On the Boundedness of the Fractional Maximal Operator, Riesz Potential and Their Commutators in Generalized Morrey Spaces

Abstract

In the paper the authors find conditions on the pair $(\varphi_{1},\varphi_{2})$ which ensure the Spanne type boundedness of the fractional maximal operator $M_{\alpha}$ and the Riesz potential operator $I_{\alpha}$ from one generalized Morrey spaces $M_{p,{\varphi_{1}}}$ to another $M_{q,{\varphi_{2}}}, 1 < p < q < \infty, 1/p-1/q = \alpha/n,$ and from $M_{1,{\varphi_{1}}}$ to the weak space W $M_{q,{\varphi_{2}}}, 1 < p < q < \infty, 1- 1/q = \alpha/n,$ We also find conditions on $\varphi$ which ensure the Adams type boundedness of the $M_{\alpha}\; {\rm and}\; I_{\alpha}\; {\rm from} \; M_{p,{\varphi}^{\frac {1}{p}}}\; \rm{to}\; M_{q,{\varphi}^{\frac {1}{q}}}\;\rm {for 1 < p < q < \infty \; and\; from\; M_{1,{\varphi}}\; to \;W\;M_{q,{\varphi}^{\frac{1}{p}}} \; for \; 1 < q < \infty.}$ As applications of those results, the boundeness of the commutators of operators $I_{\alpha} and I_{\alpha}$ on generalized Morrey spaces is also obtained. In the case $b \in BMO{\mathbb{(R)}^{n}}\; \rm and \;1 < p < q < \infty,$ we find the sufficient conditions on the pair $(\varphi_{1},\varphi_{2})$ which ensures the boundedness of the operators ${M_{b,\alpha}}\; \rm {and \;[b,I_{\alpha}] \; from \; M_{p,\varphi_{1}}\; to \; M_{q,\varphi_{2}}\; with\; 1/p - 1/q = \alpha/n.}$ We also find the sufficient conditions on $\varphi$ which ensures the boundedness of the operators ${M_{b,\alpha}}\; \rm {and \;[b,I_{\alpha}] \; from \; M_{p,{\varphi^{\frac{1}{p}}}}\; to \; M_{q,\varphi^{\frac{1}{p}}}\; for\; 1 < p < q < \infty.}$ In all cases conditions for the boundedness are given in terms of Zygmund-type integral inequalities on $\rm {(\varphi_{1},\varphi_{2}) \;and \;\varphi} ,$which do not assume any assumption on monotonicity of $\rm {\varphi_{1},\varphi_{2} \;and \;\varphi} \;\rm{in\; r} ,$ As applications, we get some estimates for Marcinkiewicz operator and fractional powers of the some analytic semigroups on generalized Morrey spaces.

Mathematics Subject Classification (2010). Primary 42B20, 42B25, 42B35.