A two weight inequality for Calderón–Zygmund operators on spaces of homogeneous type with applications

Xuan Thinh Duong; Ji Li; Eric T. Sawyer; Manasa N. Vempati; Brett D. Wick; Dongyong Yang

doi:10.1016/j.jfa.2021.109190

A two weight inequality for Calderón–Zygmund operators on spaces of homogeneous type with applications

Xuan Thinh Duong, Ji Li^*, Eric T. Sawyer, Manasa N. Vempati, Brett D. Wick, Dongyong Yang

^*Corresponding author for this work

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

7 Citations (Scopus)

Abstract

Let (X,d,μ) be a space of homogeneous type in the sense of Coifman and Weiss, i.e. d is a quasi metric on X and μ is a positive measure satisfying the doubling condition. Suppose that u and v are two locally finite positive Borel measures on (X,d,μ). Subject to the pair of weights satisfying a side condition, we characterize the boundedness of a Calderón–Zygmund operator T from L²(u) to L²(v) in terms of the A₂ condition and two testing conditions. For every cube B⊂X, we have the following testing conditions, with 1_B taken as the indicator of B

‖T(u1_B)‖_L²(B,v)≤T‖1_B‖_L²(u),

‖T^⁎(v1_B)‖_L²(B,u)≤T‖1_B‖_L²(v).

The proof uses stopping cubes and corona decompositions originating in work of Nazarov, Treil and Volberg, along with the pivotal side condition.

Original language	English
Article number	109190
Pages (from-to)	1-65
Number of pages	65
Journal	Journal of Functional Analysis
Volume	281
Issue number	9
DOIs	https://doi.org/10.1016/j.jfa.2021.109190
Publication status	Published - 1 Nov 2021

Keywords

Two weight inequality
Testing conditions
Space of homogeneous type
Calderón–Zygmund operator
Haar basis

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10.1016/j.jfa.2021.109190

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@article{5ef80bee87224c4f9953936c309a2145,

title = "A two weight inequality for Calder{\'o}n–Zygmund operators on spaces of homogeneous type with applications",

abstract = "Let (X,d,μ) be a space of homogeneous type in the sense of Coifman and Weiss, i.e. d is a quasi metric on X and μ is a positive measure satisfying the doubling condition. Suppose that u and v are two locally finite positive Borel measures on (X,d,μ). Subject to the pair of weights satisfying a side condition, we characterize the boundedness of a Calder{\'o}n–Zygmund operator T from L2(u) to L2(v) in terms of the A2 condition and two testing conditions. For every cube B⊂X, we have the following testing conditions, with 1B taken as the indicator of B ‖T(u1B)‖L2(B,v)≤T‖1B‖L2(u), ‖T⁎(v1B)‖L2(B,u)≤T‖1B‖L2(v).The proof uses stopping cubes and corona decompositions originating in work of Nazarov, Treil and Volberg, along with the pivotal side condition.",

keywords = "Two weight inequality, Testing conditions, Space of homogeneous type, Calder{\'o}n–Zygmund operator, Haar basis",

author = "Duong, \{Xuan Thinh\} and Ji Li and Sawyer, \{Eric T.\} and Vempati, \{Manasa N.\} and Wick, \{Brett D.\} and Dongyong Yang",

year = "2021",

month = nov,

day = "1",

doi = "10.1016/j.jfa.2021.109190",

language = "English",

volume = "281",

pages = "1--65",

journal = "Journal of Functional Analysis",

issn = "0022-1236",

publisher = "Academic Press Inc.",

number = "9",

}

TY - JOUR

T1 - A two weight inequality for Calderón–Zygmund operators on spaces of homogeneous type with applications

AU - Duong, Xuan Thinh

AU - Li, Ji

AU - Sawyer, Eric T.

AU - Vempati, Manasa N.

AU - Wick, Brett D.

AU - Yang, Dongyong

PY - 2021/11/1

Y1 - 2021/11/1

N2 - Let (X,d,μ) be a space of homogeneous type in the sense of Coifman and Weiss, i.e. d is a quasi metric on X and μ is a positive measure satisfying the doubling condition. Suppose that u and v are two locally finite positive Borel measures on (X,d,μ). Subject to the pair of weights satisfying a side condition, we characterize the boundedness of a Calderón–Zygmund operator T from L2(u) to L2(v) in terms of the A2 condition and two testing conditions. For every cube B⊂X, we have the following testing conditions, with 1B taken as the indicator of B ‖T(u1B)‖L2(B,v)≤T‖1B‖L2(u), ‖T⁎(v1B)‖L2(B,u)≤T‖1B‖L2(v).The proof uses stopping cubes and corona decompositions originating in work of Nazarov, Treil and Volberg, along with the pivotal side condition.

AB - Let (X,d,μ) be a space of homogeneous type in the sense of Coifman and Weiss, i.e. d is a quasi metric on X and μ is a positive measure satisfying the doubling condition. Suppose that u and v are two locally finite positive Borel measures on (X,d,μ). Subject to the pair of weights satisfying a side condition, we characterize the boundedness of a Calderón–Zygmund operator T from L2(u) to L2(v) in terms of the A2 condition and two testing conditions. For every cube B⊂X, we have the following testing conditions, with 1B taken as the indicator of B ‖T(u1B)‖L2(B,v)≤T‖1B‖L2(u), ‖T⁎(v1B)‖L2(B,u)≤T‖1B‖L2(v).The proof uses stopping cubes and corona decompositions originating in work of Nazarov, Treil and Volberg, along with the pivotal side condition.

KW - Two weight inequality

KW - Testing conditions

KW - Space of homogeneous type

KW - Calderón–Zygmund operator

KW - Haar basis

UR - https://www.scopus.com/pages/publications/85111311248

UR - http://purl.org/au-research/grants/arc/DP190100970

UR - http://purl.org/au-research/grants/arc/DP170101060

U2 - 10.1016/j.jfa.2021.109190

DO - 10.1016/j.jfa.2021.109190

M3 - Article

SN - 0022-1236

VL - 281

SP - 1

EP - 65

JO - Journal of Functional Analysis

JF - Journal of Functional Analysis

IS - 9

M1 - 109190

ER -

A two weight inequality for Calderón–Zygmund operators on spaces of homogeneous type with applications

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Harmonic analysis and dispersive partial differential equations

Harmonic analysis: function spaces and partial differential equations

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A two weight inequality for Calderón–Zygmund operators on spaces of homogeneous type with applications

Abstract

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Harmonic analysis and dispersive partial differential equations

Harmonic analysis: function spaces and partial differential equations

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