Abstract and Applied Analysis
Volume 2 (1997), Issue 1-2, Pages 121-136
Regularized functional calculi, semigroups, and cosine functions
for pseudodifferential operators
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.
Let be generators of commuting bounded strongly continuous groups, . We show that, when has sufficiently many polynomially bounded derivatives, then there
exist such that has a -regularized functional calculus. This immediately produces regularized semigroups and cosine functions with an explicit representation; in particular, when , then, for appropriate , is a Fourier-Stieltjes transform, and when , then is a Laplace-Stieltjes transform. With is a pseudodifferential operator on or .