Bilinear $\theta$-type Calder\'on-Zygmund operators and its commutator on generalized weighted Morrey spaces over RD-spaces

TL;DR

This paper proves the boundedness of bilinear $ heta$-type Calderón-Zygmund operators and their commutators on generalized weighted Morrey spaces over RD-spaces, extending harmonic analysis tools to more general metric measure spaces.

Contribution

It establishes the boundedness of these operators and their commutators on generalized weighted Morrey spaces over RD-spaces, a significant extension in harmonic analysis.

Findings

Boundedness of $T_{\theta}$ on generalized weighted Morrey spaces.

Boundedness of commutators $[b_1,b_2,T_{\theta}]$ on these spaces.

Extension of Calderón-Zygmund theory to RD-spaces.

Abstract

An RD-space $\mathcal{X}$ is a space of homogeneous type in the sense of Coifman and Weiss with the additional property that a reverse doubling property holds in $\mathcal{X}$. In this setting, the authors establish the boundedness of bilinear $\theta$-type Calder\'on-Zygmund operator $T_{\theta}$ and its commutator $[b_1,b_2,T_{\theta}]$ generated by the function $b_1,b_2\in BMO(\mu)$ and $T_{\theta}$ on generalized weighted Morrey space $\mathcal{M}^{p,\phi}(\omega)$ and generalized weighted weak Morrey space $W\mathcal{M}^{p,\phi}(\omega)$ over RD-spaces.