Boundedness of some operators on grand generalized weighted Morrey spaces on RD-spaces

TL;DR

This paper proves the boundedness of certain operators, including maximal and Calderón-Zygmund operators, on grand generalized weighted Morrey spaces over RD-spaces, extending known results to new spaces and operators.

Contribution

It establishes boundedness of key operators and their commutators on grand generalized weighted Morrey spaces over RD-spaces, a novel extension even in Euclidean domains.

Findings

Hardy-Littlewood maximal operator is bounded on these spaces.

$ heta$-type Calderón-Zygmund operators are bounded on these spaces.

Boundedness of commutators with BMO functions is proven.

Abstract

The aim of this paper is to obtain the boundedness of some operator on grand generalized weighted Morrey spaces $\mathcal{L}^{p),\phi}_{\varphi}(\omega)$ over RD-spaces. Under assumption that functions $\varphi$ and $\phi$ satisfy certain conditions, the authors prove that Hardy-Littlewood maximal operator and $\theta$-type Calder\'{o}n-Zygmund operator are bounded on grand generalized weighted Morrey spaces $\mathcal{L}^{p),\phi}_{\varphi}(\omega)$. Moreover, the boundedness of commutator $[b,T_{\theta}]$ which is generated by $\theta$-type Calder\'{o}n-Zygmund operator $T_{\theta}$ and $b\in\mathrm{BMO}(\mu)$ on spaces $\mathcal{L}^{p),\phi}_{\varphi}(\omega)$ is also established. The results regarding the grand generalized weighted Morrey spaces is new even for domains of Euclidean spaces.