On the Boundedness of Some Hardy-type Operators

Abstract

In this work, we contribute to the theory of Hardy-type operators in a number of ways on both Rn and Qn p. Firstly, we characterize the central BMO spaces with variable expo nent via the boundedness of commutators of Hardy-type operators on variable exponent Lebesgue and central Morrey spaces. Some boundedness results for the Hardy operator and its adjoint operator are also demonstrated on variable exponent Lebesgue and cen tral Morrey spaces. Furthermore, we obtaine the boundedness of variable-order fractional Hardy-type operators from grand Herz spaces to weighted spaces, subject to appropriate weight conditions. Secondly, in the framework of variable exponent, we introduce some new p-adic function spaces. The fractional p-adic Hardy-type operators on the p-adic Lebesgue and central Morrey spaces with variable exponents are shown to be bounded. We characterize some varaible p-adic function spaces by proving the boundedness of com mutators formed by p-adic Hardy-type integral operators and p-adic variable exponent λ-central BMO functions on the aforementioned spaces. Furthermore, the continuity of theses operators on p-adic variable exponent Herz-type spaces is discussed as well