Singular integral operators acting on Orlicz--Morrey spaces of the first kind

Abstract

In 2004, Nakai defined generalized Orlicz--Morrey spaces of the first kind. This is an extension of the 1994 paper by Nakai, where Nakai showed that singular integral operators are bounded from generalized Morrey spaces to weak generalized Morrey spaces. Since generalized Orlicz--Morrey spaces are shown to be embedded into generalized Morrey spaces, it makes sense to restrict them from generalized Morrey spaces to generalized Orlicz--Morrey spaces. Via the weak boundedness, the boundedness property of singular integral operators acting on generalized Orlicz--Morrey spaces of the first kind is investigated in this paper. As a byproduct, the vector-valued boundedness of the Hardy--Littlewood maximal operator is obtained for generalized Orlicz--Morrey spaces of the first kind. Our results contain the weak type boundedness of operators as well as the one of strong type. The strong type boundedness is covered by the work of Nakai in 2006, while the contributions in this paper are the weak type boundedness and the vector-valued extension of the boundedness of the Hardy--Littlewood maximal operator.