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VOL. 44 | 2006 Kato class functions of Markov processes under ultracontractivity
Kazuhiro Kuwae, Masayuki Takahashi

Editor(s) Hiroaki Aikawa, Takashi Kumagai, Yoshihiro Mizuta, Noriaki Suzuki

## Abstract

We show that $f \in L^p (X ; m)$ implies $|f| dm \in S_{K}^{1}$ for $p \gt D$ with $D \gt 0$, where $S_{K}^{1}$ is a subfamily of Kato class measures relative to a semigroup kernel $p_t (x, y)$ of a Markov process associated with a (non-symmetric) Dirichlet form on $L^2 (X ; m)$. We only assume that $p_t (x, y)$ satisfies the Nash type estimate of small time depending on $D$. No concrete expression of $p_t (x,y)$ is needed for the result.

## Information

Published: 1 January 2006
First available in Project Euclid: 16 December 2018

zbMATH: 1116.31005
MathSciNet: MR2277833

Digital Object Identifier: 10.2969/aspm/04410193

Subjects:
Primary: 31C15 , 31C25 , 31-XX , 60G52 , 60J25 , 60J45 , 60J60 , 60J65 , 60J75 , 60JXX

Keywords: Brownian motion , Brownian motion penetrating fractals , Dirichlet form , Dynkin class , Green kernel , heat kernel , Kato class , Markov process , Nash type inequality , relativistic Hamiltonian process , Resolvent kernel , semigroup kernel , Sobolev inequality , Symmetric $\alpha$-stable process , ultracontractivity

Rights: Copyright © 2006 Mathematical Society of Japan

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Vol. 44 • 1 January 2006