Sharpness transitions for weighted Triebel--Lizorkin Hardy inequalities with p < q
Abstract
For fractional difference energies with two power weights and distinct inner and outer exponents, we determine regimes in which the constant generated by the homogeneous power is optimal and regimes in which it loses optimality. In the range 1<p<q, we prove sharp Hardy inequalities on nontrivial weight intervals for every sign of D=d+2β−sp, including a two-sided interval in the balanced case D=0. For q≥2, these intervals extend strictly beyond an elementary pointwise scalar criterion. For 1<p<q<2, we determine the optimal power exponent in the scalar supporting-line inequality and obtain a larger explicit sharpness interval. The proof pairs opposite logarithmic displacements and controls the nonlinear deficit by a quadratic reference energy of the transformed function. Its translation estimate allows a positive contribution at bounded scales to absorb negative contributions at remote scales. Concentrating tests prove strict loss of homogeneous sharpness near both finite-energy endpoints, and angular variations give concrete bounds for the first sharpness threshold. We extend the method to flat singular sets and regional half-space energies, retaining the distinction between ambient finite-energy conditions and finiteness of the homogeneous candidate. We also treat the complementary order p>q, including flat-subspace inequalities and a complete alternative proof for point singularities. All sharpness sequences belong to the specified smooth test classes, and every positive optimal constant extends to their completion in the difference norm.
Related
- Preduals of weighted homogeneous Bourgain-Morrey Besov-Triebel-Lizorkin spaces associated with operators
- Besov and Triebel--Lizorkin Spaces on Filtered Lie Groups, II: Pointwise Multiplication, Localization, Differences
- Triebel - Lizorkin spaces of dominating mixed smoothness are algebras with respect to pointwise multiplication
- On Besov and Triebel-Lizorkin spaces associated with the Grushin operator
- Besov and Triebel--Lizorkin Spaces on Filtered Lie Groups, III: the Spaces F∞,qα
BibTeX
@misc{jin2026sharpness,
title = {{Sharpness transitions for weighted Triebel--Lizorkin Hardy inequalities with p < q}},
author = {Yongyang Jin and Li Tang},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.32671},
eprint = {2610.32671},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.32671},
}
RIS
TY - UNPB
TI - Sharpness transitions for weighted Triebel--Lizorkin Hardy inequalities with p < q
AU - Yongyang Jin
AU - Li Tang
PY - 2026
DA - 2026/10/23
PB - arXiv
DO - 10.48550/arxiv.2610.32671
UR - https://arxiv.org/abs/2610.32671
AB - For fractional difference energies with two power weights and distinct inner and outer exponents, we determine regimes in which the constant generated by the homogeneous power is optimal and regimes in which it loses optimality. In the range 1<p<q, we prove sharp Hardy inequalities on nontrivial weight intervals for every sign of D=d+2β−sp, including a two-sided interval in the balanced case D=0. For q≥2, these intervals extend strictly beyond an elementary pointwise scalar criterion. For 1<p<q<2, we determine the optimal power exponent in the scalar supporting-line inequality and obtain a larger explicit sharpness interval. The proof pairs opposite logarithmic displacements and controls the nonlinear deficit by a quadratic reference energy of the transformed function. Its translation estimate allows a positive contribution at bounded scales to absorb negative contributions at remote scales. Concentrating tests prove strict loss of homogeneous sharpness near both finite-energy endpoints, and angular variations give concrete bounds for the first sharpness threshold. We extend the method to flat singular sets and regional half-space energies, retaining the distinction between ambient finite-energy conditions and finiteness of the homogeneous candidate. We also treat the complementary order p>q, including flat-subspace inequalities and a complete alternative proof for point singularities. All sharpness sequences belong to the specified smooth test classes, and every positive optimal constant extends to their completion in the difference norm.
ER -