Extension of Lorentz-Shimogaki and Boyd’sresults to the weighted Lorentz spaces

E. Agora

Instituto Argentino de Matemática “Alberto P. Calderón ”IAM-CONICET Buenos Aires, Argentina

Joint work with: J. Antezana, M. J. Carro and J. Soria

bg=whiteClassical operators of Harmonic Analysis

DefinitionThe Hilbert transform H is defined

Hf (x) = limε→0+

∫|x−y|>ε

f (y)x− y

dy ,

whenever this limit exists almost everywhere.

DefinitionThe Hardy-Littlewood maximal function is defined by:

Mf (x) = supx∈I

1|I|

∫I|f (y)|dy,

where I is an interval of the real line and the supremum is considered overall intervals containing x ∈ R.

bg=whiteWeighted Lorentz spaces

Definition (Lorentz, 1950)Let p > 0, u ∈ L1

loc(R), w ∈ L1loc(R+). The weighted Lorentz spaces are:

Λpu(w) = {f ∈M(R) : ||f ||p

Λpu(w)

=∫ ∞

0(f ∗u (t))pw(t)dt <∞},

Λp,∞u (w) = {f ∈M(R) : ||f ||Λp,∞

u (w) = supt>0

W1/p(t)f ∗u (t) <∞},

where f ∗u (t) = inf{s > 0 : u({x ∈ R : |f (x)| > s}) ≤ t} and W(t) =∫ t

0 w.

Examples:

(i) If w = 1, we recover Λpu(1) = Lp(u) and Λp,∞

u (1) = Lp,∞(u);(ii) If u = 1, we obtain Λp(w) and Λp,∞(w).

bg=whiteCharacterization of H : Λpu(w)→ Λp

u(w)

Recall that given an operator T , by means of

T : Λpu(w)→ Λp

u(w),

we express||Tf ||Λp

u(w) ≤ C||f ||Λpu(w),

for some positive constant C.

ProblemLet p > 1. Find necessary and sufficient conditions on u ∈ L1

loc(R) andw ∈ L1

loc(R+) so that the boundedness of H on Λpu(w) holds, i.e.:

H : Λpu(w)→ Λp

u(w).

bg=whiteThe case of weighted Lebesgue spaces

Hunt-Muckenhoupt-Wheeden 1973, Coifman-Fefferman 1974 proved:

TheoremLet 1 < p <∞. The boundedness H : Lp(u)→ Lp(u) holds if and only if

u ∈ Ap : supI⊂R

(1|I|

∫Iu(x)dx

)(1|I|

∫Iu−1/(p−1)(x)dx

)p−1

<∞.

Muckenhoupt 1972, proved that Ap characterizes also the boundedness of M:

TheoremFor p > 1, the following statements are equivalent:

(i) u ∈ Ap;

(ii) M : Lp(u)→ Lp(u);

(iii) M : Lp(u)→ Lp,∞(u).

bg=whiteThe case of classical Lorentz spacesBoundedness of M

It is known that (Mf )∗(t) ≈ Pf ∗(t), where

Pf (t) =1t

∫ t

0f (r)dr =

∫ 1

0f (ts)ds =

∫ 1

0Es f (t)ds,

where Es f (t) = f (ts). For s ∈ (0, 1], define

‖Es‖pΛp(w)→Λp(w) = sup

‖f∗‖Lp(w)≤1‖Es f ∗‖p

Lp(w) = supt>0

W(st)W(t)

=: W(s).

By Minkowski inequality we obtain

||M||Λp(w)→Λp(w) = sup‖f‖Λp(w)≤1

‖Mf‖Λp(w) ≈ sup‖f∗‖Lp(w)≤1

‖Pf ∗‖Lp(w)

= sup‖f∗‖Lp(w)≤1

∥∥∥∥∥∫ 1

0Es f ∗ds

∥∥∥∥∥Lp(w)

≤∫ 1

0W1/p(s)ds.

Proposition

If∫ 1

0W1/p(s)ds <∞, then M : Λp(w)→ Λp(w).

bg=whiteThe case of classical Lorentz spacesLorentz-Shimogaki theorem for Λp(w)

Define the function W by

W(µ) = sups>0

W(µs)W(s)

,

and the upper Boyd index by

αΛp(w) = limµ→∞

log W1/p(µ)logµ

.

Theorem (Lorentz 1955 - Shimogaki 1965)The boundedness M : Λp(w)→ Λp(w) holds if and only if αΛp(w) < 1.

bg=whiteThe case of classical Lorentz spacesBoundedness of H

Bennett and Rudnick proved in 1980

(Hf )∗(t) . (P + Q)f ∗(t) . (Hg)∗(t), t > 0,

where f ∗ is equimeasurable with f and

Qf (t) =∫ ∞

tf (s)

dss

=∫ ∞

1f (ts)

dss

=∫ ∞

1Es f (t)

dss.

By Minkowski inequality

||H||Λp(w)→Λp(w) = sup||f ||Λp(w)≤1

||Hf ||Λp(w) . sup||f∗||Lp(w)≤1

||(P + Q)f ∗||Lp(w)

= sup||f∗||Lp(w)≤1

∣∣∣∣∣∣∣∣∣∣∫ 1

0Es f ∗ds +

∫ ∞1

Es f ∗dss

∣∣∣∣∣∣∣∣∣∣Lp(w)

.∫ 1

0W1/p(s)ds +

∫ ∞1

W1/p(s)dss.

Proposition

If∫ 1

0W1/p(s)ds +

∫ ∞1

W1/p(s)dss<∞, then H : Λp(w)→ Λp(w).

bg=whiteThe case of classical Lorentz spacesBoyd theorem for Λp(w)

Recall that

W(µ) = sups>0

W(µs)W(s)

,

and define the lower Boyd index

βΛp(w) = limλ→0

log W1/p(λ)logλ

.

Theorem (Boyd 1950)It holds that

H : Λp(w)→ Λp(w) if and only if 0 < βΛp(w) and αΛp(w) < 1.

Corollary

H : Λp(w)→ Λp(w)⇔ 1 + logst

.W(s)W(t)

.( s

t

)p−ε, t ≤ s.

bg=whiteGeneralized Lorentz Shimogaki theorem

The generalized upper Boyd index is:

αΛpu(w) = lim

µ→∞

log W1/pu (µ)

logµ,

where the function Wu : [1,∞)→ [1,∞) is defined by:

Wu(λ) := sup{

W(u(∪j∈JIj))W(u(∪j∈JSj))

: Sj ⊆ Ij and |Ij| = λ|Sj| for every j ∈ J}.

Theorem (Lerner, Pérez 2007- Carro, Raposo, Soria 2007)If p > 0, then M : Λp

u(w)→ Λpu(w) ⇔ αΛp

u(w) < 1.

Theorem (A., Antezana, Carro, Soria 2014)If p > 1, then M : Λp

u(w)→ Λp,∞u (w) ⇔ αΛp

u(w) < 1.

Remark: If w = 1, then αΛpu(w) < 1 is equivalent to the Ap condition.

bg=whiteGeneralized Boyd theoremThrough the Weighted Lorentz spaces

The generalized lower Boyd index is:

βΛpu(w) = lim

λ→0

log Wu1/p(λ)

logλ,

where the function Wu : (0, 1]→ (0, 1] is defined as follows:

Wu(λ) := sup{

W(u(∪j∈JSj))W(u(∪j∈JIj))

: Sj ⊆ Ij and |Sj| = λ|Ij|, for every j ∈ J}.

Theorem (A., Antezana, Carro, Soria 2014)If p > 1, then H : Λp

u(w)→ Λpu(w) ⇔ 0 < βΛp

u(w) and αΛpu(w) < 1.

Remark:(i) If w = 1, then 0 < βΛp

u(w) is just the A∞ condition.(ii) If w = 1, then the condition 0 < βΛp

u(w) and αΛpu(w) < 1 recover the

conditions over the classical Boyd indices.

bg=whiteSome references

[1] E. Agora, J. Antezana, M. J. Carro and J. Soria, Lorentz-Shimogaki andBoyd Theorems for weighted Lorentz spaces, London Math. Soc. 89 (2014)321-336.

[2] E. Agora, M. J. Carro and J. Soria, Complete characterization of theweak-type boundedness of the Hilbert transform on weighted Lorentzspaces, J. Fourier Anal. Appl. 19 (2013) 712-730.

[3] M. J. Carro, J. A. Raposo and J. Soria, Recent developments in the theoryof Lorentz Spaces and Weighted Inequalities, Mem. Amer. Math. Soc. 187(2007) no. 877, xii+128.

[4] A. Lerner and C. Pérez, A new characterization of the Muckenhoupt Ap

weights through an extension of the Lorentz-Shimogaki theorem, IndianaUniv. Math. J. 56 (2007) no. 6, 2697-2722.

bg=whiteThe end

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