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

Abstract

We consider generalized Orlicz-Morrey spaces $M_{\Phi ,\phi }({\mathbb{R}}^n)$ including their weak versions $W{M}_{\Phi ,\phi }({\mathbb{R}}^n)$. We find the sufficient conditions on the pairs $({\phi }_1,{\phi }_2)$ and $(\Phi ,\Psi )$ which ensures the boundedness of the fractional maximal operator $M_{\alpha }$ from $M_{\Phi ,{\phi }_1}({\mathbb{R}}^n)$ to $M_{\Psi ,{\phi }_2}({\mathbb{R}}^n)$ and from $M_{\Phi ,{\phi }_1}({\mathbb{R}}^n)$ to $W{M}_{\Psi ,{\phi }_2}({\mathbb{R}}^n)$. 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 $M_{\Phi ,\phi }({\mathbb{R}}^n)$ is also obtained. In all the cases the conditions for the boundedness are given in terms of supremal-type inequalities on weights $\phi (x,r)$, which do not assume any assumption on monotonicity of $\phi (x,r)$ on $r$.

23 pages. Complex Anal. Oper. Theory (to appear). arXiv admin note: substantial text overlap with arXiv:1310.6604