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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 $[0,\infty\rangle$.

Keywords:  Functional calculus, regularity, Riesz transform, continuous representation, induced representation, weighted subcoercive operator, $\sigma$-finite measure space

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