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.
Keywords
Mathematics Subject Classification (2010). Primary 42B20, 42B25, 42B35.
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Dedicated to the 70th birthday of Prof. S. Samko
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Guliyev, V.S., Shukurov, P.S. (2013). On the Boundedness of the Fractional Maximal Operator, Riesz Potential and Their Commutators in Generalized Morrey Spaces. In: Almeida, A., Castro, L., Speck, FO. (eds) Advances in Harmonic Analysis and Operator Theory. Operator Theory: Advances and Applications, vol 229. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-0516-2_10
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