On the Riesz potential and its commutators on generalized Orlicz-Morrey spaces

We consider generalized Orlicz-Morrey spaces M_{\Phi,\varphi}(\Rn) including their weak versions WM_{\Phi,\varphi}(\Rn). In these spaces we prove the boundedness of the Riesz potential from M_{\Phi,\varphi_1}(\Rn) to M_{\Psi,\varphi_2}(\Rn) and from M_{\Phi,\varphi_1}(\Rn) to WM_{\Psi,\varphi_2}(\Rn). As applications of those results, the boundedness of the commutators of the Riesz potential on generalized Orlicz-Morrey space is also obtained. In all the cases the conditions for the boundedness are given either in terms of Zygmund-type integral inequalities on $(\varphi_{1},\varphi_{2})$, which do not assume any assumption on monotonicity of $\varphi_{1}(x,r)$, $\varphi_{2}(x,r)$ in r.Comment: 23 pages. J. Funct. Spaces Appl.(to appear

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

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 $r$.Comment: 23 pages. Complex Anal. Oper. Theory (to appear). arXiv admin note: substantial text overlap with arXiv:1310.660