Certain bilinear operators on Morrey spaces

Abstract

In this paper, we consider that $T(f,g)$ is a bilinear operator satisfying \begin{equation*} |T(f,g)(x)|\preceq \int_{\mathbb{R}^{n}}\frac{|f(x-ty)g(x-y)|}{|y|^{n}}dy \end{equation*} for $x$ such that $0\notin {\rm supp}~(f(x-t\cdot )) \cap {\rm supp}~(g(x+\cdot ))$. We obtain the boundedness of $T(f,g)$ on the Morrey spaces with the assumption of the boundedness of the operator $T(f,g)$ on the Lebesgues spaces. As applications, we yield that many well known bilinear operators, as well as the first Calderón commutator, are bounded from the Morrey spaces $L^{q,\lambda_{1}}\times L^{r,\lambda_{2}}$ to $L^{p,\lambda}$, where $\lambda /p={\lambda_{1}}/{q}+{\lambda_{2}}/{r}$.