Bloom Type Inequality: The Off-diagonal Case

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_{\lambda_1}^1,\big[b,I_{\lambda_2}^2\big]\big] \right\|_{L^{p_2}(L^{p_1})(\mu_2^{p_2}\times\mu_1^{p_1})\rightarrow L^{q_2}(L^{q_1})(\sigma_2^{q_2}\times\sigma_1^{q_1})} % \\ %& \lesssim_{\substack{[\mu_1]_{A_{p_1,q_1}(\mathbb R^n)},[\mu_2]_{A_{p_2,q_2}(\mathbb R^m)} \\ [\sigma_1]_{A_{p_1,q_1}(\mathbb R^n)},[\sigma_2]_{A_{p_2,q_2}(\mathbb R^m)}}} \|b\|_{\BMO_{\pro}(\nu)}, \end{align*} 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, $\BMO_{\rm{prod}}(\nu)$ 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.