Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Best constants for Hardy inequalities in Triebel--Lizorkin spaces

Michał Kijaczko

TL;DR

Addresses sharp constants in fractional Hardy inequalities for weighted Triebel–Lizorkin seminorms on R^d and on half-spaces. The authors formulate and prove sharp constants C^{p,q}_{d,s,α,β} and D^{p,q}_{d,s,α,β} for general p≥q and 0<s<1, introducing a global angular kernel Φ and a detailed reduction to 1D for sharpness. The proofs rely on polar coordinates, Hölder and Minkowski inequalities, and careful finiteness lemmas, avoiding ground-state representations even for p≠q. They extend known unweighted results and prior weighted Gagliardo-type inequalities, and establish sharpness via explicit test functions; the p<q case remains open. Overall, the results deepen the understanding of nonlocal Hardy-type inequalities in weighted Triebel–Lizorkin spaces with potential applications to nonlocal PDEs and stochastic processes.

Abstract

We find sharp constants in fractional Hardy inequalities for weighted Triebel--Lizorkin seminorms on the whole space and half-spaces. Our results generalize recently obtained weighted fractional Hardy inequalities for Gagliardo seminorms, but are new even for the unweighted case.

Best constants for Hardy inequalities in Triebel--Lizorkin spaces

TL;DR

Addresses sharp constants in fractional Hardy inequalities for weighted Triebel–Lizorkin seminorms on R^d and on half-spaces. The authors formulate and prove sharp constants C^{p,q}_{d,s,α,β} and D^{p,q}_{d,s,α,β} for general p≥q and 0<s<1, introducing a global angular kernel Φ and a detailed reduction to 1D for sharpness. The proofs rely on polar coordinates, Hölder and Minkowski inequalities, and careful finiteness lemmas, avoiding ground-state representations even for p≠q. They extend known unweighted results and prior weighted Gagliardo-type inequalities, and establish sharpness via explicit test functions; the p<q case remains open. Overall, the results deepen the understanding of nonlocal Hardy-type inequalities in weighted Triebel–Lizorkin spaces with potential applications to nonlocal PDEs and stochastic processes.

Abstract

We find sharp constants in fractional Hardy inequalities for weighted Triebel--Lizorkin seminorms on the whole space and half-spaces. Our results generalize recently obtained weighted fractional Hardy inequalities for Gagliardo seminorms, but are new even for the unweighted case.

Paper Structure

This paper contains 4 sections, 3 theorems, 75 equations.

Table of Contents

  1. Introduction
  2. Hardy inequality with power weights on $\mathbb{R}^d$
  3. Hardy inequality on a half-space
  4. Proofs

Key Result

Theorem 1

Let $0<s<1$, $p\geq q$ and $d\geq 1$. Assume that the parameters $\alpha,\beta$ satisfy Let $u\in C_c^1(\mathbb{R}^d)$ if $sq-\alpha-\tfrac{q\beta}{p}<d$ and $u\in C_c^1(\mathbb{R}^d\setminus\{0\})$ if $sq-\alpha-\tfrac{q\beta}{p}>d$. Then, the following Hardy inequality holds, The constant $\mathcal{C}^{p,q}_{d,s,\alpha,\beta}$ is sharp and equals where

Theorems & Definitions (6)

  • Theorem 1
  • Theorem 2
  • Lemma 3
  • proof
  • proof : Proof of Theorem \ref{['thm1']}
  • proof : Proof of Theorem \ref{['thm2']}