Weak type estimates for singular integrals related to a dual problem of Muckenhoupt-Wheeden

Andrei K. Lerner; Sheldy Ombrosi; Carlos Pérez

doi:10.1007/s00041-008-9032-2

Weak type estimates for singular integrals related to a dual problem of Muckenhoupt-Wheeden

Andrei K. Lerner
, Sheldy Ombrosi
, Carlos Pérez

Research output: Contribution to journal › Article › peer-review

20 Scopus citations

Abstract

A well-known open problem of Muckenhoupt-Wheeden says that any Calderón-Zygmund singular integral operator T is of weak type (1,1) with respect to a couple of weights (w,Mw). In this paper, we consider a somewhat "dual" problem: sup_{λ > 0} λ w {x ∈ ℝⁿ: |Tf(x)|/Mw > λ} ≤ c ∫ℝⁿ |f|,dx. We prove a weaker version of this inequality with M ³ w instead of Mw. Also we study a related question about the behavior of the constant in terms of the A ₁ characteristic of w.

Original language	English
Pages (from-to)	394-403
Number of pages	10
Journal	Journal of Fourier Analysis and Applications
Volume	15
Issue number	3
DOIs	https://doi.org/10.1007/s00041-008-9032-2
State	Published - Jun 2009
Externally published	Yes

Keywords

Calderón-Zygmund operators
Weights

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10.1007/s00041-008-9032-2

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abstract = "A well-known open problem of Muckenhoupt-Wheeden says that any Calder{\'o}n-Zygmund singular integral operator T is of weak type (1,1) with respect to a couple of weights (w,Mw). In this paper, we consider a somewhat {"}dual{"} problem: supλ > 0 λ w \{x ∈ ℝn: |Tf(x)|/Mw > λ\} ≤ c ∫ℝn |f|,dx. We prove a weaker version of this inequality with M 3 w instead of Mw. Also we study a related question about the behavior of the constant in terms of the A 1 characteristic of w.",

keywords = "Calder{\'o}n-Zygmund operators, Weights",

author = "Lerner, \{Andrei K.\} and Sheldy Ombrosi and Carlos P{\'e}rez",

year = "2009",

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doi = "10.1007/s00041-008-9032-2",

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AU - Lerner, Andrei K.

AU - Ombrosi, Sheldy

AU - Pérez, Carlos

PY - 2009/6

Y1 - 2009/6

N2 - A well-known open problem of Muckenhoupt-Wheeden says that any Calderón-Zygmund singular integral operator T is of weak type (1,1) with respect to a couple of weights (w,Mw). In this paper, we consider a somewhat "dual" problem: supλ > 0 λ w {x ∈ ℝn: |Tf(x)|/Mw > λ} ≤ c ∫ℝn |f|,dx. We prove a weaker version of this inequality with M 3 w instead of Mw. Also we study a related question about the behavior of the constant in terms of the A 1 characteristic of w.

AB - A well-known open problem of Muckenhoupt-Wheeden says that any Calderón-Zygmund singular integral operator T is of weak type (1,1) with respect to a couple of weights (w,Mw). In this paper, we consider a somewhat "dual" problem: supλ > 0 λ w {x ∈ ℝn: |Tf(x)|/Mw > λ} ≤ c ∫ℝn |f|,dx. We prove a weaker version of this inequality with M 3 w instead of Mw. Also we study a related question about the behavior of the constant in terms of the A 1 characteristic of w.

KW - Calderón-Zygmund operators

KW - Weights

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SN - 1069-5869

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JO - Journal of Fourier Analysis and Applications

JF - Journal of Fourier Analysis and Applications

IS - 3

ER -

Weak type estimates for singular integrals related to a dual problem of Muckenhoupt-Wheeden

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