Boundedness of fractional maximal operator and its commutators on generalized Orlicz-Morrey spaces

TL;DR

This paper establishes conditions under which the fractional maximal operator and its commutators are bounded on generalized Orlicz-Morrey spaces, expanding understanding of their behavior without requiring monotonicity of the involved functions.

Contribution

It provides new boundedness criteria for fractional maximal operators and their commutators on generalized Orlicz-Morrey spaces, including weak versions, without monotonicity assumptions.

Findings

Boundedness conditions for fractional maximal operator on these spaces.

Boundedness of commutators with BMO functions on the spaces.

Conditions expressed via supremal-type inequalities on weights.

Abstract

We consider generalized Orlicz-Morrey spaces $M_{\Phi,\varphi}(\mathbb{R}^{n})$ including their weak versions $WM_{\Phi,\varphi}(\mathbb{R}^{n})$. We find the sufficient conditions on the pairs $(\varphi_{1},\varphi_{2})$ and $(\Phi, \Psi)$ which ensures the boundedness of the fractional maximal operator $M_{\alpha}$ from $M_{\Phi,\varphi_1}(\mathbb{R}^{n})$ to $M_{\Psi,\varphi_2}(\mathbb{R}^{n})$ and from $M_{\Phi,\varphi_1}(\mathbb{R}^{n})$ to $WM_{\Psi,\varphi_2}(\mathbb{R}^{n})$. As applications of those results, the boundedness of the commutators of the fractional maximal operator $M_{b,\alpha}$ with $b \in BMO(\mathbb{R}^{n})$ on the spaces $M_{\Phi,\varphi}(\mathbb{R}^{n})$ is also obtained. In all the cases the conditions for the boundedness are given in terms of supremal-type inequalities on weights $\varphi(x,r)$, which do not assume any assumption on monotonicity of $\varphi(x,r)$ on $r$.