Sharp Hardy inequalities for the Vladimirov--Taibleson operator on ${\mathbb{Q}}_p^n$

Abstract

We prove sharp Hardy inequalities on ${\mathbb{Q}}_p^n$ for the quadratic form of the Vladimirov--Taibleson operator, the $L^q$ norm of its fractional derivative, and a nonlinear difference energy. Under the self-dual Fourier normalization, the sharp quadratic-form constant for $0<\alpha <n$ is $[(1-p^{-(n-\alpha )/2})/(1-p^{-(n+\alpha )/2}){]}^2$, with a remainder given by the ground state representation. For $1<q<\infty$, $q^{\prime }=q/(q-1)$, and $0<s<n/q$, a weighted Schur estimate determines the exact $L^q$ norm of the weighted Riesz potential $\Vert \mathbf{x}{\Vert }_p^{-s}{I}_s$, giving the sharp linear Hardy constant $[{\Gamma }_p^{(n)}(n/q^{\prime }+s)/{\Gamma }_p^{(n)}(n/q^{\prime }){]}^q$. The two-point Picone inequality and shell decomposition give the sharp constant $2(1-p^{-n}){\sum }_{m\ge 1}{p}^{-sqm}(1-p^{-(n/q-s)m}{)}^q$ for the Gagliardo-type energy with kernel $\Vert \mathbf{x}-\mathbf{y}{\Vert }_p^{-n-sq}$. Annular truncations of the critical powers give explicit minimizing sequences with error estimates. Equality is attained only by the zero function in each stated class. For $q=2$, both $L^q$ inequalities reduce to the quadratic-form inequality. Uniform local constancy makes the difference energy finite for every Bruhat--Schwartz function and every $s>0$. The Hardy inequalities on the full test-function space therefore hold precisely when $sq<n$, without the restriction $s<1$.