Weighted Sobolev spaces on curves
type: Item
instance of: scholarly article
title: Weighted Sobolev spaces on curves
zbMATH Open document ID: 1019.46026
author: Venancio Álvarez
author: D. Pestana
author: E. Romera
author: José M. Rodríguez
published in: Journal of Approximation Theory
publication date: 2003-04-02T00:00:00Z
full work available at URL: http://hdl.handle.net/10016/6486
review text: The aim of this paper is to define weighted Sobolev spaces \(W^{k,p}(\gamma,\mu)\) on a simple curve that has a locally absolutely continuous parametrization, \(\gamma\colon I\to\mathbb{C}\). The authors need to introduce the classes \(\text{AC}^{(k)}(\gamma), {\text{AC}}^{(k)}_{\text{loc}}(\gamma)\) of \(k\) times differentiable functions on \(\gamma\) in order to define the derivatives along \(\gamma\). If \(1\leq p\leq\infty\), for every arc \([z_1,z_2]\subset\gamma\), by means of a Muckenhoupt-type weighted norm inequality they also define the class \(B_p([z_1,z_2])\) of weights \(w\), which contains the classical \(A_p\) class when \(\gamma\) assumes values in \({\mathbb R}\). Next, if \(\mu=(\mu_0,\mu_1,\dots,\mu_k)\) is a given vector measure on \(\gamma\), they denote by \(\Omega_j\) the set of those points in \(\gamma\) with a neighborhood \(V\subset\gamma\) such that \(w_j\in B_p(V)\). Then they introduce the completion of \(\mu\) and the set \(\Omega^{(j)}\) of \(j\)-regular points \(z_0\in\gamma\), \(1\leq j\leq k\). These sets are used in the definition of a \(p\)-admissible vector measure \(\mu\), and the corresponding weighted Sobolev space \(W^{k,p}(\gamma,\mu)\) contains all the equivalence classes of functions \(f\colon\gamma\to\mathbb{C}\) such that \(f^{(j)}\in{\text{AC}}^{(1)}_{\text{loc}}(\Omega^{(j)})\) for \(0\leq j<k\) and \(\|f^{(j)}\|_{L^p(\gamma,\mu_j)}<\infty\) for \(0\leq j\leq k\), with respect to the seminorm \(\|f\|_{W^{k,p}(\gamma,\mu)}:=\sum^k_{j=0}\|f^{(j)}\|_{L^p(\gamma,\mu_j)}\). With these definitions, the authors find conditions under which these new spaces are complete, they study the density of the polynomials and the problem of the boundedness of the multiplication operator \(f\mapsto zf\).
Mathematics Subject Classification ID: 46E35
Mathematics Subject Classification ID: 42B35
zbMATH DE Number: 1891506
zbMATH Keywords: Sobolev space on a curve
zbMATH Keywords: density of polynomials
zbMATH Keywords: multiplication operator
reviewed by: Joan Cerdà
MaRDI profile type: Publication
OpenAlex ID: W2051516705
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DOI: 10.1006/JATH.2002.3709
description: scientific article; zbMATH DE number 1891506