$H^\infty$-calculus for submarkovian generators
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- by Peer Christian Kunstmann and Željko Štrkalj PDF
- Proc. Amer. Math. Soc. 131 (2003), 2081-2088 Request permission
Abstract:
Let $-A$ be the generator of a symmetric submarkovian semigroup in $L_2(\Omega )$. In this note we show that on $L_p(\Omega ), 1<p<\infty ,$ the operator $A$ admits a bounded $H^\infty$ functional calculus on the sector $\Sigma (\phi )=\{z\in \mathbb {C}\setminus \{0\}:|\mbox {arg} z|<\phi \}$ for each $\phi >\psi _p^*$ with \[ \psi _p^*=\frac {\pi }{2}|\frac {1}{p}-\frac {1}{2}| +(1-|\frac {1}{p}-\frac {1}{2}|)\arcsin (\frac {|p-2|}{2p-|p-2|}). \] This improves a result due to M. Cowling. We apply our result to obtain maximal regularity for parabolic equations and evolutionary integral equations.References
- Philippe Clément and Jan Prüss, An operator-valued transference principle and maximal regularity on vector-valued $L_p$-spaces, Evolution equations and their applications in physical and life sciences (Bad Herrenalb, 1998) Lecture Notes in Pure and Appl. Math., vol. 215, Dekker, New York, 2001, pp. 67–87. MR 1816437
- Michael G. Cowling, Harmonic analysis on semigroups, Ann. of Math. (2) 117 (1983), no. 2, 267–283. MR 690846, DOI 10.2307/2007077
- Michael Cowling, Ian Doust, Alan McIntosh, and Atsushi Yagi, Banach space operators with a bounded $H^\infty$ functional calculus, J. Austral. Math. Soc. Ser. A 60 (1996), no. 1, 51–89. MR 1364554
- Xuan T. Duong and Derek W. Robinson, Semigroup kernels, Poisson bounds, and holomorphic functional calculus, J. Funct. Anal. 142 (1996), no. 1, 89–128. MR 1419418, DOI 10.1006/jfan.1996.0145
- Gero Fendler, On dilations and transference for continuous one-parameter semigroups of positive contractions on ${\scr L}^p$-spaces, Ann. Univ. Sarav. Ser. Math. 9 (1998), no. 1, iv+97. MR 1664244
- José García-Cuerva, Giancarlo Mauceri, Stefano Meda, Peter Sjögren, and José Luis Torrea, Functional calculus for the Ornstein-Uhlenbeck operator, J. Funct. Anal. 183 (2001), no. 2, 413–450. MR 1844213, DOI 10.1006/jfan.2001.3757
- N.J. Kalton and L. Weis, The $H^\infty$-calculus and sums of closed operators, Math. Ann. 321 (2001), 319-345.
- P.C. Kunstmann, $L_p$-spectral properties of the Neumann Laplacian on horns, comets and stars, to appear in Math. Z.
- Damien Lamberton, Équations d’évolution linéaires associées à des semi-groupes de contractions dans les espaces $L^p$, J. Funct. Anal. 72 (1987), no. 2, 252–262 (French). MR 886813, DOI 10.1016/0022-1236(87)90088-7
- V. A. Liskevich and M. A. Perel′muter, Analyticity of sub-Markovian semigroups, Proc. Amer. Math. Soc. 123 (1995), no. 4, 1097–1104. MR 1224619, DOI 10.1090/S0002-9939-1995-1224619-1
- A. Pazy, Semigroups of linear operators and applications to partial differential equations, Applied Mathematical Sciences, vol. 44, Springer-Verlag, New York, 1983. MR 710486, DOI 10.1007/978-1-4612-5561-1
- Akbar Kh. Begmatov, On a class of problems in integral geometry in the plane, Dokl. Akad. Nauk 331 (1993), no. 3, 261–262 (Russian); English transl., Russian Acad. Sci. Dokl. Math. 48 (1994), no. 1, 56–58. MR 1242669
- Lutz Weis, Operator-valued Fourier multiplier theorems and maximal $L_p$-regularity, Math. Ann. 319 (2001), no. 4, 735–758. MR 1825406, DOI 10.1007/PL00004457
- Lutz Weis, A new approach to maximal $L_p$-regularity, Evolution equations and their applications in physical and life sciences (Bad Herrenalb, 1998) Lecture Notes in Pure and Appl. Math., vol. 215, Dekker, New York, 2001, pp. 195–214. MR 1818002
Additional Information
- Peer Christian Kunstmann
- Affiliation: Institute of Mathematics I, University of Karlsruhe, Englerstrasse 2, D-76128 Karlsruhe, Germany
- Email: peer.kunstmann@math.uni-karlsruhe.de
- Željko Štrkalj
- Affiliation: Department of Mathematics, 202 Mathematical Sciences Building, University of Missouri, Columbia, Missouri 65211
- Address at time of publication: Institute of Mathematics I, University of Karlsruhe, Englerstrasse 2, D-76128 Karlsruhe, Germany
- Email: zeljko.strkalj@math.uni-karlsruhe.de
- Received by editor(s): March 19, 2001
- Received by editor(s) in revised form: December 12, 2001
- Published electronically: February 5, 2003
- Additional Notes: This work has been partially supported by the “Landesforschungsschwerpunkt Evolutionsgleichungen” of the Land Baden-Württemberg
The second author acknowledges support from DAAD. Die Arbeit wurde mit Unterstützung eines Stipendiums im Rahmen des Gemeinsamen Hochschulsonderprogramms III von Bund und Ländern über den DAAD ermöglicht - Communicated by: Joseph A. Ball
- © Copyright 2003 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 131 (2003), 2081-2088
- MSC (2000): Primary 47A60, 47D03, 47D07
- DOI: https://doi.org/10.1090/S0002-9939-03-06956-9
- MathSciNet review: 1963753