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On sharp constants in higher order Adams-Cianchi inequalities

Prasun Roychowdhury, Daniel Spector

Abstract

The main results of this paper are the establishment of sharp constants for several families of critical Sobolev embeddings. These inequalities were pioneered by David R. Adams, while the sharp constant in the first order case is due to Andrea Cianchi. We also prove a trace improvement of an inequality obtained independently by K. Hansson and H. Brezis and S. Wainger.

On sharp constants in higher order Adams-Cianchi inequalities

Abstract

The main results of this paper are the establishment of sharp constants for several families of critical Sobolev embeddings. These inequalities were pioneered by David R. Adams, while the sharp constant in the first order case is due to Andrea Cianchi. We also prove a trace improvement of an inequality obtained independently by K. Hansson and H. Brezis and S. Wainger.

Paper Structure

This paper contains 5 sections, 15 theorems, 147 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Proofs of the Main Results
  4. The Weak-Type Endpoint
  5. Trace Hansson-Brezis-Wainger

Key Result

Theorem 1.1

Let $\Omega$ be an open bounded subset of $\mathbb{R}^n$, $n\geq 2$. Let $k \in \mathbb{N}$, $1\leq k<n$, $1<q<\infty$, and set $q'=q/(q-1)$. Let $\nu$ be any positive Borel measure on $\Omega$ which satisfies bgc and condition-1 for some $d\in (0,n]$. Then there exists positive constant $C=C(n,k,\O for every $\beta \leq d n^{\frac{q'}{q}}\,\omega_n^{\frac{k}{n}q'}\,\sqrt{\ell^k_n}^{q'}$ and every

Theorems & Definitions (26)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Theorem 1.8: Trace Hansson-Brezis-Wainger
  • Corollary 1.9
  • Remark 1.10
  • Lemma 2.1
  • ...and 16 more