Regularized functional calculi, semigroups, and cosine functions for pseudodifferential operators
type: Item
instance of: scholarly article
title: Regularized functional calculi, semigroups, and cosine functions for pseudodifferential operators
zbMATH Open document ID: 0937.47015
DOI: 10.1155/S1085337597000304
author: Yansong Lei
author: R. deLaubenfels
published in: Abstract and Applied Analysis
publication date: 2000-06-13T00:00:00Z
full work available at URL: https://eudml.org/doc/48413
full work available at URL: issue-1-2.html
review text: Summary: Let \(iA_j\) \((1\leq j\leq n)\) be generators of commuting bounded strongly continuous groups, \(A\equiv (A_1,A_2,\dots, A_n)\). We show that, when \(f\) has sufficiently many polynomially bounded derivatives, then there exist \(k\), \(r>0\) such that \(f(A)\) has a \((1+|A|^2)^{-r}\)-regularized \(BC^k(f(\mathbb{R}^n))\) functional calculus. This immediately produces regularized semigroups and cosine functions with an explicit representation; in particular, when \(f(\mathbb{R}^n)\subseteq \mathbb{R}\), then, for appropriate \(k,r,t\mapsto(1- it)^{-k} e^{-itf(A)}(1+|A|^2)^{-r}\) is a Fourier-Stieltjes transform, and when \(f(\mathbb{R}^n)\subseteq [0,\infty)\), then \(t\mapsto(1+ t)^{-k}e^{-tf(A)}(1+|A|^2)^{-r}\) is a Laplace-Stieltjes transform. With \(A\equiv i(D_1,\dots, D_n)\), \(f(A)\) is a pseudo-differential operator on \(L^p(\mathbb{R}^n)\) \((1\leq p<\infty)\) or \(BUC(\mathbb{R}^n)\).
Mathematics Subject Classification ID: 47A60
Mathematics Subject Classification ID: 47D03
Mathematics Subject Classification ID: 47D06
Mathematics Subject Classification ID: 47D09
Mathematics Subject Classification ID: 47F05
zbMATH DE Number: 1370674
zbMATH Keywords: regularized functional calculus
zbMATH Keywords: generators
zbMATH Keywords: bounded strongly continuous groups
zbMATH Keywords: polynomially bounded derivatives
zbMATH Keywords: cosine functions
zbMATH Keywords: Fourier-Stieltjes transform
zbMATH Keywords: Laplace-Stieltjes transform
zbMATH Keywords: pseudo-differential operator
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 1370674