Hardy-Sobolev-type inequalities with monomial weights
type: Item
instance of: scholarly article
title: Hardy-Sobolev-type inequalities with monomial weights
zbMATH Open document ID: 1370.26033
author: Hernán Castro
published in: Annali di Matematica Pura ed Applicata. Serie Quarta
publication date: 2017-04-13T00:00:00Z
review text: In this paper, the author gives some conditions on \(A,B \in \mathbb{R}^N\) and \(p \geq 1\) for the existence of \(p^*\) and a constant \(C>0\) such that the Hardy-Sobolev-type inequality \[ \Big(\int_{\mathbb{R}^N}| x^B u(x)|^{p^{*}}dx\Big)^{\frac{1}{p^*}} \leq \Big(\int_{\mathbb{R}^N}| x^A \nabla u(x)|^{p}dx\Big)^{\frac{1}{p}} \] holds for all \(u\in C^{\infty}_{C} (\mathbb{R}^N)\) and \(N \geq 1\). In addition, the author gives an alternative technique to prove the above Hardy-Sobolev-type inequality with monomial weight using the elementary calculus tools instead of the Alexandroff-Bakelman-Pucci (ABP) method, the theory of Muckhenhoupt weights or other general techniques in the literature on weighted Sobolev inequalities. Furthermore, it is shown that this elementary method in comparison with the ABP method works not only for domains of the form \(\mathbb{R}^{N}_{A}\), that is, a domain of the form \[ \mathbb{R}^{N}_{I,+}=\{(x_1, x_2,\dots,x_N)\in \mathbb{R}^{N}:x_i \geq 0 \text{ for } i\in I\}, \] or \[ \mathbb{R}^{N}_{I,-}=\{(x_1, x_2,\dots,x_N)\in \mathbb{R}^{N}:x_i \leq 0 \text{ for } i\in I\}, \] where \(I\) is any subset of \(\{1,2,\dots,N\}\). Finally, a weighted trace inequality related to the fractional Laplacian was derived and proved.
reviewed by: James A. Oguntuase
Mathematics Subject Classification ID: 26D10
Mathematics Subject Classification ID: 35A25
Mathematics Subject Classification ID: 46E35
zbMATH DE Number: 6705698
zbMATH Keywords: weighted Sobolev inequality
zbMATH Keywords: Caffarelli-Kohn-Nirenberg inequality
zbMATH Keywords: Hardy inequality
zbMATH Keywords: monomial weights
MaRDI profile type: Publication
full work available at URL: s10231-016-0587-2
OpenAlex ID: W2466800396
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DOI: 10.1007/S10231-016-0587-2
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description: scientific article; zbMATH DE number 6705698