometric polynomial.

Comez (2006) - There is a bounded Besicovitch sequence {ak} such that the limit fails to exist.

A. Akhmedov & D. Comez, (2014)- Let Mα = {a : ∑n −n |ak | = O( nα−1

logα n )},

1 < α ≤ 2. If a ∈ Mα, then it is universally good for the eHt in L1.

There is a subclass Bα of bounded Besicovitch sequences s.t. if a ∈ Bα, then it is universally good for the eHt in L2.

Question. Does eHt exist along moving averages sequences? We will return!

Dogan Comez Existence and non-existence of ergodic Hilbert transform for admissible processes

Set up History.

Sketch of the proof eHt along moving averages sequences

Results on the modulated eHt

Some other eHt results:

C. Demeter, M. Lacey, T. Tao & C. Thiele (2007)- Return time theorem for the eHt (discrete).

M. Lacey & E. Terwilleger (2008)- Wiener-Wintner type theorem for the eHt (local and discrete).

Corollary: limN→∞ ∑

induced by a trigonometric polynomial.

Comez (2006) - There is a bounded Besicovitch sequence {ak} such that the limit fails to exist.

A. Akhmedov & D. Comez, (2014)- Let Mα = {a : ∑n −n |ak | = O( nα−1

logα n )},

1 < α ≤ 2. If a ∈ Mα, then it is universally good for the eHt in L1.

There is a subclass Bα of bounded Besicovitch sequences s.t. if a ∈ Bα, then it is universally good for the eHt in L2.

Question. Does eHt exist along moving averages sequences? We will return!

Dogan Comez Existence and non-existence of ergodic Hilbert transform for admissible processes

Set up History.

Sketch of the proof eHt along moving averages sequences

Results on the modulated eHt

Some other eHt results:

C. Demeter, M. Lacey, T. Tao & C. Thiele (2007)- Return time theorem for the eHt (discrete).

M. Lacey & E. Terwilleger (2008)- Wiener-Wintner type theorem for the eHt (local and discrete).

Corollary: limN→∞ ∑

induced by a trigonometric polynomial.

Comez (2006) - There is a bounded Besicovitch sequence {ak} such that the limit fails to exist.

A. Akhmedov & D. Comez, (2014)- Let Mα = {a : ∑n −n |ak | = O( nα−1

logα n )},

1 < α ≤ 2. If a ∈ Mα, then it is universally good for the eHt in L1.

There is a subclass Bα of bounded Besicovitch sequences s.t. if a ∈ Bα, then it is universally good for the eHt in L2.

Question. Does eHt exist along moving averages sequences? We will return!

Dogan Comez Existence and non-existence of ergodic Hilbert transform for admissible processes

Set up History.

Sketch of the proof eHt along moving averages sequences

Results on the modulated eHt

Some other eHt results:

C. Demeter, M. Lacey, T. Tao & C. Thiele (2007)- Return time theorem for the eHt (discrete).

M. Lacey & E. Terwilleger (2008)- Wiener-Wintner type theorem for the eHt (local and discrete).

Corollary: limN→∞ ∑

induced by a trigonometric polynomial.

Comez (2006) - There is a bounded Besicovitch sequence {ak} such that the limit fails to exist.

A. Akhmedov & D. Comez, (2014)- Let Mα = {a : ∑n −n |ak | = O( nα−1

logα n )},

1 < α ≤ 2. If a ∈ Mα, then it is universally good for the eHt in L1.

There is a subclass Bα of bounded Besicovitch sequences s.t. if a ∈ Bα, then it is universally good for the eHt in L2.

Question. Does eHt exist along moving averages sequences? We will return!

Dogan Comez Existence and non-existence of ergodic Hilbert transform for admissible processes

Set up History.

Sketch of the proof eHt along moving averages sequences

Results on the modulated eHt

Some other eHt results:

C. Demeter, M. Lacey, T. Tao & C. Thiele (2007)- Return time theorem for the eHt (discrete).

M. Lacey & E. Terwilleger (2008)- Wiener-Wintner type theorem for the eHt (local and discrete).

Corollary: limN→∞ ∑

induced by a trigonometric polynomial.

Comez (2006) - There is a bounded Besicovitch sequence {ak} such that the limit fails to exist.

A. Akhmedov & D. Comez, (2014)- Let Mα = {a : ∑n −n |ak | = O( nα−1

logα n )},

1 < α ≤ 2. If a ∈ Mα, then it is universally good for the eHt in L1.

There is a subclass Bα of bounded Besicovitch sequences s.t. if a ∈ Bα, then it is universally good for the eHt in L2.

Question. Does eHt exist along moving averages sequences? We will return!

Dogan Comez Existence and non-existence of ergodic Hilbert transform for admissible processes

Set up History.

Sketch of the proof eHt along moving averages sequences

Results on the modulated eHt

Some other eHt results:

C. Demeter, M. Lacey, T. Tao & C. Thiele (2007)- Return time theorem for the eHt (discrete).

M. Lacey & E. Terwilleger (2008)- Wiener-Wintner type theorem for the eHt (local and discrete).

Corollary: limN→∞ ∑

induced by a trigonometric polynomial.

Comez (2006) - There is a bounded Besicovitch sequence {ak} such that the limit fails to exist.

A. Akhmedov & D. Comez, (2014)- Let Mα = {a : ∑n −n |ak | = O( nα−1

logα n )},

1 < α ≤ 2. If a ∈ Mα, then it is universally good for the eHt in L1.

There is a subclass Bα of bounded Besicovitch sequences s.t. if a ∈ Bα, then it is universally good for the eHt in L2.

Question. Does eHt exist along moving averages sequences? We will return!

Dogan Comez Existence and non-existence of ergodic Hilbert transform for admissible processes

Set up History.

Sketch of the proof eHt along moving averages sequences

Results on the modulated eHt

Some other eHt results:

C. Demeter, M. Lacey, T. Tao & C. Thiele (2007)- Return time theorem for the eHt (discrete).

M. Lacey & E. Terwilleger (2008)- Wiener-Wintner type theorem for the eHt (local and discrete).

Corollary: limN→∞ ∑

induced by a trigonometric polynomial.

Comez (2006) - There is a bounded Besicovitch sequence {ak} such that the limit fails to exist.

A. Akhmedov & D. Comez, (2014)- Let Mα = {a : ∑n −n |ak | = O( nα−1

logα n )},

1 < α ≤ 2. If a ∈ Mα, then it is universally good for the eHt in L1.

There is a subclass Bα of bounded Besicovitch sequences s.t. if a ∈ Bα, then it is universally good for the eHt in L2.

Question. Does eHt exist along moving averages sequences? We will return!

Dogan Comez Existence and non-existence of ergodic Hilbert transform for admissible processes

Set up History.

Sketch of the proof eHt along moving averages sequences

Results on the modulated eHt

Some other eHt results:

C. Demeter, M. Lacey, T. Tao & C. Thiele (2007)- Return time theorem for the eHt (discrete).

M. Lacey & E. Terwilleger (2008)- Wiener-Wintner type theorem for the eHt (local and discrete).

Corollary: limN→∞ ∑

induced by a trigonometric polynomial.

Comez (2006) - There is a bounded Besicovitch sequence {ak} such that the limit fails to exist.

A. Akhmedov & D. Comez, (2014)- Let Mα = {a : ∑n −n |ak | = O( nα−1

logα n )},

1 < α ≤ 2. If a ∈ Mα, then it is universally good for the eHt in L1.

There is a subclass Bα of bounded Besicovitch sequences s.t. if a ∈ Bα, then it is universally good for the eHt in L2.

Question. Does eHt exist along moving averages sequences?

We will return!

Dogan Comez Existence and non-existence of ergodic Hilbert transform for admissible processes

Set up History.

Sketch of the proof eHt along moving averages sequences

Results on the modulated eHt

Some other eHt results:

C. Demeter, M. Lacey, T. Tao & C. Thiele (2007)- Return time theorem for the eHt (discrete).

M. Lacey & E. Terwilleger (2008)- Wiener-Wintner type theorem for the eHt (local and discrete).

Corollary: limN→∞ ∑

induced by a trigonometric polynomial.

Comez (2006) - There is a bounded Besicovitch sequence {ak} such that the limit fails to exist.

A. Akhmedov & D. Comez, (2014)- Let Mα = {a : ∑n −n |ak | = O( nα−1

logα n )},

1 < α ≤ 2. If a ∈ Mα, then it is universally good for the eHt in L1.

There is a subclass Bα of bounded Besicovitch sequences s.t. if a ∈ Bα, then it is universally good for the eHt in L2.

Question. Does eHt exist along moving averages sequences? We will return!

Dogan Comez Existence and non-existence of ergodic Hilbert transform for admissible processes

Set up History.

Sketch of the proof eHt along moving averages sequences

Admissible processes

F = {ft}t∈R ⊂ Lp is called a τ -admissible process on R if Ts ft ≤ fs+t and T−s f−t ≤ f−s−t for t, s ≥ 0.

Discrete version: F = {fn}n∈Z is T -admissible on Z if Tfk ≤ fk+1 and T−1f−k ≤ f−k−1 for k ∈ Z+.

An admissible process F = {ft}t∈R is called

positive if ft ≥ 0 for all t ∈ R, strongly bounded if γF := supt∈R ftp <∞, symmetric if T2t f−t = ft for all t ∈ R+.

D. Comez (2006) - If T : X → X is an i.m.p.t and F = {fn}n∈Z ⊂ L1 is a strongly

bounded symmetric T -admissible process, then limn ∑

1≤|k|≤n fk (x) k

exists a.e.

Question. How about extending the local eHt result to the admissible processes?

Dogan Comez Existence and non-existence of ergodic Hilbert transform for admissible processes

Set up History.

Sketch of the proof eHt along moving averages sequences

Admissible processes

F = {ft}t∈R ⊂ Lp is called a τ -admissible process on R if Ts ft ≤ fs+t and T−s f−t ≤ f−s−t for t, s ≥ 0.

Discrete version: F = {fn}n∈Z is T -admissible on Z if Tfk ≤ fk+1 and T−1f−k ≤ f−k−1 for k ∈ Z+.

An admissible process F = {ft}t∈R is called

positive if ft ≥ 0 for all t ∈ R, strongly bounded if γF := supt∈R ftp <∞, symmetric if T2t f−t = ft for all t ∈ R+.

D. Comez (2006) - If T : X → X is an i.m.p.t and F = {fn}n∈Z ⊂ L1 is a strongly

bounded symmetric T -admissible process, then limn ∑

1≤|k|≤n fk (x) k

exists a.e.

Question. How about extending the local eHt result to the admissible processes?

Dogan Comez Existence and non-existence of ergodic Hilbert transform for admissible processes

Set up History.

Sketch of the proof eHt along moving averages sequences

Admissible processes

F = {ft}t∈R ⊂ Lp is called a τ -admissible process on R if Ts ft ≤ fs+t and T−s f−t ≤ f−s−t for t, s ≥ 0.

Discrete version: F = {fn}n∈Z is T -admissible on Z if Tfk ≤ fk+1 and T−1f−k ≤ f−k−1 for k ∈ Z+.

An admissible process F = {ft}t∈R is called

positive if ft ≥ 0 for all t ∈ R, strongly bounded if γF := supt∈R ftp <∞, symmetric if T2t f−t = ft for all t ∈ R+.

D. Comez (2006) - If T : X → X is an i.m.p.t and F = {fn}n∈Z ⊂ L1 is a strongly

bounded symmetric T -admissible process, then limn ∑

1≤|k|≤n fk (x) k

exists a.e.

Question. How about extending the local eHt result to the admissible processes?

Dogan Comez Existence and non-existence of ergodic Hilbert transform for admissible processes

Set up History.

Sketch of the proof eHt along moving averages sequences

Admissible processes

F = {ft}t∈R ⊂ Lp is called a τ -admissible process on R if Ts ft ≤ fs+t and T−s f−t ≤ f−s−t for t, s ≥ 0.

Discrete version: F = {fn}n∈Z is T -admissible on Z if Tfk ≤ fk+1 and T−1f−k ≤ f−k−1 for k ∈ Z+.

An admissible process F = {ft}t∈R is called

positive if ft ≥ 0 for all t ∈ R, strongly bounded if γF := supt∈R ftp <∞, symmetric if T2t f−t = ft for all t ∈ R+.

D. Comez (2006) - If T : X → X is an i.m.p.t and F = {fn}n∈Z ⊂ L1 is a strongly

bounded symmetric T -admissible process, then limn ∑

1≤|k|≤n fk (x) k

exists a.e.

Question. How about extending the local eHt result to the admissible processes?

Dogan Comez Existence and non-existence of ergodic Hilbert transform for admissible processes

Set up History.

Sketch of the proof eHt along moving averages sequences

Admissible processes

F = {ft}t∈R ⊂ Lp is called a τ -admissible process on R if Ts ft ≤ fs+t and T−s f−t ≤ f−s−t for t, s ≥ 0.

Discrete version: F = {fn}n∈Z is T -admissible on Z if Tfk ≤ fk+1 and T−1f−k ≤ f−k−1 for k ∈ Z+.

An admissible process F = {ft}t∈R is called

positive if ft ≥ 0 for all t ∈ R, strongly bounded if γF := supt∈R ftp <∞, symmetric if T2t f−t = ft for all t ∈ R+.

D. Comez (2006) - If T : X → X is an i.m.p.t and F = {fn}n∈Z ⊂ L1 is a strongly

bounded symmetric T -admissible process, then limn ∑

1≤|k|≤n fk (x) k

exists a.e.

Question. How about extending the local eHt result to the admissible processes?

Dogan Comez Existence and non-existence of ergodic Hilbert transform for admissible processes

Set up History.

Sketch of the proof eHt along moving averages sequences

LeHt for admissible processes-set up

Recall: Local eHt of f is Hf (x) = limε→0

∫ ε≤|t|≤1/ε

ds.

Local ergodic Hilbert transform of a symmetric admissible process F = {ft}t∈R :

HF (x) = limε→0

ds.

Let F = {ft} be a positive, symmetric, strongly bounded τ -admissible process. Then:

M.A. Akcoglu & U. Krengel (1981) - ∃ a maximal τ -additive process {g Tt} such that g Tt(x) ≤ ft(x) a.e. for all t ∈ R. (Hence, we can assume a given admissible F a positive process.)

∃ a family of non-negative integrable functions {ut} s.t. ft = Ttu|t| for all t ∈ R ∃ δ ∈ L1 s.t. ft ≤ Ttδ, ∀ t ∈ R, and δ1 = supt∈R ft1 (exact dominant of F ).

Dogan Comez Existence and non-existence of ergodic Hilbert transform for admissible processes

Set up History.

Sketch of the proof eHt along moving averages sequences

LeHt for admissible processes-set up

Recall: Local eHt of f is Hf (x) = limε→0

∫ ε≤|t|≤1/ε

ds.

Local ergodic Hilbert transform of a symmetric admissible process F = {ft}t∈R :

HF (x) = limε→0

ds.

Let F = {ft} be a positive, symmetric, strongly bounded τ -admissible process. Then:

M.A. Akcoglu & U. Krengel (1981) - ∃ a maximal τ -additive process {g Tt} such that g Tt(x) ≤ ft(x) a.e. for all t ∈ R. (Hence, we can assume a given admissible F a positive process.)

∃ a family of non-negative integrable functions {ut} s.t. ft = Ttu|t| for all t ∈ R ∃ δ ∈ L1 s.t. ft ≤ Ttδ, ∀ t ∈ R, and δ1 = supt∈R ft1 (exact dominant of F ).

Dogan Comez Existence and non-existence of ergodic Hilbert transform for admissible processes

Set up History.

Sketch of the proof eHt along moving averages sequences

LeHt for admissible processes-set up

Recall: Local eHt of f is Hf (x) = limε→0

∫ ε≤|t|≤1/ε

ds.

Local ergodic Hilbert transform of a symmetric admissible process F = {ft}t∈R :

HF (x) = limε→0

ds.

Let F = {ft} be a positive, symmetric, strongly bounded τ -admissible process. Then:

M.A. Akcoglu & U. Krengel (1981) - ∃ a maximal τ -additive process {g Tt} such that g Tt(x) ≤ ft(x) a.e. for all t ∈ R. (Hence, we can assume a given admissible F a positive process.)

∃ a family of non-negative integrable functions {ut} s.t. ft = Ttu|t| for all t ∈ R ∃ δ ∈ L1 s.t. ft ≤ Ttδ, ∀ t ∈ R, and δ1 = supt∈R ft1 (exact dominant of F ).

Dogan Comez Existence and non-existence of ergodic Hilbert transform for admissible processes

Set up History.

Sketch of the proof eHt along moving averages sequences

LeHt for admissible processes-set up

Recall: Local eHt of f is Hf (x) = limε→0

∫ ε≤|t|≤1/ε

ds.

Local ergodic Hilbert transform of a symmetric admissible process F = {ft}t∈R :

HF (x) = limε→0

ds.

Let F = {ft} be a positive, symmetric, strongly bounded τ -admissible process. Then:

M.A. Akcoglu & U. Krengel (1981) - ∃ a maximal τ -additive process {g Tt} such that g Tt(x) ≤ ft(x) a.e. for all t ∈ R. (Hence, we can assume a given admissible F a positive process.)

∃ a family of non-negative integrable functions {ut} s.t. ft = Ttu|t| for all t ∈ R

∃ δ ∈ L1 s.t. ft ≤ Ttδ, ∀ t ∈ R, and δ1 = supt∈R ft1 (exact dominant of F ).

Dogan Comez Existence and non-existence of ergodic Hilbert transform for admissible processes

Set up History.

Sketch of the proof eHt along moving averages sequences

LeHt for admissible processes-set up

Recall: Local eHt of f is Hf (x) = limε→0

∫ ε≤|t|≤1/ε

ds.

Local ergodic Hilbert transform of a symmetric admissible process F = {ft}t∈R :

HF (x) = limε→0

ds.

Let F = {ft} be a positive, symmetric, strongly bounded τ -admissible process. Then:

M.A. Akcoglu & U. Krengel (1981) - ∃ a maximal τ -additive process {g Tt} such that g Tt(x) ≤ ft(x) a.e. for all t ∈ R. (Hence, we can assume a given admissible F a positive process.)

∃ a family of non-negative integrable functions {ut} s.t. ft = Ttu|t| for all t ∈ R ∃ δ ∈ L1 s.t. ft ≤ Ttδ, ∀ t ∈ R, and δ1 = supt∈R ft1 (exact dominant of F ).

Dogan Comez Existence and non-existence of ergodic Hilbert transform for admissible processes

Set up History.

Sketch of the proof eHt along moving averages sequences

LeHt for admissible processes-results

Let τ = {Tt}t∈R be an i.m.p. flow and F = {ft} ⊂ L1 be a symmetric, strongly bounded τ -admissible process with exact dominant δ.

Theorem.1 (Maximal Inequality)

For any λ > 0, there exists a constant C such that,

µ{x : sup q>0 | ∫ q≤|s|≤1/q

fs(x)

HF (x) = lim q→0

∫ q≤|s|≤1/q

Dogan Comez Existence and non-existence of ergodic Hilbert transform for admissible processes

Set up History.

Sketch of the proof eHt along moving averages sequences

LeHt for admissible processes-results

Let τ = {Tt}t∈R be an i.m.p. flow and F = {ft} ⊂ L1 be a symmetric, strongly bounded τ -admissible process with exact dominant δ.

Theorem.1 (Maximal Inequality)

For any λ > 0, there exists a constant C such that,

µ{x : sup q>0 | ∫ q≤|s|≤1/q

fs(x)

HF (x) = lim q→0

∫ q≤|s|≤1/q

Dogan Comez Existence and non-existence of ergodic Hilbert transform for admissible processes

Set up History.

Sketch of the proof eHt along moving averages sequences

LeHt for admissible processes-results

Let τ = {Tt}t∈R be an i.m.p. flow and F = {ft} ⊂ L1 be a symmetric, strongly bounded τ -admissible process with exact dominant δ.

Theorem.1 (Maximal Inequality)

For any λ > 0, there exists a constant C such that,

µ{x : sup q>0 | ∫ q≤|s|≤1/q

fs(x)

HF (x) = lim q→0

∫ q≤|s|≤1/q

Dogan Comez Existence and non-existence of ergodic Hilbert transform for admissible processes

Set up History.

Sketch of the proof eHt along moving averages sequences

One-sided leHt

∫ q<|s|<1/q

One-sided variant: limq→0

Ergodic version:

∫ q<|s|≤a

s ds exists a.e. for all f ∈ L1, a > 0.

Hence: existence of the local eHt and (*) implies that

(∗∗) H l f (x) := lim q→0

∫ a≤|s|<1/q

s ds exists a.e. for all f ∈ L1, a > 0.

Dogan Comez Existence and non-existence of ergodic Hilbert transform for admissible processes

Set up History.

Sketch of the proof eHt along moving averages sequences

One-sided leHt

∫ q<|s|<1/q

One-sided variant: limq→0

Ergodic version:

∫ q<|s|≤a

s ds exists a.e. for all f ∈ L1, a > 0.

Hence: existence of the local eHt and (*) implies that

(∗∗) H l f (x) := lim q→0

∫ a≤|s|<1/q

s ds exists a.e. for all f ∈ L1, a > 0.

Dogan Comez Existence and non-existence of ergodic Hilbert transform for admissible processes

Set up History.

Sketch of the proof eHt along moving averages sequences

One-sided leHt

∫ q<|s|<1/q

One-sided variant: limq→0

Ergodic version:

∫ q<|s|≤a

s ds exists a.e. for all f ∈ L1, a > 0.

Hence: existence of the local eHt and (*) implies that

(∗∗) H l f (x) := lim q→0

∫ a≤|s|<1/q

s ds exists a.e. for all f ∈ L1, a > 0.

Dogan Comez Existence and non-existence of ergodic Hilbert transform for admissible processes

Set up History.

Sketch of the proof eHt along moving averages sequences

One-sided leHt

∫ q<|s|<1/q

One-sided variant: limq→0

Ergodic version:

∫ q<|s|≤a

s ds exists a.e. for all f ∈ L1, a > 0.

Hence: existence of the local eHt and (*) implies that

(∗∗) H l f (x) := lim q→0

∫ a≤|s|<1/q

s ds exists a.e. for all f ∈ L1, a > 0.

Dogan Comez Existence and non-existence of ergodic Hilbert transform for admissible processes

Set up History.

Sketch of the proof eHt along moving averages sequences

One-sided leHt

∫ q<|s|<1/q

One-sided variant: limq→0

Ergodic version:

∫ q<|s|≤a

s ds exists a.e. for all f ∈ L1, a > 0.

Hence: existence of the local eHt and (*) implies that

(∗∗) H l f (x) := lim q→0

∫ a≤|s|<1/q

s ds exists a.e. for all f ∈ L1, a > 0.

Dogan Comez Existence and non-existence of ergodic Hilbert transform for admissible processes

Set up History.

Sketch of the proof eHt along moving averages sequences

Sketch of proof

Sketch of proof of Theorem.2 (assuming Theorem.1):

We can assume that ft ≥ 0 for each t. Recall ft = Ttu|t|, and ut ≥ 0, for all t ∈ R.

Let G1 = {Ttu0}t∈R. Then G1 t =

∫ t 0 Tsu0ds ≤ Ft for all t ∈ R.

Fix ε > 0, and pick t0 > 0 such that ut01 > δ1 − ε, and let G2 = {Tt ft0}. Then G2

t = ∫ t

0 Tsut0ds ≤ Ft for all t ≥ t0 and γG2 = ut01 > γF − ε.

Define the process G = {gt} by

gt =

Ttut0 when t0 ≤ t.

Then: G is τ -admissible, Gt ≤ Ft for all t ∈ R, and γG > γF − ε.

Dogan Comez Existence and non-existence of ergodic Hilbert transform for admissible processes

Set up History.

Sketch of the proof eHt along moving averages sequences

Sketch of proof

Sketch of proof of Theorem.2 (assuming Theorem.1):

We can assume that ft ≥ 0 for each t. Recall ft = Ttu|t|, and ut ≥ 0, for all t ∈ R.

Let G1 = {Ttu0}t∈R. Then G1 t =

∫ t 0 Tsu0ds ≤ Ft for all t ∈ R.

Fix ε > 0, and pick t0 > 0 such that ut01 > δ1 − ε, and let G2 = {Tt ft0}. Then G2

t = ∫ t

0 Tsut0ds ≤ Ft for all t ≥ t0 and γG2 = ut01 > γF − ε.

Define the process G = {gt} by

gt =

Ttut0 when t0 ≤ t.

Then: G is τ -admissible, Gt ≤ Ft for all t ∈ R, and γG > γF − ε.

Dogan Comez Existence and non-existence of ergodic Hilbert transform for admissible processes

Set up History.

Sketch of the proof eHt along moving averages sequences

Sketch of proof

Sketch of proof of Theorem.2 (assuming Theorem.1):

We can assume that ft ≥ 0 for each t. Recall ft = Ttu|t|, and ut ≥ 0, for all t ∈ R.

Let G1 = {Ttu0}t∈R. Then G1 t =

∫ t 0 Tsu0ds ≤ Ft for all t ∈ R.

Fix ε > 0, and pick t0 > 0 such that ut01 > δ1 − ε, and let G2 = {Tt ft0}. Then G2

t = ∫ t

0 Tsut0ds ≤ Ft for all t ≥ t0 and γG2 = ut01 > γF − ε.

Define the process G = {gt} by

gt =

Ttut0 when t0 ≤ t.

Then: G is τ -admissible, Gt ≤ Ft for all t ∈ R, and γG > γF − ε.

Dogan Comez Existence and non-existence of ergodic Hilbert transform for admissible processes

Set up History.

Sketch of the proof eHt along moving averages sequences

Sketch of proof

Sketch of proof of Theorem.2 (assuming Theorem.1):

We can assume that ft ≥ 0 for each t. Recall ft = Ttu|t|, and ut ≥ 0, for all t ∈ R.

Let G1 = {Ttu0}t∈R. Then G1 t =

∫ t 0 Tsu0ds ≤ Ft for all t ∈ R.

Fix ε > 0, and pick t0 > 0 such that ut01 > δ1 − ε, and let G2 = {Tt ft0}. Then G2

t = ∫ t

γG2 = ut01 > γF − ε.

Define the process G = {gt} by

gt =

Ttut0 when t0 ≤ t.

Then: G is τ -admissible, Gt ≤ Ft for all t ∈ R, and γG > γF − ε.

Dogan Comez Existence and non-existence of ergodic Hilbert transform for admissible processes

Set up History.

Sketch of the proof eHt along moving averages sequences

Sketch of proof

Sketch of proof of Theorem.2 (assuming Theorem.1):

We can assume that ft ≥ 0 for each t. Recall ft = Ttu|t|, and ut ≥ 0, for all t ∈ R.

Let G1 = {Ttu0}t∈R. Then G1 t =

∫ t 0 Tsu0ds ≤ Ft for all t ∈ R.

Fix ε > 0, and pick t0 > 0 such that ut01 > δ1 − ε, and let G2 = {Tt ft0}. Then G2

t = ∫ t

0 Tsut0ds ≤ Ft for all t ≥ t0 and γG2 = ut01 > γF − ε.

Define the process G = {gt} by

gt =

Ttut0 when t0 ≤ t.

Then: G is τ -admissible, Gt ≤ Ft for all t ∈ R, and γG > γF − ε.

Dogan Comez Existence and non-existence of ergodic Hilbert transform for admissible processes

Set up History.

Sketch of the proof eHt along moving averages sequences

Sketch of proof

Sketch of proof of Theorem.2 (assuming Theorem.1):

We can assume that ft ≥ 0 for each t. Recall ft = Ttu|t|, and ut ≥ 0, for all t ∈ R.

Let G1 = {Ttu0}t∈R. Then G1 t =

∫ t 0 Tsu0ds ≤ Ft for all t ∈ R.

Fix ε > 0, and pick t0 > 0 such that ut01 > δ1 − ε, and let G2 = {Tt ft0}. Then G2

t = ∫ t

0 Tsut0ds ≤ Ft for all t ≥ t0 and γG2 = ut01 > γF − ε.

Define the process G = {gt} by

gt =

Ttut0 when t0 ≤ t.

Then: G is τ -admissible, Gt ≤ Ft for all t ∈ R, and γG > γF − ε.

Dogan Comez Existence and non-existence of ergodic Hilbert transform for admissible processes

Set up History.

Sketch of the proof eHt along moving averages sequences

Sketch of proof

Sketch of proof of Theorem.2 (assuming Theorem.1):

We can assume that ft ≥ 0 for each t. Recall ft = Ttu|t|, and ut ≥ 0, for all t ∈ R.

Let G1 = {Ttu0}t∈R. Then G1 t =

∫ t 0 Tsu0ds ≤ Ft for all t ∈ R.

Fix ε > 0, and pick t0 > 0 such that ut01 > δ1 − ε, and let G2 = {Tt ft0}. Then G2

t = ∫ t

0 Tsut0ds ≤ Ft for all t ≥ t0 and γG2 = ut01 > γF − ε.

Define the process G = {gt} by

gt =

Ttut0 when t0 ≤ t.

Then: G is τ -admissible, Gt ≤ Ft for all t ∈ R, and γG > γF − ε.

Dogan Comez Existence and non-existence of ergodic Hilbert transform for admissible processes

Set up History.

Sketch of the proof eHt along moving averages sequences

Sketch of proof

Define a new process Z = {zt} by zt(x) = ft(x)− Ttg(x) for all t ∈ R.

Then: 0 < zt(x) ≤ Tt(δ − g)(x), and δ − g1 < ε.

Furthermore, Z = {zt} is a strongly bounded, symmetric τ -admissible process with exact dominant δ − g such that γZ = δ − g < ε.

Also, HqF (x) = HqZ(x) + ∫ q≤|s|≤t0

u0(Ts x) s

s ds.

Since the last two limits exist a.e. by (*) and (**), it remains to prove that limq→0 HqZ(x) exists a.e.

Dogan Comez Existence and non-existence of ergodic Hilbert transform for admissible processes

Set up History.

Sketch of the proof eHt along moving averages sequences

Sketch of proof

Define a new process Z = {zt} by zt(x) = ft(x)− Ttg(x) for all t ∈ R. Then: 0 < zt(x) ≤ Tt(δ − g)(x), and δ − g1 < ε.

Furthermore, Z = {zt} is a strongly bounded, symmetric τ -admissible process with exact dominant δ − g such that γZ = δ − g < ε.

Also, HqF (x) = HqZ(x) + ∫ q≤|s|≤t0

u0(Ts x) s

s ds.

Since the last two limits exist a.e. by (*) and (**), it remains to prove that limq→0 HqZ(x) exists a.e.

Dogan Comez Existence and non-existence of ergodic Hilbert transform for admissible processes

Set up History.

Sketch of the proof eHt along moving averages sequences

Sketch of proof

Define a new process Z = {zt} by zt(x) = ft(x)− Ttg(x) for all t ∈ R. Then: 0 < zt(x) ≤ Tt(δ − g)(x), and δ − g1 < ε.

Furthermore, Z = {zt} is a strongly bounded, symmetric τ -admissible process with exact dominant δ − g such that γZ = δ − g < ε.

Also, HqF (x) = HqZ(x) + ∫ q≤|s|≤t0

u0(Ts x) s

s ds.

Since the last two limits exist a.e. by (*) and (**), it remains to prove that limq→0 HqZ(x) exists a.e.

Dogan Comez Existence and non-existence of ergodic Hilbert transform for admissible processes

Set up History.

Sketch of the proof eHt along moving averages sequences

Sketch of proof

Define a new process Z = {zt} by zt(x) = ft(x)− Ttg(x) for all t ∈ R. Then: 0 < zt(x) ≤ Tt(δ − g)(x), and δ − g1 < ε.

Furthermore, Z = {zt} is a strongly bounded, symmetric τ -admissible process with exact dominant δ − g such that γZ = δ − g < ε.

Also, HqF (x) = HqZ(x) + ∫ q≤|s|≤t0

u0(Ts x) s

s ds.

Since the last two limits exist a.e. by (*) and (**), it remains to prove that limq→0 HqZ(x) exists a.e.

Dogan Comez Existence and non-existence of ergodic Hilbert transform for admissible processes

Set up History.

Sketch of the proof eHt along moving averages sequences

Sketch of proof

Define a new process Z = {zt} by zt(x) = ft(x)− Ttg(x) for all t ∈ R. Then: 0 < zt(x) ≤ Tt(δ − g)(x), and δ − g1 < ε.

Furthermore, Z = {zt} is a strongly bounded, symmetric τ -admissible process with exact dominant δ − g such that γZ = δ − g < ε.

Also, HqF (x) = HqZ(x) + ∫ q≤|s|≤t0

u0(Ts x) s

s ds.

Since the last two limits exist a.e. by (*) and (**), it remains to prove that limq→0 HqZ(x) exists a.e.

Dogan Comez Existence and non-existence of ergodic Hilbert transform for admissible processes

Set up History.

Sketch of the proof eHt along moving averages sequences

Sketch of proof

Let E = {x : lim supq HqF (x)− lim infq HqF (x) > λ}, then

E ⊂ {x : sup q |HqZ(x)| >

λ

2 }.

Since ε > 0 is arbitrary, so µ(E) = 0.

Thus HF (x) = limq→0 HqF (x) exists a.e.

Dogan Comez Existence and non-existence of ergodic Hilbert transform for admissible processes

Set up History.

Sketch of the proof eHt along moving averages sequences

Sketch of proof

Let E = {x : lim supq HqF (x)− lim infq HqF (x) > λ}, then

E ⊂ {x : sup q |HqZ(x)| >

λ

2 }.

Since ε > 0 is arbitrary, so µ(E) = 0.

Thus HF (x) = limq→0 HqF (x) exists a.e.

Dogan Comez Existence and non-existence of ergodic Hilbert transform for admissible processes

Set up History.

Sketch of the proof eHt along moving averages sequences

Sketch of proof

Let E = {x : lim supq HqF (x)− lim infq HqF (x) > λ}, then

E ⊂ {x : sup q |HqZ(x)| >

λ

2 }.

Since ε > 0 is arbitrary, so µ(E) = 0.

Thus HF (x) = limq→0 HqF (x) exists a.e.

Dogan Comez Existence and non-existence of ergodic Hilbert transform for admissible processes

Set up History.

Sketch of the proof eHt along moving averages sequences

Moving averages

Moving averages sequences.

A sequence w = {(vn, rn)} in Z× Z+, where rn > 0, satisfies the cone condition (CC) if ∃ C such that ∀ s ∈ R+, |α(s)| ≤ Cs, where

α = {(t, s) : ∃n, |t − vn| ≤ α(s − rn)}, and α(s) = {x ∈ R : (x , s) ∈ α}. Examples: {(n, n)} and {(22n ,

√ 22n )} satisfy CC, but {(n,

√ n)} does not.

M.A. Akcoglu and A. delJunco (1975): limn 1√ n

∑√n k=0 f (T n+kx) fails to exist a.e.

Notice: {(n, √ n)} does not satisfy cone condition.

A. Bellow, R. Jones and J. Rosenblatt (1990): {(vn, rn)} satisfy CC is necessary and sufficient for the moving average 1

rn

∑rn k=0 f (T vn+kx) to converge a.e.

S. Ferrando (1995): extended it to superadditive processes setting.

Question. How about the analogous results for the eHt?

Dogan Comez Existence and non-existence of ergodic Hilbert transform for admissible processes

Set up History.

Sketch of the proof eHt along moving averages sequences

Moving averages

Moving averages sequences.

A sequence w = {(vn, rn)} in Z× Z+, where rn > 0, satisfies the cone condition (CC) if ∃ C such that ∀ s ∈ R+, |α(s)| ≤ Cs, where

α = {(t, s) : ∃n, |t − vn| ≤ α(s − rn)}, and α(s) = {x ∈ R : (x , s) ∈ α}.

Examples: {(n, n)} and {(22n , √

22n )} satisfy CC, but {(n, √ n)} does not.

M.A. Akcoglu and A. delJunco (1975): limn 1√ n

∑√n k=0 f (T n+kx) fails to exist a.e.

Notice: {(n, √ n)} does not satisfy cone condition.

A. Bellow, R. Jones and J. Rosenblatt (1990): {(vn, rn)} satisfy CC is necessary and sufficient for the moving average 1

rn

∑rn k=0 f (T vn+kx) to converge a.e.

S. Ferrando (1995): extended it to superadditive processes setting.

Question. How about the analogous results for the eHt?

Dogan Comez Existence and non-existence of ergodic Hilbert transform for admissible processes

Set up History.

Sketch of the proof eHt along moving averages sequences

Moving averages

Moving averages sequences.

A sequence w = {(vn, rn)} in Z× Z+, where rn > 0, satisfies the cone condition (CC) if ∃ C such that ∀ s ∈ R+, |α(s)| ≤ Cs, where

α = {(t, s) : ∃n, |t − vn| ≤ α(s − rn)}, and α(s) = {x ∈ R : (x , s) ∈ α}. Examples: {(n, n)} and {(22n ,

√ 22n )} satisfy CC, but {(n,

√ n)} does not.

M.A. Akcoglu and A. delJunco (1975): limn 1√ n

∑√n k=0 f (T n+kx) fails to exist a.e.

Notice: {(n, √ n)} does not satisfy cone condition.

A. Bellow, R. Jones and J. Rosenblatt (1990): {(vn, rn)} satisfy CC is necessary and sufficient for the moving average 1

rn

∑rn k=0 f (T vn+kx) to converge a.e.

S. Ferrando (1995): extended it to superadditive processes setting.

Question. How about the analogous results for the eHt?

Dogan Comez Existence and non-existence of ergodic Hilbert transform for admissible processes

Set up History.

Sketch of the proof eHt along moving averages sequences

Moving averages

Moving averages sequences.

A sequence w = {(vn, rn)} in Z× Z+, where rn > 0, satisfies the cone condition (CC) if ∃ C such that ∀ s ∈ R+, |α(s)| ≤ Cs, where

α = {(t, s) : ∃n, |t − vn| ≤ α(s − rn)}, and α(s) = {x ∈ R : (x , s) ∈ α}. Examples: {(n, n)} and {(22n ,

√ 22n )} satisfy CC, but {(n,

√ n)} does not.

M.A. Akcoglu and A. delJunco (1975): limn 1√ n

∑√n k=0 f (T n+kx) fails to exist a.e.

Notice: {(n, √ n)} does not satisfy cone condition.

A. Bellow, R. Jones and J. Rosenblatt (1990): {(vn, rn)} satisfy CC is necessary and sufficient for the moving average 1

rn

∑rn k=0 f (T vn+kx) to converge a.e.

S. Ferrando (1995): extended it to superadditive processes setting.

Question. How about the analogous results for the eHt?

Dogan Comez Existence and non-existence of ergodic Hilbert transform for admissible processes

Set up History.

Sketch of the proof eHt along moving averages sequences

Moving averages

Moving averages sequences.

A sequence w = {(vn, rn)} in Z× Z+, where rn > 0, satisfies the cone condition (CC) if ∃ C such that ∀ s ∈ R+, |α(s)| ≤ Cs, where

α = {(t, s) : ∃n, |t − vn| ≤ α(s − rn)}, and α(s) = {x ∈ R : (x , s) ∈ α}. Examples: {(n, n)} and {(22n ,

√ 22n )} satisfy CC, but {(n,

√ n)} does not.

M.A. Akcoglu and A. delJunco (1975): limn 1√ n

∑√n k=0 f (T n+kx) fails to exist a.e.

Notice: {(n, √ n)} does not satisfy cone condition.

A. Bellow, R. Jones and J. Rosenblatt (1990): {(vn, rn)} satisfy CC is necessary and sufficient for the moving average 1

rn

∑rn k=0 f (T vn+kx) to converge a.e.

S. Ferrando (1995): extended it to superadditive processes setting.

Question. How about the analogous results for the eHt?

Dogan Comez Existence and non-existence of ergodic Hilbert transform for admissible processes

Set up History.

Sketch of the proof eHt along moving averages sequences

Moving averages

Moving averages sequences.

A sequence w = {(vn, rn)} in Z× Z+, where rn > 0, satisfies the cone condition (CC) if ∃ C such that ∀ s ∈ R+, |α(s)| ≤ Cs, where

α = {(t, s) : ∃n, |t − vn| ≤ α(s − rn)}, and α(s) = {x ∈ R : (x , s) ∈ α}. Examples: {(n, n)} and {(22n ,

√ 22n )} satisfy CC, but {(n,

√ n)} does not.

M.A. Akcoglu and A. delJunco (1975): limn 1√ n

∑√n k=0 f (T n+kx) fails to exist a.e.

Notice: {(n, √ n)} does not satisfy cone condition.

A. Bellow, R. Jones and J. Rosenblatt (1990): {(vn, rn)} satisfy CC is necessary and sufficient for the moving average 1

rn

∑rn k=0 f (T vn+kx) to converge a.e.

S. Ferrando (1995): extended it to superadditive processes setting.

Question. How about the analogous results for the eHt?

Dogan Comez Existence and non-existence of ergodic Hilbert transform for admissible processes

Set up History.

Sketch of the proof eHt along moving averages sequences

Moving averages

Moving averages sequences.

A sequence w = {(vn, rn)} in Z× Z+, where rn > 0, satisfies the cone condition (CC) if ∃ C such that ∀ s ∈ R+, |α(s)| ≤ Cs, where

α = {(t, s) : ∃n, |t − vn| ≤ α(s − rn)}, and α(s) = {x ∈ R : (x , s) ∈ α}. Examples: {(n, n)} and {(22n ,

√ 22n )} satisfy CC, but {(n,

√ n)} does not.

M.A. Akcoglu and A. delJunco (1975): limn 1√ n

∑√n k=0 f (T n+kx) fails to exist a.e.

Notice: {(n, √ n)} does not satisfy cone condition.

A. Bellow, R. Jones and J. Rosenblatt (1990): {(vn, rn)} satisfy CC is necessary and sufficient for the moving average 1

rn

∑rn k=0 f (T vn+kx) to converge a.e.

S. Ferrando (1995): extended it to superadditive processes setting.

Question. How about the analogous results for the eHt?

Dogan Comez Existence and non-existence of ergodic Hilbert transform for admissible processes

Set up History.

Sketch of the proof eHt along moving averages sequences

Moving averages

Moving averages sequences.

A sequence w = {(vn, rn)} in Z× Z+, where rn > 0, satisfies the cone condition (CC) if ∃ C such that ∀ s ∈ R+, |α(s)| ≤ Cs, where

α = {(t, s) : ∃n, |t − vn| ≤ α(s − rn)}, and α(s) = {x ∈ R : (x , s) ∈ α}. Examples: {(n, n)} and {(22n ,

√ 22n )} satisfy CC, but {(n,

√ n)} does not.

M.A. Akcoglu and A. delJunco (1975): limn 1√ n

∑√n k=0 f (T n+kx) fails to exist a.e.

Notice: {(n, √ n)} does not satisfy cone condition.

A. Bellow, R. Jones and J. Rosenblatt (1990): {(vn, rn)} satisfy CC is necessary and sufficient for the moving average 1

rn

∑rn k=0 f (T vn+kx) to converge a.e.

S. Ferrando (1995): extended it to superadditive processes setting.

Question. How about the analogous results for the eHt?

Dogan Comez Existence and non-existence of ergodic Hilbert transform for admissible processes

Set up History.

Sketch of the proof eHt along moving averages sequences

Moving eHt

Let T : X → X be an i.m.p.t. and w = {(vn, rn)}. Define the moving eHt of f by

HT w f = limn

, if exists.

As usual, one needs the maximal inequality and the a.e. existence of HT w f for all f in a

dense subset of Lp .

Maximal inequality is OK: For continuous parameter additive processes (Ferrando, Jones and Reinhold, 1995). For (discrete) admissible processes (Comez, 2015).

Convergence on a dense subset of Lp?

Dogan Comez Existence and non-existence of ergodic Hilbert transform for admissible processes

Set up History.

Sketch of the proof eHt along moving averages sequences

Moving eHt

Let T : X → X be an i.m.p.t. and w = {(vn, rn)}. Define the moving eHt of f by

HT w f = limn

, if exists.

As usual, one needs the maximal inequality and the a.e. existence of HT w f for all f in a

dense subset of Lp .

Maximal inequality is OK: For continuous parameter additive processes (Ferrando, Jones and Reinhold, 1995). For (discrete) admissible processes (Comez, 2015).

Convergence on a dense subset of Lp?

Dogan Comez Existence and non-existence of ergodic Hilbert transform for admissible processes

Set up History.

Sketch of the proof eHt along moving averages sequences

Moving eHt

Let T : X → X be an i.m.p.t. and w = {(vn, rn)}. Define the moving eHt of f by

HT w f = limn

, if exists.

As usual, one needs the maximal inequality and the a.e. existence of HT w f for all f in a

dense subset of Lp .

Maximal inequality is OK: For continuous parameter additive processes (Ferrando, Jones and Reinhold, 1995). For (discrete) admissible processes (Comez, 2015).

Convergence on a dense subset of Lp?

Dogan Comez Existence and non-existence of ergodic Hilbert transform for admissible processes

Set up History.

Sketch of the proof eHt along moving averages sequences

Moving eHt

Let T : X → X be an i.m.p.t. and w = {(vn, rn)}. Define the moving eHt of f by

HT w f = limn

, if exists.

As usual, one needs the maximal inequality and the a.e. existence of HT w f for all f in a

dense subset of Lp .

Maximal inequality is OK: For continuous parameter additive processes (Ferrando, Jones and Reinhold, 1995). For (discrete) admissible processes (Comez, 2015).

Convergence on a dense subset of Lp?

Dogan Comez Existence and non-existence of ergodic Hilbert transform for admissible processes

Set up History.

Sketch of the proof eHt along moving averages sequences

Example

Example. Let (X ,F , µ,T ), where X = {0, 1, 2}, F = P(X ), µ({i}) = 1/3, 0 ≤ i ≤ 2,

and T (0) = 1, T (1) = 2, T (2) = 0.

Let w = {(n, n)}. w satisfies the cone condition and vn = n = rn →∞.

Consider two subsequences: w1 consists of those that are multiples of 3, w2 consists of those that are of the form n = 3k + 1.

Let f : X → R be such that f (0) = f (2) = 0 and f (1) = 1.

Hf (1) = ∑ k 6=0

f (T k1)

f (T k0)

Dogan Comez Existence and non-existence of ergodic Hilbert transform for admissible processes

Set up History.

Sketch of the proof eHt along moving averages sequences

Example

Example. Let (X ,F , µ,T ), where X = {0, 1, 2}, F = P(X ), µ({i}) = 1/3, 0 ≤ i ≤ 2, and T (0) = 1, T (1) = 2, T (2) = 0.

Let w = {(n, n)}. w satisfies the cone condition and vn = n = rn →∞.

Consider two subsequences: w1 consists of those that are multiples of 3, w2 consists of those that are of the form n = 3k + 1.

Let f : X → R be such that f (0) = f (2) = 0 and f (1) = 1.

Hf (1) = ∑ k 6=0

f (T k1)

f (T k0)

Dogan Comez Existence and non-existence of ergodic Hilbert transform for admissible processes

Set up History.

Sketch of the proof eHt along moving averages sequences

Example

Example. Let (X ,F , µ,T ), where X = {0, 1, 2}, F = P(X ), µ({i}) = 1/3, 0 ≤ i ≤ 2, and T (0) = 1, T (1) = 2, T (2) = 0.

Let w = {(n, n)}.

w satisfies the cone condition and vn = n = rn →∞.

Consider two subsequences: w1 consists of those that are multiples of 3, w2 consists of those that are of the form n = 3k + 1.

Let f : X → R be such that f (0) = f (2) = 0 and f (1) = 1.

Hf (1) = ∑ k 6=0

f (T k1)

f (T k0)

Dogan Comez Existence and non-existence of ergodic Hilbert transform for admissible processes

Set up History.

Sketch of the proof eHt along moving averages sequences

Example

Example. Let (X ,F , µ,T ), where X = {0, 1, 2}, F = P(X ), µ({i}) = 1/3, 0 ≤ i ≤ 2, and T (0) = 1, T (1) = 2, T (2) = 0.

Let w = {(n, n)}. w satisfies the cone condition and vn = n = rn →∞.

Consider two subsequences: w1 consists of those that are multiples of 3, w2 consists of those that are of the form n = 3k + 1.

Let f : X → R be such that f (0) = f (2) = 0 and f (1) = 1.

Hf (1) = ∑ k 6=0

f (T k1)

f (T k0)

Dogan Comez Existence and non-existence of ergodic Hilbert transform for admissible processes

Set up History.

Sketch of the proof eHt along moving averages sequences

Example

Example. Let (X ,F , µ,T ), where X = {0, 1, 2}, F = P(X ), µ({i}) = 1/3, 0 ≤ i ≤ 2, and T (0) = 1, T (1) = 2, T (2) = 0.

Let w = {(n, n)}. w satisfies the cone condition and vn = n = rn →∞.

Consider two subsequences:

w1 consists of those that are multiples of 3, w2 consists of those that are of the form n = 3k + 1.

Let f : X → R be such that f (0) = f (2) = 0 and f (1) = 1.

Hf (1) = ∑ k 6=0

f (T k1)

f (T k0)

Dogan Comez Existence and non-existence of ergodic Hilbert transform for admissible processes

Set up History.

Sketch of the proof eHt along moving averages sequences

Example

Example. Let (X ,F , µ,T ), where X = {0, 1, 2}, F = P(X ), µ({i}) = 1/3, 0 ≤ i ≤ 2, and T (0) = 1, T (1) = 2, T (2) = 0.

Let w = {(n, n)}. w satisfies the cone condition and vn = n = rn →∞.

Consider two subsequences: w1 consists of those that are multiples of 3,

w2 consists of those that are of the form n = 3k + 1.

Let f : X → R be such that f (0) = f (2) = 0 and f (1) = 1.

Hf (1) = ∑ k 6=0

f (T k1)

f (T k0)

Dogan Comez Existence and non-existence of ergodic Hilbert transform for admissible processes

Set up History.

Sketch of the proof eHt along moving averages sequences

Example

Example. Let (X ,F , µ,T ), where X = {0, 1, 2}, F = P(X ), µ({i}) = 1/3, 0 ≤ i ≤ 2, and T (0) = 1, T (1) = 2, T (2) = 0.

Let w = {(n, n)}. w satisfies the cone condition and vn = n = rn →∞.

Consider two subsequences: w1 consists of those that are multiples of 3, w2 consists of those that are of the form n = 3k + 1.

Let f : X → R be such that f (0) = f (2) = 0 and f (1) = 1.

Hf (1) = ∑ k 6=0

f (T k1)

f (T k0)

Dogan Comez Existence and non-existence of ergodic Hilbert transform for admissible processes

Set up History.

Sketch of the proof eHt along moving averages sequences

Example

Example. Let (X ,F , µ,T ), where X = {0, 1, 2}, F = P(X ), µ({i}) = 1/3, 0 ≤ i ≤ 2, and T (0) = 1, T (1) = 2, T (2) = 0.

Let w = {(n, n)}. w satisfies the cone condition and vn = n = rn →∞.

Consider two subsequences: w1 consists of those that are multiples of 3, w2 consists of those that are of the form n = 3k + 1.

Let f : X → R be such that f (0) = f (2) = 0 and f (1) = 1.

Hf (1) = ∑ k 6=0

f (T k1)

f (T k0)

Dogan Comez Existence and non-existence of ergodic Hilbert transform for admissible processes

Set up History.

Sketch of the proof eHt along moving averages sequences

Example

Example. Let (X ,F , µ,T ), where X = {0, 1, 2}, F = P(X ), µ({i}) = 1/3, 0 ≤ i ≤ 2, and T (0) = 1, T (1) = 2, T (2) = 0.

Let w = {(n, n)}. w satisfies the cone condition and vn = n = rn →∞.

Consider two subsequences: w1 consists of those that are multiples of 3, w2 consists of those that are of the form n = 3k + 1.

Let f : X → R be such that f (0) = f (2) = 0 and f (1) = 1.

Hf (1) = ∑ k 6=0

f (T k1)

f (T k0)

Dogan Comez Existence and non-existence of ergodic Hilbert transform for admissible processes

Set up History.

Sketch of the proof eHt along moving averages sequences

Example

Along w2,

Hence limn H(vn,rn)f (0) does not exist!

Question. Is it possible to have an affirmative answer under some additional conditions?

Dogan Comez Existence and non-existence of ergodic Hilbert transform for admissible processes

Set up History.

Sketch of the proof eHt along moving averages sequences

Example

Along w2,

Hence limn H(vn,rn)f (0) does not exist!

Question. Is it possible to have an affirmative answer under some additional conditions?

Dogan Comez Existence and non-existence of ergodic Hilbert transform for admissible processes

Set up History.

Sketch of the proof eHt along moving averages sequences

Example

Hence limn H(vn,rn)f (0) does not exist!

Question. Is it possible to have an affirmative answer under some additional conditions?

Dogan Comez Existence and non-existence of ergodic Hilbert transform for admissible processes

Set up History.

Sketch of the proof eHt along moving averages sequences

Example

Along w2,

Hence limn H(vn,rn)f (0) does not exist!

Question. Is it possible to have an affirmative answer under some additional conditions?

Dogan Comez Existence and non-existence of ergodic Hilbert transform for admissible processes

Set up History.

Sketch of the proof eHt along moving averages sequences

Example

Along w2,

Hence limn H(vn,rn)f (0) does not exist!

Question. Is it possible to have an affirmative answer under some additional conditions?

Dogan Comez Existence and non-existence of ergodic Hilbert transform for admissible processes

Set up History.

Sketch of the proof eHt along moving averages sequences

Example

Along w2,

Hence limn H(vn,rn)f (0) does not exist!

Question. Is it possible to have an affirmative answer under some additional conditions?

Dogan Comez Existence and non-existence of ergodic Hilbert transform for admissible processes

Set up History.

Sketch of the proof eHt along moving averages sequences

Example

Along w2,

Hence limn H(vn,rn)f (0) does not exist!

Question. Is it possible to have an affirmative answer under some additional conditions?

Dogan Comez Existence and non-existence of ergodic Hilbert transform for admissible processes

Set up History.

Sketch of the proof eHt along moving averages sequences

Example

Along w2,

Hence limn H(vn,rn)f (0) does not exist!

Question. Is it possible to have an affirmative answer under some additional conditions?

Dogan Comez Existence and non-existence of ergodic Hilbert transform for admissible processes

Set up History.

Sketch of the proof eHt along moving averages sequences

Example

Along w2,

Hence limn H(vn,rn)f (0) does not exist!

Question. Is it possible to have an affirmative answer under some additional conditions?

Dogan Comez Existence and non-existence of ergodic Hilbert transform for admissible processes

Set up History.

Sketch of the proof eHt along moving averages sequences

Example

Along w2,

Hence limn H(vn,rn)f (0) does not exist!

Question. Is it possible to have an affirmative answer under some additional conditions?

Dogan Comez Existence and non-existence of ergodic Hilbert transform for admissible processes

Set up History.

Sketch of the proof eHt along moving averages sequences

Some special dynamical systems

Since, moving eHt of an integrable function fails to exist for arbitrary i.m.p. dynamical system; positive results are possible if one considers:

restricted classes of functions, or

some special classes of dynamical systems, or

further restrictions of the sequence {(vn, rn)}.

T has Lebesgue spectrum if L0 2(X ) = ⊕jHj , where Hj = span{T k fj} for some fj ∈ L0

2

such that < fj ,T k fj >= 0 if k 6= 0.

Theorem (Comez, 2016) Let {(vn, rn)} satisfy CC. If T has Lebesgue spectrum, then, for all f ∈ L0

2, limn H(vn,rn)f (x) exists a.e.

Dogan Comez Existence and non-existence of ergodic Hilbert transform for admissible processes

Set up History.

Sketch of the proof eHt along moving averages sequences

Some special dynamical systems

Since, moving eHt of an integrable function fails to exist for arbitrary i.m.p. dynamical system; positive results are possible if one considers:

restricted classes of functions, or

some special classes of dynamical systems, or

further restrictions of the sequence {(vn, rn)}.

T has Lebesgue spectrum if L0 2(X ) = ⊕jHj , where Hj = span{T k fj} for some fj ∈ L0

2

such that < fj ,T k fj >= 0 if k 6= 0.

Theorem (Comez, 2016) Let {(vn, rn)} satisfy CC. If T has Lebesgue spectrum, then, for all f ∈ L0

2, limn H(vn,rn)f (x) exists a.e.

Dogan Comez Existence and non-existence of ergodic Hilbert transform for admissible processes

Set up History.

Sketch of the proof eHt along moving averages sequences

Some special dynamical systems

Since, moving eHt of an integrable function fails to exist for arbitrary i.m.p. dynamical system; positive results are possible if one considers:

restricted classes of functions, or

some special classes of dynamical systems, or

further restrictions of the sequence {(vn, rn)}.

T has Lebesgue spectrum if L0 2(X ) = ⊕jHj , where Hj = span{T k fj} for some fj ∈ L0

2

such that < fj ,T k fj >= 0 if k 6= 0.

Theorem (Comez, 2016) Let {(vn, rn)} satisfy CC. If T has Lebesgue spectrum, then, for all f ∈ L0

2, limn H(vn,rn)f (x) exists a.e.

Dogan Comez Existence and non-existence of ergodic Hilbert transform for admissible processes

Set up History.

Sketch of the proof eHt along moving averages sequences

Some special dynamical systems

Since, moving eHt of an integrable function fails to exist for arbitrary i.m.p. dynamical system; positive results are possible if one considers:

restricted classes of functions, or

some special classes of dynamical systems, or

further restrictions of the sequence {(vn, rn)}.

T has Lebesgue spectrum if L0 2(X ) = ⊕jHj , where Hj = span{T k fj} for some fj ∈ L0

2

such that < fj ,T k fj >= 0 if k 6= 0.

Theorem (Comez, 2016) Let {(vn, rn)} satisfy CC. If T has Lebesgue spectrum, then, for all f ∈ L0

2, limn H(vn,rn)f (x) exists a.e.

Dogan Comez Existence and non-existence of ergodic Hilbert transform for admissible processes

Set up History.

Sketch of the proof eHt along moving averages sequences

Some special dynamical systems

Since, moving eHt of an integrable function fails to exist for arbitrary i.m.p. dynamical system; positive results are possible if one considers:

restricted classes of functions, or

some special classes of dynamical systems, or

further restrictions of the sequence {(vn, rn)}.

T has Lebesgue spectrum if L0 2(X ) = ⊕jHj , where Hj = span{T k fj} for some fj ∈ L0

2

such that < fj ,T k fj >= 0 if k 6= 0.

Theorem (Comez, 2016) Let {(vn, rn)} satisfy CC. If T has Lebesgue spectrum, then, for all f ∈ L0

2, limn H(vn,rn)f (x) exists a.e.

Dogan Comez Existence and non-existence of ergodic Hilbert transform for admissible processes

Set up History.

Sketch of the proof eHt along moving averages sequences

THANK YOU!

Dogan Comez Existence and non-existence of ergodic Hilbert transform for admissible processes

Set up

Sketch of the proof