Bilinear $$\theta $$-type Calderón–Zygmund operator and its commutator on non-homogeneous weighted Morrey spaces

Abstract

The main purpose of this paper is to establish the boundedness of bilinear $$\theta $$ -type Calderon–Zygmund operator $$T_{\theta }$$ and its commutator $$[b_{1},b_{2},T_{\theta }]$$ generated by the function $$b_{i}\in \widetilde{\mathrm {RBMO}}(\mu )$$ with $$i=1,2$$ and $$T_{\theta }$$ on weighted Morrey space $$L^{p,\kappa ,\varrho }(\omega )$$ and weighted weak Morrey space $$WL^{p,\kappa ,\varrho }(\omega )$$ over non-homogeneous metric measure space. Under assumption that $$\omega $$ satisfies weighted integral conditions, the author proves that $$T_{\theta }$$ is bounded from weighted weak Morrey space $$WL^{p,\kappa ,\varrho }(\omega )$$ into weighted Morrey space $$L^{p,\kappa ,\varrho }(\omega )$$ with $$1\le p<\infty $$ . In addition, via the sharp maximal function, the boundedness of the commutator $$[b_{1},b_{2},T_{\theta }]$$ on the weighted Morrey space $$L^{p,\kappa ,\varrho }(\omega )$$ is also obtained.