Boundedness of the fractional maximal operator in generalized Morrey space on the Heisenberg group

WOS: 000318731900005In this paper we study the fractional maximal operator M (alpha) , 0 a parts per thousand currency sign alpha < Q on the Heisenberg group a"i (n) in the generalized Morrey spaces M (p, I center dot)(a"i (n) ), where Q = 2n + 2 is the homogeneous dimension of a"i (n) . We find the conditions on the pair (I center dot (1), I center dot (2)) which ensures the boundedness of the operator M (alpha) from one generalized Morrey space M (p, I center dot 1)(a"i (n) ) to another M (q, I center dot 2)(a"i (n) ), 1 < p < q < a, 1/p-1/q = alpha/Q, and from the space M (1, I center dot 1)(a"i (n) ) to the weak space WM (q, I center dot 2)(a"i (n) ), 1 < q < a, 1 - 1/q = alpha/Q. We also find conditions on the phi which ensure the Adams type boundedness of M (alpha) from to for 1 < p < q < a and from M (1, I center dot)(a"i (n) ) to for 1 < q < a. As applications we establish the boundedness of some Schrodinger type operators on generalized Morrey spaces related to certain nonnegative potentials V belonging to the reverse Holder class B (a)(" (n) ).Science Development Foundation under Republic of AzerbaijanScience Development Foundation (SDF) - Azerbaijan [EIF-2010-1(1)-40/06-1]; Scientific and Technological Research Council of Turkey (TUBITAK)Turkiye Bilimsel ve Teknolojik Arastirma Kurumu (TUBITAK) [110T695]; Ahi Evran UniversityAhi Evran University [PYO-FEN 4001.12.18]The research of V. Guliyev was partially supported by the grant of Science Development Foundation under the President of the Republic of Azerbaijan project EIF-2010-1(1)-40/06-1, by the Scientific and Technological Research Council of Turkey (TUBITAK Project No: 110T695) and by the grant of 2012-Ahi Evran University Scientific Research Projects (PYO-FEN 4001.12.18)