Bloom Type Inequality: The Off-Diagonal Case

Abstract

In this paper, we establish a representation formula for fractional integrals. As a consequence, we get a Bloom type inequality for the Ferguson-Lacey type commutator involved with fractional integrals. Our results extend similar results for singular integral operators. The main difference is that mixed-norm spaces are invoked when we study the off-diagonal case of Bloom type inequalities for fractional integral operators. Specifically, for two fractional integral operators $I_{\lambda _1}$ and $I_{\lambda _2}$, we prove a Bloom type inequality

where the indices satisfy $1<p_1<q_1<\infty$, $1<p_2<q_2<\infty$, $1/q_1+1/p_1'=\lambda _1/n$ and $1/q_2+1/p_2'=\lambda _2/m$, 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, ${\text {BMO}}_{\mathrm{{prod}}}(\nu )$ is a weighted product ${\text {BMO}}$ 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.