Bloom Type Inequality: The Off-diagonal Case

Abstract

In this paper, we establish a representation formula for fractional integrals. As a consequence, for two fractional integral operators $I_{{\lambda }_1}$ and $I_{{\lambda }_2}$, we prove a Bloom type inequality \begin{align*} \mbox{\hbox to 8em{}}& \hskip -8em \left\|\big[I_{λ_1}^1,\big[b,I_{λ_2}^2\big]\big] \right\|_{L^{p_2}(L^{p_1})(μ_2^{p_2}\timesμ_1^{p_1})\rightarrow L^{q_2}(L^{q_1})(σ_2^{q_2}\timesσ_1^{q_1})} % \\ %& \lesssim_{\substack{[μ_1]_{A_{p_1,q_1}(\mathbb R^n)},[μ_2]_{A_{p_2,q_2}(\mathbb R^m)} \\ [σ_1]_{A_{p_1,q_1}(\mathbb R^n)},[σ_2]_{A_{p_2,q_2}(\mathbb R^m)}}} \|b\|_{\BMO_{\pro}(ν)}, \end{align*} where the indices satisfy $1<p_1<q_1<\infty$, $1<p_2<q_2<\infty$, $1/q_1+1/p_1^{\prime }={\lambda }_1/n$ and $1/q_2+1/p_2^{\prime }={\lambda }_2/m$, the weights ${\mu }_1,{\sigma }_1\in A_{p_1,q_1}({\mathbb{R}}^n)$, ${\mu }_2,{\sigma }_2\in A_{p_2,q_2}({\mathbb{R}}^m)$ and $\nu :={\mu }_1{\sigma }_1^{-1}\otimes {\mu }_2{\sigma }_2^{-1}$, $I_{{\lambda }_1}^1$ stands for $I_{{\lambda }_1}$ acting on the first variable and $I_{{\lambda }_2}^2$ stands for $I_{{\lambda }_2}$ acting on the second variable, $\BMO_{\rm{prod}}(ν)$ is a weighted product $\BMO$ space and $L^{p_2}(L^{p_1})({\mu }_2^{p_2}\times {\mu }_1^{p_1})$ and $L^{q_2}(L^{q_1})({\sigma }_2^{q_2}\times {\sigma }_1^{q_1})$ are mixed-norm spaces.

27 pages