$$L^{p}(\mu ) \rightarrow L^{q}(\nu )$$ L p ( μ ) → L q ( ν ) Characterization for Well Localized Operators

Vuorinen, Emil

doi:10.1007/s00041-015-9453-7

Cited by 13 publications

(15 citation statements)

References 11 publications

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“…Very interesting recent papers [20], [21] give necessary and sufficient conditions for the two weight estimates of the well localized operators and of the Haar shifts, but again the conditions were more restrictive than the Sawyer type ones: again, for each interval I one has to check the roundedness on infinitely many vector-valued functions.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

Two weight L estimates for paraproducts in non-homogeneous settings

Lai

Treil

2018

Journal of Functional Analysis

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Abstract. We give a necessary and sufficient condition for the two weight L p -estimates for paraproducts in non-homogeneous settings, 1 < p < ∞. We are mainly interested in the case p = 2, since the case p = 2 is a well-known and easy corollary of the Carleson embedding theorem. The necessary and sufficient condition is given in terms of testing conditions of Sawyer type: for p ≤ 2 only one ("direct") testing condition is required, but for p > 2 both "direct" and "adjoint" testing conditions are needed. An interesting feature is that the "adjoint" testing condition is that it is a testing condition not for the adjoint of the paraproduct, but for some auxiliary operator.

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Section: Introduction and Main Resultsmentioning

confidence: 99%

Two weight L estimates for paraproducts in non-homogeneous settings

Lai

Treil

2018

Journal of Functional Analysis

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“…The definition of a well localized operator depends on a parameter r which measures how "well" the operator is localized. The constant C in the two weight inequality proved in [14] and [17] depends on r and the constants in the Sawyer type testing conditions. In [7] the dyadic shifts were looked at from a little different perspective.…”

Section: Introductionmentioning

confidence: 98%

Two-weight $L^{p}$-inequalities for dyadic shifts and the dyadic square function

Vuorinen

2017

Studia Math.

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We consider two weight L p → L q -inequalities for dyadic shifts and the dyadic square function with general exponents 1 < p, q < ∞. It is shown that if a so-called quadratic A p,q -condition related to the measures holds, then a family of dyadic shifts satisfies the two weight estimate in an R-bounded sense if and only if it satisfies the direct-and the dual quadratic testing condition. In the case p = q = 2 this reduces to the result by T. Hytönen, C. Pérez, S. Treil and A. Volberg [7].The dyadic square function satisfies the two weight estimate if and only if it satisfies the quadratic testing condition and the quadratic A p,q -condition holds. Again in the case p = q = 2 we recover the result by F. Nazarov, S. Treil and A. Volberg [13].An example shows that in general the quadratic A p,q -condition is stronger than the Muckenhoupt type A p,q -condition.2010 Mathematics Subject Classification. Primary 42B20; Secondary 42B25.

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“…Operators of interest are the maximal function [S1, Moe, PzR, V], fractional and Poisson integrals [S2,Cr], the Hilbert transform [CS1,CS2,KP,NTV1,LSSU,L3] and general Calderón-Zygmund singular integral operators and their commutators [CrRV,CrMoe,CrMPz2,NRTV], the square functions [LLi1,LLi2,CLiX,HLi], paraproducts and their dyadic counterparts [M,HoLWic1,HoLWic2,IKP,Be3]. Necessary and sufficient conditions are only known for the maximal function, fractional and Poisson integrals [S1], square functions [LLi1] and the Hilbert transform [L3,LSSU], and among the dyadic operators for the martingale transform, the dyadic square functions, positive and well localized dyadic operators [Wil1,NTV1,NTV3,T,Ha,HaHLi,HL,LSU2,Ta,Vu1,Vu2]. If the weights u and v are assumed to be in A d 2 , then necessary and sufficient conditions for boundedness of dyadic paraproducts and commutators in terms of Bloom's BM O are known [HoLWic1,HoLWic2].…”

Section: Introductionmentioning

confidence: 99%

On Two Weight Estimates for Dyadic Operators

Beznosova

Chung

Moraes

et al. 2017

Association for Women in Mathematics Series

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We provide a quantitative two weight estimate for the dyadic paraproduct π b under certain conditions on a pair of weights (u, v) and b in Carlu,v, a new class of functions that we show coincides with BM O when u = v ∈ A d 2 . We obtain quantitative two weight estimates for the dyadic square function and the martingale transforms under the assumption that the maximal function is bounded from L 2 (u) into L 2 (v) and v ∈ RH d 1 . Finally we obtain a quantitative two weight estimate from L 2 (u) into L 2 (v) for the dyadic square function under the assumption that the pair (u, v) is in joint A d 2 and u −1 ∈ RH d 1 , this is sharp in the sense that when u = v the conditions reduce to u ∈ A d 2 and the estimate is the known linear mixed estimate.2010 Mathematics Subject Classification. Primary 42B20, 42B25 ; Secondary 47B38.where ∆ I v := m I + v − m I − v, and I ± are the right and left children of I.Then π b , the dyadic paraproduct associated to b, is bounded from L 2 (u) into L 2 (v). Moreover, there exists a constant C > 0 such that for all f ∈ L 2 (u)where π b f := I∈D m I f b, h I h I .

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$$L^{p}(\mu ) \rightarrow L^{q}(\nu )$$ L p ( μ ) → L q ( ν ) Characterization for Well Localized Operators

Cited by 13 publications

References 11 publications

Two weight L estimates for paraproducts in non-homogeneous settings

Two weight L estimates for paraproducts in non-homogeneous settings

Two-weight $L^{p}$-inequalities for dyadic shifts and the dyadic square function

On Two Weight Estimates for Dyadic Operators

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