Hostname: page-component-7c8c6479df-24hb2 Total loading time: 0 Render date: 2024-03-28T14:56:56.402Z Has data issue: false hasContentIssue false

POINTWISE CONVERGENCE AND SEMIGROUPS ACTING ON VECTOR-VALUED FUNCTIONS

Published online by Cambridge University Press:  22 March 2011

MICHAEL G. COWLING*
Affiliation:
School of Mathematics and Statistics, University of New South Wales, UNSW Sydney 2052, Australia (email: m.cowling@unsw.edu.au)
MICHAEL LEINERT
Affiliation:
Institut für angewandte Mathematik, Im Neuenheimer Ruprechts-Karl-Universität Heidelberg, D-69120 Heidelberg, Germany (email: leinert@math.uni-heidelberg.de)
*
For correspondence; e-mail: m.cowling@unsw.edu.au
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

A submarkovian C0 semigroup (Tt)t∈ℝ+ acting on the scale of complex-valued functions Lp(X,ℂ) extends to a semigroup of operators on the scale of vector-valued function spaces Lp(X,E), when E is a Banach space. It is known that, if fLp(X,ℂ), where 1<p<, then Ttff pointwise almost everywhere. We show that the same holds when fLp(X,E) .

Type
Research Article
Copyright
Copyright © Australian Mathematical Publishing Association Inc. 2011

Footnotes

It is a pleasure for the first-named author to acknowledge the generous support of an Alexander von Humboldt Foundation Research Prize, and the hospitality of the University of Heidelberg.

References

[1]Cowling, M. G., ‘Harmonic analysis on semigroups’, Ann. of Math. (2) 117 (1983), 267283.CrossRefGoogle Scholar
[2]Kunstmann, P. C. and Weis, L., ‘Maximal L p-regularity for parabolic equations, Fourier multiplier theorems and H -functional calculus’, in: Functional Analytic Methods for Evolution Equations, Lecture Notes in Mathematics, 1855 (Springer, Berlin, 2004), pp. 65311.CrossRefGoogle Scholar
[3]Liskevich, V. A. and Perelmuter, M. A., ‘Analyticity of submarkovian semigroups’, Proc. Amer. Math. Soc. 123 (1995), 10971104.Google Scholar
[4]Stein, E. M., Topics in Harmonic Analysis Related to the Littlewood–Paley Theory (Princeton University Press, Princeton, NJ, 1970).CrossRefGoogle Scholar
[5]Taggart, R. J., ‘Pointwise convergence for semigroups in vector-valued L p spaces’, Math. Z. 261 (2009), 933949.CrossRefGoogle Scholar