Bloom Type Inequality: The Off-diagonal Case

2019/07/17 by Junren Pan, Pan, Junren, Wenchang Sun +1
Mathematics · #42B20 #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #math.CA #msc:42B20

paper · pdf · doi:10.48550/arxiv.1907.07292

27 pages

arxiv created 2019/07/17 · arxiv updated 2019/07/18

Abstract

In this paper, we establish a representation formula for fractional integrals. As a consequence, for two fractional integral operators I_λ1 and I_λ2, we prove a Bloom type inequality \mbox\hbox to 8em \hskip -8em ‖[I_λ1¹,[b,I_λ2²]] ‖_{L^p2(L^p1)(μ2^p2×μ1^p1)→ L^q2(L^q1)(σ2^q2×σ1^q1)} %
% \lesssim_\substack[μ₁]_A_p1,q1(\mathbb Rⁿ),[μ₂]_A_p2,q2(\mathbb R^m)
[σ₁]_A_p1,q1(\mathbb Rⁿ),[σ₂]_A_p2,q2(\mathbb R^m) ‖b‖_\BMO_\pro(ν), where the indices satisfy 1<p₁<q₁<∞, 1<p₂<q₂<∞, 1/q₁+1/p₁'=λ₁/n and 1/q₂+1/p₂'=λ₂/m, the weights μ₁,σ₁ ∈ A_p1,q1(\mathbb Rⁿ), μ₂,σ₂ ∈ A_p2,q2(\mathbb R^m) and ν:=μ₁σ₁^-1⊗ μ₂σ₂^-1, I_λ1¹ stands for I_λ1 acting on the first variable and I_λ2² stands for I_λ2 acting on the second variable, \BMO_\rmprod(ν) is a weighted product \BMO space and L^p₂(L^p₁)(μ₂^p₂×μ₁^p₁) and L^q₂(L^q₁)(σ₂^q₂×σ₁^q₁) are mixed-norm spaces.

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