Adams' inequality and limiting Sobolev embeddings into Zygmund spaces
type: Item
instance of: scholarly article
title: Adams' inequality and limiting Sobolev embeddings into Zygmund spaces
zbMATH Open document ID: 1279.46025
published in: Potential Analysis
publication date: 2013-01-14T00:00:00Z
review text: The author exhibits sharp embedding constants for Sobolev spaces of any order into Zygmund spaces, obtained as the product of sharp embedding constants for second-order Sobolev spaces into Lorentz spaces. One of the excellent results reads as follows. Let \(n> 2\), \(q> 1\), and \(\Omega\) be a bounded domain in \(\mathbb{R}^n\). Then, the following embedding holds: \[ \begin{gathered} W^2 L^{n/2,q}(\Omega)\cap W^1_0 L^{n/2,q}(\Omega)\to Z^{q-1/q}(\Omega),\text{ namely,}\\ \| u\|_{Z^{q-1/q}}\leq \gamma_{n,2}\|\Delta u\|_{n/2, q}\end{gathered}\tag{1} \] for any \(u\in W^2 L^{n/2,q}(\Omega)\cap W^1_0 L^{n/2,q}(\Omega)\), where \(\gamma_{n,2}= \omega^{{-2\over n}}_n[n(n-2)]^{-1}\) and \(\omega_n= \pi^{{n\over 2}}/\Gamma({n\over 2}+ 1)\) is the volume of the unit ball in \(\mathbb{R}^n\). Moreover, the constant appearing in equation (1) is sharp. As an easy consequence of this result, the author then gives a new proof of Adams' inequality, which holds under the weaker hypothesis of homogeneous Navier boundary conditions.
Mathematics Subject Classification ID: 46E35
Mathematics Subject Classification ID: 35A23
zbMATH DE Number: 6125021
zbMATH Keywords: limiting Sobolev embeddings
zbMATH Keywords: Trudinger-Moser inequalities
zbMATH Keywords: Adams' inequality
zbMATH Keywords: best constants
zbMATH Keywords: Zygmund spaces
zbMATH Keywords: Lorentz spaces
author: Cristina Tarsi
reviewed by: John Schmeelk
MaRDI profile type: Publication
full work available at URL: s11118-011-9259-4
OpenAlex ID: W2025334692
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DOI: 10.1007/S11118-011-9259-4
description: scientific article; zbMATH DE number 6125021