Cauchy integrals, Calderón projectors, and Toeplitz operators on uniformly rectifiable domains
type: Item
instance of: scholarly article
title: Cauchy integrals, Calderón projectors, and Toeplitz operators on uniformly rectifiable domains
zbMATH Open document ID: 1305.31003
author: Irina Mitrea
author: Marius Mitrea
author: Michael Taylor
published in: Advances in Mathematics
publication date: 2014-11-17T00:00:00Z
review text: The paper deals with first-order elliptic systems of differential operators \(D\) on bounded domains \(\Omega\) in a Riemannian manifold \(M\). The boundaries \(\partial\Omega\) are permitted to be rough. Theorem 2.2.1 gives a reproducing formula of Cauchy-Pompeiu type: \[ u(x) = i\int_{\partial\Omega} E(x,y)\sigma_{\mathcal{D}}(y,\nu(y))u(y)\,d\sigma(y) + \int_{\Omega} E(x,y)\mathcal{D}u(y)\,dV(y), \] where \(\Omega\) is Ahlfors regular, \(\mathcal{D}u\) is integrable, and \(u\) has nontangential boundary limit and \(L^p\)-bounded maximal function for some \(p>1\). Here \(\mathcal{D}\) is a \(2\times 2\)-system with Fredholm index zero. The off-diagonal elements are \(D^*\) and \(D\), and the diagonal elements are suitable multiplication operators. It is assumed that the unique continuation property (UCP) holds for \(D\) and \(D^*\). Then \(\mathcal{D}\) is invertible, and \(E\) denotes the integral kernel of the inverse. Moreover \(\sigma_{\mathcal{D}}\) denotes the principal symbol of \(\mathcal{D}\), and \(\nu\) is the outward pointing unit normal. From the reproducing formula the Calderón projector \(\mathcal{P}_{\mathcal{D}}\) is obtained, which is a projection operator onto the Hardy space of boundary traces of solutions of \(\mathcal{D}u=0\). The Calderón projector is shown to be \(L^p(\partial\Omega)\)-bounded for \(1<p<\infty\). If \(\Omega\) is a uniformly rectifiable (UR) domain, and if some additional assumptions hold, then the orthogonal projector onto the Hardy space, the Szegő projector, is an \(L^p(\partial\Omega)\)-bounded operator \(S_{\mathcal{D}}=\mathcal{P}_{\mathcal{D}}(I+A)^{-1}\). A Toeplitz operator associated with \(D\) is the \(L^p\)-bounded operator \[ \mathfrak{T}_{\mathcal{D}}:=\mathcal{P}_{\mathcal{D}}\Phi\mathcal{P}_{\mathcal{D}}+(I-\mathcal{P}_{\mathcal{D}}), \] where the coefficient \(\Phi\) is multiplication by a map, also denoted \(\Phi\), from \(\partial\Omega\) into the endomorphisms of \(\mathbb{C}^\ell\). If \(\Phi\) is an \(…
reviewed by: Sönke Hansen
Mathematics Subject Classification ID: 31B10
Mathematics Subject Classification ID: 35J46
Mathematics Subject Classification ID: 45B05
Mathematics Subject Classification ID: 45E05
Mathematics Subject Classification ID: 49Q15
Mathematics Subject Classification ID: 47B35
zbMATH DE Number: 6370015
zbMATH Keywords: Cauchy integral
zbMATH Keywords: Calderón projector
zbMATH Keywords: Toeplitz operator
zbMATH Keywords: uniformly rectifiable domain
MaRDI profile type: Publication
full work available at URL: j.aim.2014.09.020
OpenAlex ID: W2044294206
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DOI: 10.1016/J.AIM.2014.09.020
description: scientific article; zbMATH DE number 6370015