A small example of non-local operators having no transmission property.
type: Item
instance of: scholarly article
title: A small example of non-local operators having no transmission property.
zbMATH Open document ID: 1035.47026
author: Yasushi Ishikawa
published in: Tsukuba Journal of Mathematics
publication date: 2002-10-31T00:00:00Z
review text: The transmission property of a linear operator \(P\) on \(\mathbb{R}^d\) with respect to the boundary \(\mathbb{R}^{d-1}\) means that the transformation \[ {\mathcal C}^\infty_0 (\overline{\mathbb{R}^d_+})\partial u \mapsto(Pu) |_{\mathbb{R}^d} \in{\mathcal C}^\infty (\mathbb{R}^d_+),\;u:=\overline u \text{ on }\mathbb{R}^d_+,\;\overline u:=0\text{ on }\mathbb{R}^d_-, \] is continuous. The author shows that the pseudodifferential operator \(\sigma(n,D)\) having the symbol \(\sigma(n,\gamma)\), which is the composition of a general continuous negative definite function \(\psi(x,\gamma)\) of quadratic type with a complete Bernstein function \(f(\xi)\), does not have the transmission property. This result should be interesting since the generator of a Feller semigroup is a pseudodifferential operator \(\sigma (x,D)\) for which the symbol \(\sigma(x,\xi)\) is a general negative definite symbol.
Mathematics Subject Classification ID: 47G30
Mathematics Subject Classification ID: 35S05
Mathematics Subject Classification ID: 47D07
zbMATH DE Number: 1822094
zbMATH Keywords: pseudodifferential operator
zbMATH Keywords: transmission property
zbMATH Keywords: Feller semigroup
zbMATH Keywords: symbol
reviewed by: Viorel Iftimie
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.21099/tkbjm/1496164296
OpenAlex ID: W633505159
DOI: 10.21099/TKBJM/1496164296
description: scientific article; zbMATH DE number 1822094