Existence and non-existence of ergodic Hilbert transform ...

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.
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
∫∞ −∞
(x−s)2+y2 ds.
v(x , y) = 1
∫∞ −∞
s ds.
Set up History.
Set up
∂2u ∂x2 + ∂2u
∫∞ −∞
(x−s)2+y2 ds.
v(x , y) = 1
∫∞ −∞
s ds.
Set up History.
Set up
∂2u ∂x2 + ∂2u
∫∞ −∞
(x−s)2+y2 ds.
v(x , y) = 1
∫∞ −∞
s ds.
Set up History.
Set up
∂2u ∂x2 + ∂2u
∫∞ −∞
(x−s)2+y2 ds.
v(x , y) = 1
∫∞ −∞
s ds.
Set up History.
Set up
∂2u ∂x2 + ∂2u
∫∞ −∞
(x−s)2+y2 ds.
v(x , y) = 1
∫∞ −∞
s ds.
Set up History.
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)
Set up History.
Ergodic setting
Hf (x) = lim ε→0
∫ ε<|s|<1/ε
f (x − s)
s ds = lim
f (Usx)
s dt.
∫ ε<|s|<1/ε
f (Tsx)
s ds.
∑ 0<|k|≤n
f (T kx)
Set up History.
Ergodic setting
Hf (x) = lim ε→0
∫ ε<|s|<1/ε
f (x − s)
s ds = lim
f (Usx)
s dt.
∫ ε<|s|<1/ε
f (Tsx)
s ds.
∑ 0<|k|≤n
f (T kx)
Set up History.
Ergodic setting
Hf (x) = lim ε→0
∫ ε<|s|<1/ε
f (x − s)
s ds = lim
f (Usx)
s dt.
∫ ε<|s|<1/ε
f (Tsx)
s ds.
∑ 0<|k|≤n
f (T kx)
Set up History.
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?
Set up History.
Ergodic theorems
∫ t 0 f (Tsx)ds = f (x) a.e.
Set up History.
Ergodic theorems
∫ t 0 f (Tsx)ds = f (x) a.e.
Set up History.
Ergodic theorems
∫ t 0 f (Tsx)ds = f (x) a.e.
Set up History.
Existence of eHt
∑ 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).
Set up History.
Existence of eHt
∑ 0<|k|≤n
f (T kx)
ε→0
M. Cotlar (1955)
Set up History.
Existence of eHt
∑ 0<|k|≤n
f (T kx)
ε→0
M. Cotlar (1955)
Set up History.
Existence of eHt
∑ 0<|k|≤n
f (T kx)
ε→0
M. Cotlar (1955)
Set up History.
Existence of eHt
∑ 0<|k|≤n
f (T kx)
ε→0
M. Cotlar (1955)
Set up History.
Existence of eHt
∑ 0<|k|≤n
f (T kx)
ε→0
M. Cotlar (1955)
Set up History.
Existence of eHt
∑ 0<|k|≤n
f (T kx)
ε→0
M. Cotlar (1955)
Set up History.
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!
Set up History.
logα n )},
Set up History.
logα n )},
Set up History.
logα n )},
Set up History.
logα n )},
Set up History.
logα n )},
Set up History.
logα n )},
Set up History.
logα n )},
Question. Does eHt exist along moving averages sequences?
We will return!
Set up History.
logα n )},
Set up History.
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?
Set up History.
1≤|k|≤n fk (x) k
exists a.e.
Set up History.
1≤|k|≤n fk (x) k
exists a.e.
Set up History.
1≤|k|≤n fk (x) k
exists a.e.
Set up History.
1≤|k|≤n fk (x) k
exists a.e.
Set up History.
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 ).
Set up History.
∫ ε≤|t|≤1/ε
ds.
HF (x) = limε→0
ds.
Set up History.
∫ ε≤|t|≤1/ε
ds.
HF (x) = limε→0
ds.
Set up History.
∫ ε≤|t|≤1/ε
ds.
HF (x) = limε→0
ds.
∃ 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 ).
Set up History.
∫ ε≤|t|≤1/ε
ds.
HF (x) = limε→0
ds.
Set up History.
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
Set up History.
µ{x : sup q>0 | ∫ q≤|s|≤1/q
fs(x)
HF (x) = lim q→0
∫ q≤|s|≤1/q
Set up History.
µ{x : sup q>0 | ∫ q≤|s|≤1/q
fs(x)
HF (x) = lim q→0
∫ q≤|s|≤1/q
Set up History.
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
Set up History.
One-sided leHt
∫ q<|s|<1/q
Ergodic version:
∫ q<|s|≤a
(∗∗) H l f (x) := lim q→0
∫ a≤|s|<1/q
Set up History.
One-sided leHt
∫ q<|s|<1/q
Ergodic version:
∫ q<|s|≤a
(∗∗) H l f (x) := lim q→0
∫ a≤|s|<1/q
Set up History.
One-sided leHt
∫ q<|s|<1/q
Ergodic version:
∫ q<|s|≤a
(∗∗) H l f (x) := lim q→0
∫ a≤|s|<1/q
Set up History.
One-sided leHt
∫ q<|s|<1/q
Ergodic version:
∫ q<|s|≤a
(∗∗) H l f (x) := lim q→0
∫ a≤|s|<1/q
Set up History.
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.
Set up History.
Sketch of proof
t = ∫ t
gt =
Set up History.
Sketch of proof
t = ∫ t
gt =
Set up History.
Sketch of proof
t = ∫ t
γG2 = ut01 > γF − ε.
gt =
Set up History.
Sketch of proof
t = ∫ t
gt =
Set up History.
Sketch of proof
t = ∫ t
gt =
Set up History.
Sketch of proof
t = ∫ t
gt =
Set up History.
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.
Set up History.
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.
u0(Ts x) s
s ds.
Set up History.
Sketch of proof
u0(Ts x) s
s ds.
Set up History.
Sketch of proof
u0(Ts x) s
s ds.
Set up History.
Sketch of proof
u0(Ts x) s
s ds.
Set up History.
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.
Set up History.
Sketch of proof
λ
2 }.
Set up History.
Sketch of proof
λ
2 }.
Set up History.
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?
Set up History.
Moving averages
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.
rn
Set up History.
Moving averages
√ n)} does not.
rn
Set up History.
Moving averages
√ n)} does not.
rn
Set up History.
Moving averages
√ n)} does not.
rn
Set up History.
Moving averages
√ n)} does not.
rn
Set up History.
Moving averages
√ n)} does not.
rn
Set up History.
Moving averages
√ n)} does not.
rn
Set up History.
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?
Set up History.
Moving eHt
HT w f = limn
, if exists.
Set up History.
Moving eHt
HT w f = limn
, if exists.
Set up History.
Moving eHt
HT w f = limn
, if exists.
Set up History.
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)
Set up History.
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.
Hf (1) = ∑ k 6=0
f (T k1)
f (T k0)
Set up History.
Example
Let w = {(n, n)}.
w satisfies the cone condition and vn = n = rn →∞.
Hf (1) = ∑ k 6=0
f (T k1)
f (T k0)
Set up History.
Example
Hf (1) = ∑ k 6=0
f (T k1)
f (T k0)
Set up History.
Example
Consider two subsequences:
w1 consists of those that are multiples of 3, w2 consists of those that are of the form n = 3k + 1.
Hf (1) = ∑ k 6=0
f (T k1)
f (T k0)
Set up History.
Example
Consider two subsequences: w1 consists of those that are multiples of 3,
w2 consists of those that are of the form n = 3k + 1.
Hf (1) = ∑ k 6=0
f (T k1)
f (T k0)
Set up History.
Example
Hf (1) = ∑ k 6=0
f (T k1)
f (T k0)
Set up History.
Example
Hf (1) = ∑ k 6=0
f (T k1)
f (T k0)
Set up History.
Example
Hf (1) = ∑ k 6=0
f (T k1)
f (T k0)
Set up History.
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?
Set up History.
Example
Along w2,
Set up History.
Example
Set up History.
Example
Along w2,
Set up History.
Example
Along w2,
Set up History.
Example
Along w2,
Set up History.
Example
Along w2,
Set up History.
Example
Along w2,
Set up History.
Example
Along w2,
Set up History.
Example
Along w2,
Set up History.
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.
Set up History.
2
Set up History.
2
Set up History.
2
Set up History.
2
Set up History.
THANK YOU!
Set up
Sketch of the proof

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