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

Conference paper
Part of the Operator Theory: Advances and Applications book series (OT, volume 229)

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

Fractional maximal operator Riesz potential operator generalized Morrey space commutator BMO space. 

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Copyright information

© Springer Basel 2013

Authors and Affiliations

  1. 1.Department of MathematicsAhi Evran UniversityKirsehirTurkey
  2. 2.Institute of Mathematics and MechanicsAcademy of Sciences of AzerbaijanBakuAzerbaijan

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