Bloom Type Upper Bounds in the Product BMO Setting

Kangwei Li; Henri Martikainen; Emil Vuorinen

doi:10.1007/s12220-019-00194-3

Bloom Type Upper Bounds in the Product BMO Setting

Kangwei Li
, Henri Martikainen
, Emil Vuorinen

Department of Mathematics

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

13 Scopus citations

Abstract

We prove some Bloom type estimates in the product BMO setting. More specifically, for a bounded singular integral T_n in Rⁿ and a bounded singular integral T_m in R^m we prove that ‖[Tn1,[b,Tm2]]‖Lp(μ)→Lp(λ)≲[μ]Ap,[λ]Ap‖b‖BMOprod(ν),where p∈ (1 , ∞) , μ, λ∈ A_p and ν: = μ¹ ^/ ^pλ^- ¹ ^/ ^p is the Bloom weight. Here Tn1 is T_n acting on the first variable, Tm2 is T_m acting on the second variable, A_p stands for the bi-parameter weights of Rⁿ× R^m and BMO _prod(ν) is a weighted product BMO space.

Original language	English
Pages (from-to)	3181-3203
Number of pages	23
Journal	Journal of Geometric Analysis
Volume	30
Issue number	3
DOIs	https://doi.org/10.1007/s12220-019-00194-3
State	Published - Jul 1 2020

Keywords

Bloom’s inequality
Iterated commutators
Product BMO
Weighted BMO

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title = "Bloom Type Upper Bounds in the Product BMO Setting",

abstract = "We prove some Bloom type estimates in the product BMO setting. More specifically, for a bounded singular integral Tn in Rn and a bounded singular integral Tm in Rm we prove that ‖[Tn1,[b,Tm2]]‖Lp(μ)→Lp(λ)≲[μ]Ap,[λ]Ap‖b‖BMOprod(ν),where p∈ (1 , ∞) , μ, λ∈ Ap and ν: = μ1 / pλ- 1 / p is the Bloom weight. Here Tn1 is Tn acting on the first variable, Tm2 is Tm acting on the second variable, Ap stands for the bi-parameter weights of Rn× Rm and BMO prod(ν) is a weighted product BMO space.",

keywords = "Bloom{\textquoteright}s inequality, Iterated commutators, Product BMO, Weighted BMO",

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T1 - Bloom Type Upper Bounds in the Product BMO Setting

AU - Li, Kangwei

AU - Martikainen, Henri

AU - Vuorinen, Emil

PY - 2020/7/1

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N2 - We prove some Bloom type estimates in the product BMO setting. More specifically, for a bounded singular integral Tn in Rn and a bounded singular integral Tm in Rm we prove that ‖[Tn1,[b,Tm2]]‖Lp(μ)→Lp(λ)≲[μ]Ap,[λ]Ap‖b‖BMOprod(ν),where p∈ (1 , ∞) , μ, λ∈ Ap and ν: = μ1 / pλ- 1 / p is the Bloom weight. Here Tn1 is Tn acting on the first variable, Tm2 is Tm acting on the second variable, Ap stands for the bi-parameter weights of Rn× Rm and BMO prod(ν) is a weighted product BMO space.

AB - We prove some Bloom type estimates in the product BMO setting. More specifically, for a bounded singular integral Tn in Rn and a bounded singular integral Tm in Rm we prove that ‖[Tn1,[b,Tm2]]‖Lp(μ)→Lp(λ)≲[μ]Ap,[λ]Ap‖b‖BMOprod(ν),where p∈ (1 , ∞) , μ, λ∈ Ap and ν: = μ1 / pλ- 1 / p is the Bloom weight. Here Tn1 is Tn acting on the first variable, Tm2 is Tm acting on the second variable, Ap stands for the bi-parameter weights of Rn× Rm and BMO prod(ν) is a weighted product BMO space.

KW - Bloom’s inequality

KW - Iterated commutators

KW - Product BMO

KW - Weighted BMO

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DO - 10.1007/s12220-019-00194-3

M3 - Article

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SP - 3181

EP - 3203

JO - Journal of Geometric Analysis

JF - Journal of Geometric Analysis

IS - 3

ER -

Bloom Type Upper Bounds in the Product BMO Setting

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