Sharp Hardy inequalities for the Vladimirov--Taibleson operator on ℚpn
Abstract
We prove sharp Hardy inequalities on ℚpn for the quadratic form of the Vladimirov--Taibleson operator, the Lq norm of its fractional derivative, and a nonlinear difference energy. Under the self-dual Fourier normalization, the sharp quadratic-form constant for 0<α<n is [(1−p−(n−α)/2)/(1−p−(n+α)/2)]2, with a remainder given by the ground state representation. For 1<q<∞, q'=q/(q−1), and 0<s<n/q, a weighted Schur estimate determines the exact Lq norm of the weighted Riesz potential |x|p−sIs, giving the sharp linear Hardy constant [Γp(n)(n/q'+s)/Γp(n)(n/q')]q. The two-point Picone inequality and shell decomposition give the sharp constant 2(1−p−n)∑m≥1p−sqm(1−p−(n/q−s)m)q for the Gagliardo-type energy with kernel |x−y|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 Lq 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.
Related
- Weyl's law and Pólya's conjecture for the Vladimirov-Taibleson operator
- Extension Problems for the Vladimirov--Taibleson Operator and the Hierarchical Laplacian
- Sharp Weighted Hardy-Type Inequalities for Homogeneous Magnetic Potentials
- Optimal Hardy Inequalities for Fractional p-Laplacians on the Integers
- On improved Hardy-type inequalities in Log-Weighted Sobolev Spaces
BibTeX
@misc{sun2026sharp,
title = {{Sharp Hardy inequalities for the Vladimirov--Taibleson operator on $\mathbbQ_p^n$}},
author = {Yaojia Sun},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.04463},
eprint = {2610.04463},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.04463},
}
RIS
TY - UNPB
TI - Sharp Hardy inequalities for the Vladimirov--Taibleson operator on ℚ_p^n
AU - Yaojia Sun
PY - 2026
DA - 2026/10/03
PB - arXiv
DO - 10.48550/arxiv.2610.04463
UR - https://arxiv.org/abs/2610.04463
AB - We prove sharp Hardy inequalities on 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<α<n is [(1−p^−(n−α)/2)/(1−p^−(n+α)/2)]^2, with a remainder given by the ground state representation. For 1<q<∞, q'=q/(q−1), and 0<s<n/q, a weighted Schur estimate determines the exact L^q norm of the weighted Riesz potential |x|_p^−sI_s, giving the sharp linear Hardy constant [Γ_p^(n)(n/q'+s)/Γ_p^(n)(n/q')]^q. The two-point Picone inequality and shell decomposition give the sharp constant 2(1−p^−n)∑_m≥1p^−sqm(1−p^−(n/q−s)m)^q for the Gagliardo-type energy with kernel |x−y|_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.
ER -