Journal of Mathematics of Kyoto University

Perturbation theorems for supercontractive semigroups

Wicharn Lewkeeratiyutkul

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Let $\mu$ be a probability measure on a Riemannian manifold. It is known that if the semigroup $e^{-t\nabla *\nabla}$ is hypercontractive, then any function $g$ for which $\|\nabla g\|_{\infty}\leq 1$ will satisfy a Herbst inequality, $\int \exp (\alpha g^{2})d\mu < \infty$, for small $\alpha > 0$. If the semigroup is supercontractive, then the above inequality will hold for all $\alpha > 0$. For any $\alpha > 0$ for which $Z = \int \exp(\alpha g^{2})d\mu < \infty$, we define a measure $\mu _{g}$ by $d\mu _{g}=Z^{-1}\exp (\alpha g^{2})d\mu$. We show that if $\mu$ is hyper- or supercontractive, then so is $\mu _{g}$. Moreover, under standard conditions on logarithmic Sobolev inequalities which yield ultracontractivity of the semigroup, Gross and Rothaus have shown that $Z = \int \exp (\alpha g^{2}|\log |g||^{c})d\mu < \infty$ for some constants $\alpha ,c$. We in addition show that the perturbed measure $d\mu _{g} = Z^{-1} \exp (\alpha g^{2}|\log |g||^{c})d\mu$ is ultracontractive.

Article information

J. Math. Kyoto Univ., Volume 39, Number 4 (1999), 649-673.

First available in Project Euclid: 17 August 2009

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 31C25: Dirichlet spaces
Secondary: 47D07: Markov semigroups and applications to diffusion processes {For Markov processes, see 60Jxx}


Lewkeeratiyutkul, Wicharn. Perturbation theorems for supercontractive semigroups. J. Math. Kyoto Univ. 39 (1999), no. 4, 649--673. doi:10.1215/kjm/1250517819.

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