Certain bilinear operators on power-weighted Morrey spaces

In this paper, we consider the boundedness properties of two classes of bilinear operators on Morrey spaces with power weights. The first operator is the bilinear maximal operator $T^{\ast }(f,g)(x)=\sup_{j}|T_{j}(f,g)(x)|$, where $T_{j}(f,g)$ is a bilinear operator with the kernel $K_{j}$ satisfying the uniform estimate $$ \left\vert K_{j}(x,y_{1},y_{2})\right\vert \preceq \frac{1}{(|x-y_{1}|+|x-y_{2}|)^{2n}}, $$ where $x$, $y_{1}$, $y_{2}\in \mathbb{R}^{n}$ with $x\neq y_{k}$ for some $k\in\{1,2\}$. The second operator is $\mathcal{T}(f,g)$, which, being a bilinear operator, satisfies $$ |\mathcal{T}(f,g)(x)|\preceq \int_{\mathbb{R}^{n}}\frac{|f(x-ty)g(x-y)|}{|y|^{n}}dy $$ for $x\in\mathbb{R}^{n}$ and $0＜|t|\leq1$ such that $0\notin \operatorname{supp}\ (f(x-t\cdot ))\ \cap\ \operatorname{supp}\ (g(x+\cdot ))$. We obtain that these two operators are bounded operators from the product weighted Morrey spaces $L^{q,\lambda _{1}}(\mathbb{R} ^{n},|x|^{\beta }dx)\times L^{r,\lambda _{2}}(\mathbb{R}^{n},|x|^{\tau }dx)$ to the weighted Morrey spaces $L^{p,\lambda }(\mathbb{R}^{n},|x|^{\alpha }dx)$ with the assumption of the boundedness on Lebesgue spaces. As applications, we yield that many well-known bilinear operators are bounded on power-weighted Morrey spaces.