On the Theory of Multilinear Singular Operators with Rough Kernels on the Weighted Morrey Spaces

Abstract

We study some multilinear operators with rough kernels. For the multilinear fractional integral operators $T_{\mathrm{\Omega },\alpha }^A$and the multilinear fractional maximal integral operators $M_{\mathrm{\Omega },\alpha }^A$, we obtain their boundedness on weighted Morrey spaces with two weights $L^{p,\kappa }(u,v)$when $D^{\gamma }A\in {\dot{\mathrm{\Lambda }}}_{\beta }(|\gamma |=m-\mathrm{1})$or $D^{\gamma }A\in \mathrm{B}\mathrm{M}\mathrm{O}(|\gamma |=m-\mathrm{1})$. For the multilinear singular integral operators $T_{\mathrm{\Omega }}^A$and the multilinear maximal singular integral operators $M_{\mathrm{\Omega }}^A$, we show they are bounded on weighted Morrey spaces with two weights $L^{p,\kappa }(u,v)$if $D^{\gamma }A\in {\dot{\mathrm{\Lambda }}}_{\beta }(|\gamma |=m-\mathrm{1})$and bounded on weighted Morrey spaces with one weight $L^{p,\kappa }(w)$if $D^{\gamma }A\in \mathrm{B}\mathrm{M}\mathrm{O}(|\gamma |=m-\mathrm{1})$for $m=\mathrm{1,2}$.