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
Tianyang He, Zhiwen Liu, Ting Yu
TL;DR
This paper proves sharp norm bounds for $p$-adic averaging operators on weighted function spaces. It first derives a sharp weak-type bound for the $p$-adic fractional Hardy operator $H_\alpha$ from $L^{p_1}(\mathbb{Q}_p^n,|x|_p^{\beta})$ to $L^{q,\infty}(\mathbb{Q}_p^n,|x|_p^{\gamma})$ under the scale condition $(\gamma+n)/q+\alpha=(\beta+n)/p_1$ and provides an exact constant, with a endpoint $L^1 \to L^{(n+\gamma)/(n-\alpha),\infty}$ bound. It then studies the sharp bounds for the $m$-linear $n$-dimensional Hardy and Hilbert operators with kernels on $p$-adic weighted spaces $H_{\alpha}^{\infty}(\mathbb{Q}_p^n)$ by introducing a general homogeneous kernel operator $T^p$ and obtaining its sharp norm $C^p$, including explicit results for Hardy-type $T_1^p$ and Hilbert-type $T_2^p$ as corollaries. The work further extends to a sharp bound for a $p$-adic integral operator with a kernel and to a sharp bound for the $p$-adic Hausdorff operator on weighted spaces, with norms expressed in terms of kernel integrals $C^p$ and $C_{\Phi}^{p}$, respectively. Overall, the results furnish precise mapping properties and embeddings in the $p$-adic harmonic analysis setting, generalizing several classical real-analytic bounds to the $p$-adic context.
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 $H_α^{\infty}( \mathbb{Q} _{p}^{n} )$. 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.