Composition operators in weighted Sobolev spaces on Carnot groups
type: Item
instance of: scholarly article
title: Composition operators in weighted Sobolev spaces on Carnot groups
zbMATH Open document ID: 1249.46035
published in: Acta Mathematica Hungarica
publication date: 2011-12-20T00:00:00Z
review text: Let \(\Omega\) be an open set in the Carnot group \({\mathbb G}\). Let \(w\) be a Muckenhoupt weight. Let \(L^1_q (\Omega, w)\) be the weighted homogeneous Sobolev space and let \(W^1_q (\Omega, w)\) be the usual weighted Sobolev space for any \(1<q<\infty\). Let \(\varphi^*: f \mapsto f \circ \varphi\) be a composition operator. The first main result of the paper characterizes which \(\varphi\) generate bounded composition operators \[ \varphi^*: \quad L^1_p (\Omega', v) \hookrightarrow L^1_q (\Omega, w), \qquad 1<q\leq p < \infty. \] The second main result studies under which circumstances the embedding \[ \text{id}: \quad W^1_p (\Omega', v) \hookrightarrow L_s (\Omega', v) \] is bounded or, in addition, compact.
reviewed by: Hans Triebel
Mathematics Subject Classification ID: 46E35
Mathematics Subject Classification ID: 58J05
zbMATH DE Number: 5990916
zbMATH Keywords: weighted Sobolev spaces
zbMATH Keywords: composition operator
zbMATH Keywords: Carnot group
author: A. Ukhlov
MaRDI profile type: Publication
full work available at URL: s10474-011-0104-4
OpenAlex ID: W2007126266
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DOI: 10.1007/S10474-011-0104-4
description: scientific article; zbMATH DE number 5990916