Boundedness of $p$-adic Hardy--Hilbert and Erdélyi--Kober fractional integral operators on $p$-adic Cesàro function Spaces

Abstract

In this paper, we introduce Cesàro function spaces over $p$-adic fields and investigate their fundamental properties, such as the dilation operator and the Minkowski-type integral inequality. We establish boundedness result for $p$-adic Hardy--Hilbert-type integral operators acting on $p$-adic Cesàro function spaces, and as an application we derive $p$-adic analogue of the Hardy inequality, the Hilbert inequality, and the Hardy-Littlewood-Pólya inequality. Furthermore, we define the $p$-adic analogue of the Erdélyi--Kober fractional integral operators and prove their boundedness on $p$-adic Cesàro function spaces with the help of the obtained boundedness result.