Estimate for bilinear Calder\'{o}n-Zygmund operator and its commutator on homogeneous variable exponent spaces

Abstract

Let $(X,d,\mu)$ be a space of homogeneous type in the sense of Coifman and and Weiss. In this setting, the author proves that bilinear Calder\'{o}n-Zygmund operators are bounded from the product of variable exponent Lebesgue spaces $L^{p_{1}(\cdot)}(X)\times L^{p_{2}(\cdot)}(X)$ into spaces $L^{p(\cdot)}(X)$, and bounded from product of variable exponent generalized Morrey spaces $\mathcal{L}^{p_{1}(\cdot),\varphi_{1}}(X)\times \mathcal{L}^{p_{2}(\cdot),\varphi_{2}}(X)$ into spaces $\mathcal{L}^{p(\cdot),\varphi}(X)$, where the Lebesgue measure functions $\varphi(\cdot,\cdot), \varphi_{1}(\cdot,\cdot)$ and $\varphi_{2}(\cdot,\cdot)$ satisfy $\varphi_{1}\times\varphi_{2}=\varphi$, and $\frac{1}{p(\cdot)}=\frac{1}{p_{1}(\cdot)}+\frac{1}{p_{2}(\cdot)}$. Furthermore, by establishing sharp maximal estimate for the commutator $[b_{1},b_{2},BT]$ generated by $b_{1}, b_{2}\in\mathrm{BMO}(X)$ and the $BT$, the author shows that the $[b_{1},b_{2},BT]$ is bounded from product of spaces $L^{p_{1}(\cdot)}(X)\times L^{p_{2}(\cdot)}(X)$ into spaces $L^{p(\cdot)}(X)$, and also bounded from product of spaces $\mathcal{L}^{p_{1}(\cdot),\varphi_{1}}(X)\times \mathcal{L}^{p_{2}(\cdot),\varphi_{2}}(X)$ into spaces $L^{p(\cdot),\varphi}(X)$.