Boundedness of some operators on grand generalized Morrey spaces over non-homogeneous spaces

Abstract

The aim of this paper is to obtain the boundedness of some operator on grand generalized Morrey space $\mathcal{L}^{p),\varphi,\phi}_{\mu}(G)$ over non-homogeneous spaces, where $G\subset$ $\mathbb{R}^{n}$ is a bounded domain. Under assumption that functions $\varphi$ and $\phi$ satisfy certain conditions, the authors prove that the Hardy-Littlewood maximal operator, fractional integral operators and $\theta$-type Calder\'{o}n-Zygmund operators are bounded on the non-homogeneous grand generalized Morrey space $\mathcal{L}^{p),\varphi,\phi}_{\mu}(G)$. Moreover, the boundedness of commutator $[b,T^{G}_{\theta}]$ which is generated by $\theta$-type Calder\'{o}n-Zygmund operator $T_{\theta}$ and $b\in\mathrm{RBMO}(\mu)$ on spaces $\mathcal{L}^{p),\varphi,\phi}_{\mu}(G)$ is also established.

Keywords

• non-doubling measure
• maximal operator
• fractional integral operator
• θ-type calderón-zygmund operator
• grand generalized morrey space