## Abstract and Applied Analysis

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

#### Abstract

Let $iA_j(1\leq j\leq n)$ be generators of commuting bounded strongly continuous groups, $A\equiv(A_1,A_2,\ldots,A_n)$. 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 $BC^{k}(f(\mathbf{R}^n))$ functional calculus. This immediately produces regularized semigroups and cosine functions with an explicit representation; in particular, when $f(\mathbf{R}^n)\subseteq\mathbf{R}$, then, for appropriate $k,r$, $t\mapsto(1-it)^{-k}e^{-itf(A)}(1+|A|^2)^{-r}$ is a Fourier-Stieltjes transform, and when $f(\mathbf{R}^n)\subseteq[0,\infty)$, then $t\mapsto(1+t)^{-k}e^{-tf(A)}(1+|A|^2)^{-r}$ is a Laplace-Stieltjes transform. With $A\equiv i(D_1,\ldots,D_n),f(A)$ is a pseudodifferential operator on $L^{p}(\mathbf{R}^n) (1\leq p < \infty)$ or $BUC(\mathbf{R}^n)$.

#### Article information

Source
Abstr. Appl. Anal., Volume 2, Number 1-2 (1997), 121-136.

Dates
First available in Project Euclid: 7 April 2003

https://projecteuclid.org/euclid.aaa/1049737246

Digital Object Identifier
doi:10.1155/S1085337597000304

Mathematical Reviews number (MathSciNet)
MR1604169

Zentralblatt MATH identifier
0937.47015

#### Citation

Delaubenfels, Ralph; Lei, Yansong. Regularized functional calculi, semigroups, and cosine functions for pseudodifferential operators. Abstr. Appl. Anal. 2 (1997), no. 1-2, 121--136. doi:10.1155/S1085337597000304. https://projecteuclid.org/euclid.aaa/1049737246