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Reverse Hypercontractivity for Subharmonic Functions

Published online by Cambridge University Press:  20 November 2018

Leonard Gross
Affiliation:
Department of Mathematics, Cornell University, Ithaca, NY 14853, U.S.A., e-mail: gross@math.cornell.edu
Martin Grothaus
Affiliation:
Fachbereich Mathematik, Universität Kaiserslautern, 67663 Kaiserslautern, Germany, e-mail: grothaus@mathematik.uni-kl.de
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Abstract

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Contractivity and hypercontractivity properties of semigroups are now well understood when the generator, $A$, is a Dirichlet form operator. It has been shown that in some holomorphic function spaces the semigroup operators, ${{e}^{-tA}}$, can be bounded below from ${{L}^{p}}$ to ${{L}^{q}}$ when $p,\,q$ and $t$ are suitably related. We will show that such lower boundedness occurs also in spaces of subharmonic functions.

Type
Research Article
Copyright
Copyright © Canadian Mathematical Society 2005

References

[BH] Bouleau, N. and Hirsch, F., Dirichlet Forms and Analysis on Wiener Space. de Gruyter Studies in Mathematics 14, Gruyter, W. de, Berlin, 1991.Google Scholar
[Ca] Carlen, E. A., Some integral identities and inequalities for entire functions and their applications to the coherent state transform. J. Funct. Anal. 97(1991), 231249.Google Scholar
[Da80] Davies, E. B., One-parameter Semigroups. LondonMathematical Society Monographs 15, Academic Press, London, 1980.Google Scholar
[Da89] Davies, E. B., Heat Kernels and Spectral Theory. Cambridge Tracts in Mathematics 92, Cambridge University Press, Cambridge, 1989.Google Scholar
[Fu80] Fukushima, M., Dirichlet Forms and Markov Processes. North-Holland, Amsterdam, 1980.Google Scholar
[GGS] Galaz-Fontes, F., Gross, L., and Sontz, S. B., Reverse hypercontractivity over manifolds. Ark. Mat. 39(2001), 283309.Google Scholar
[GS] Galaz-Fontes, F. and Sontz, S. B., On two reverse inequalities in the Segal-Bargmann space. In: Proceedings of the Symposium onMathematical Physics and Quantum Field Theory (Berkeley, CA), Electron. J. Differ. Equ. Conf. 4(2000), 103111.Google Scholar
[G75] Gross, L., Logarithmic Sobolev inequalities. Amer. J. Math. 97(1975), 10611083.Google Scholar
[G99] Gross, L., Hypercontractivity over complex manifolds. Acta Math. 182(1999), 159206.Google Scholar
[G02] Gross, L., Strong hypercontractivity and relative subharmonicity. J. Funct. Anal. 190(2002), 3892.Google Scholar
[GM] Gross, L. and Malliavin, P., Hall's transform and the Segal-Bargmann map. In: It ô's Stochastic Calculus and Probability Theory (Ikeda, N., Watanabe, S., Fukushima, M., and Kunita, H., eds.), Springer, Tokyo, 1996, pp. 73116.Google Scholar
[HP] Hille, E. and Phillips, R. S., Functional Analysis and Semi-Groups. American Mathematical Society, Providence, RI, 1957.Google Scholar
[MR] Ma, Z.-M. and Röckner, M.. Introduction to the Theory of (Non-Symmetric) Dirichlet Forms. Springer, Berlin, New York, 1992.Google Scholar
[Si] Silverstein, M. L., Symmetric Markov Processes, Lecture Notes in Mathematics 426, Springer-Verlag, Berlin, 1974.Google Scholar
[So] Sontz, S. B., On some reverse inequalities in the Segal-Bargmann space. In: Differential Equations and Mathematical Physics (Birmingham, AL,), AMS/IP Stud. Adv. Math. 16, Amer. Math. Soc., Providence, RI, 2000. pp. 361373.Google Scholar