Sharp bounds for the $\boldsymbol{p}$-adic $\boldsymbol{n}$-dimensional fractional Hardy operator and a class of integral operators on $\boldsymbol{p}$-adic function spaces

Abstract

In this paper, we first study the sharp weak estimate for the $p$-adic $n$-dimensional fractional Hardy operator from $L^p$ to $L^{q,\infty}$. Secondly, we study the sharp bounds for the $m$-linear $n$-dimensional $p$-adic integral operator with a kernel on $p$-adic weighted spaces $H_{\alpha}^{\infty}( \mathbb{Q} _{p}^{n} )$. As an application, the sharp bounds for $p$-adic Hardy and Hilbert operators on $p$-adic weighted spaces are obtained. Finally, we also find the sharp bound for the Hausdorff operator on $p$-adic weighted spaces, which generalizes the previous results.