Abstract and Applied Analysis
Volume 2 (1997), Issue 1-2, Pages 121-136
doi:10.1155/S1085337597000304

Regularized functional calculi, semigroups, and cosine functions for pseudodifferential operators

Ralph Delaubenfels1 and Yansong Lei2

1Scientia Research Institute, P.O. Box 988, Athens 45701, Ohio, USA
2Mathematics Department, Ohio University, Athens 45701, Ohio, USA

Received 27 May 1996

Copyright © 1997 Ralph Delaubenfels and Yansong Lei. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

Let iAj(1≤j≤n) be generators of commuting bounded strongly continuous groups, A≡(A1,A2,…,An). 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 BCk(f(Rn)) functional calculus. This immediately produces regularized semigroups and cosine functions with an explicit representation; in particular, when f(Rn)⫅R, then, for appropriate k,r, t↦(1−it)−ke−itf(A)(1+|A|2)−r is a Fourier-Stieltjes transform, and when f(Rn)⫅[0,∞), then t↦(1+t)−ke−tf(A)(1+|A|2)−r is a Laplace-Stieltjes transform. With A≡i(D1,…,Dn),f(A) is a pseudodifferential operator on Lp(Rn)(1≤p<∞) or BUC(Rn).