Bilinear pseudo-differential operators on product of weighted spaces

Abstract

Let $\widetilde{T}_{\sigma }$ be a bilinear pseudo-differential operator and $\vec {\omega }=(\omega _{1},\omega _{2})\in A_{\vec {P}}$, where $\vec {P}=(p_{1},p_{2})$, $p_{i}\in [1,\infty )$ with $i=1,\ 2$, and $A_{\vec {P}}$ denotes the multiple weight class. In this paper, we will prove that $\widetilde{T}_{\sigma }$ is bounded from the product of weighted Lebesgue spaces $L^{p_{1}}_{\omega _{1}}(\mathbb {R}^{n})\times L^{p_{2}}_{\omega _{2}}(\mathbb {R}^{n})$ into the $L^{p}_{\vec {\nu }_{\omega }}(\mathbb {R}^{n})$, where $\frac{1}{p}=\frac{1}{p_{1}}+\frac{1}{p_{2}}$ and $\vec {\nu }_{\omega }(x)=\mathop {\prod }\nolimits ^{2}_{i=1}[\omega _{i}(x)]^{\frac{p}{p_{i}}}$, and bounded from the weighted Morrey spaces $L^{p_{1},\kappa ,\rho }_{\omega _{1}}(\mathbb {R}^{n})\times L^{p_{2},\kappa ,\rho }_{\omega _{2}}(\mathbb {R}^{n})$ into the $L^{p,\kappa ,\rho }_{\vec {\nu }_{\omega }}(\mathbb {R}^{n})$. Furthermore, the endpoint estimate for $\widetilde{T}_{\sigma }$ on the weighted Hardy space $H^{1}_{\omega }(\mathbb {R}^{n})$ is also obtained.