Boundedness of $p$-adic Hardy-Hilbert type integral operator on Block spaces

Abstract

In this paper, we estimate an operator norm of dilation operators on block spaces (${\mathfrak{B}}_{r,\alpha }({\mathbb{Q}}_p)$) over $p$-adic field. With this estimate, we establish the boundedness of $p$-adic Hardy-Hilbert type integral operator on ${\mathfrak{B}}_{r,\alpha }({\mathbb{Q}}_p)$. Moreover as application to our result, we obtain the $p$-adic Hilbert inequality, $p$-adic Hardy inequality and $p$-adic Hardy-Littlewood-Pólya inequality on ${\mathfrak{B}}_{r,\alpha }({\mathbb{Q}}_p)$.