Journal of Operator Theory

Volume 49, Issue 2, Spring 2003 pp. 263-283.

Functional calculus, regularity and Riesz transforms of weighted subcoercive operators on

$\sigma$ -finite measure spaces

Authors: C.M.P.A. Smulders
Author institution: Department of Mathematics and Computational Sciences, Eindhoven University of Technology, P.O. Box 513, 5600 MB Eindhoven, The Netherlands

Summary: If

$H$ is an

$n$ -th order weighted subcoercive operator associated to a continuous representation

$U$ of a

$d$ -dimensional connected Lie group

$G$ in

$L_{p}(\cm;\mu)$ , where

$p\in\langle 1,\infty\rangle$ and

$(\cm ;\mu)$ is a

$\sigma$ -finite measure space, then we show that

$\nu I+\overline{H}$ has a bounded

$H_{\infty}$ functional calculus if

$\RRe\nu$ is large enough. Moreover, the domain

$D((\nu I+\overline{H})^{m/n})$ of the fractional power equals the space of

$m$ times differentiable vectors in

$L_{p}$ -se nse if

$\RRe\nu$ is large enough and

$m$ is in a suitable subset of

$[0,\infty\rangle$ .

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