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-P\'olya inequality on $\mathfrak{B}_{r,\alpha}(\mathbb{Q}_p)$.