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 .
"Regularized functional calculi, semigroups, and cosine functions for pseudodifferential operators." Abstr. Appl. Anal. 2 (1-2) 121 - 136, 1997. https://doi.org/10.1155/S1085337597000304