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

Abstract

Let $i{A}_j\left(1\le j\le n\right)$ be generators of commuting bounded strongly continuous groups, $A\equiv \left(A_1,A_2,\dots ,A_n\right)$. We show that, when $f$ has sufficiently many polynomially bounded derivatives, then there exist $k,r>0$ such that $f\left(A\right)$ has a ${\left(1+{\left|A\right|}^2\right)}^{-r}$-regularized $B{C}^k\left(f\left({\mathbf{R}}^n\right)\right)$ functional calculus. This immediately produces regularized semigroups and cosine functions with an explicit representation; in particular, when $f\left({\mathbf{R}}^n\right)\subseteq \mathbf{R}$, then, for appropriate $k,r$, $t\mapsto {\left(1-it\right)}^{-k}{e}^{-itf\left(A\right)}{\left(1+{\left|A\right|}^2\right)}^{-r}$ is a Fourier-Stieltjes transform, and when $f\left({\mathbf{R}}^n\right)\subseteq \left[0,\infty \right)$, then $t\mapsto {\left(1+t\right)}^{-k}{e}^{-tf\left(A\right)}{\left(1+{\left|A\right|}^2\right)}^{-r}$ is a Laplace-Stieltjes transform. With $A\equiv i\left(D_1,\dots ,D_n\right),f\left(A\right)$ is a pseudodifferential operator on or $BUC\left({\mathbf{R}}^n\right)$.