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

6 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

Access to Document

10.1016/j.jfa.2021.109190

Cite this

@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 = "Elsevier",

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 - http://www.scopus.com/inward/record.url?scp=85111311248&partnerID=8YFLogxK

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

Abstract

Keywords

Access to Document

Other files and links

Fingerprint

Harmonic analysis and dispersive partial differential equations

Harmonic analysis: function spaces and partial differential equations

Cite this

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

Abstract

Keywords

Access to Document

Other files and links

Fingerprint

Projects

Harmonic analysis and dispersive partial differential equations

Harmonic analysis: function spaces and partial differential equations

Cite this