\(L^p\)-gradient estimates of symmetric Markov semigroups for \(1<p\leq2\)
type: Item
instance of: scholarly article
title: \(L^p\)-gradient estimates of symmetric Markov semigroups for \(1<p\leq2\)
zbMATH Open document ID: 1107.60040
author: Xicheng Zhang
author: Ana Bela Cruzeiro
publication date: 2006-10-24T00:00:00Z
review text: The authors consider a strongly continuous, positive Markov semigroup \((T_t)_{t>0}\) on \(L^2\). A Dirichlet form is assumed to be associated with this semigroup and \(L\) denotes the Dirichlet operator. Moreover, it is supposed that the semigroup admits a carré du champ operator \(\Gamma \). The main result asserts the following \(L^p\)-gradient estimates (for \(1 < p < 2\), \(q > 2\) and \(\frac 12 < \alpha < 1\)): \[ \| \Gamma^{1/2} (T_t f)\| _p \leq \frac{C_p}{\sqrt{t}}\| f\| _p,\quad \| \Gamma^{1/2}((I - L)^{-\alpha}f)\| _p \leq C_{p,\alpha} \| f\| _p, \] \[ c_{q,\alpha} \| f\| _q \leq \| \Gamma^{1/2}((I - L)^{-(1-\alpha )}f)\| _q + \| ( I - L)^{-(1-\alpha )}f\| _q. \]
Mathematics Subject Classification ID: 60J45
Mathematics Subject Classification ID: 60H07
Mathematics Subject Classification ID: 47D07
Mathematics Subject Classification ID: 31C12
Mathematics Subject Classification ID: 31C05
zbMATH DE Number: 5066703
zbMATH Keywords: Carré du champ
zbMATH Keywords: path space
reviewed by: Liliana Popa
MaRDI profile type: Publication
full work available at URL: s10114-005-0538-0
OpenAlex ID: W2060703789
cites work: Dirichlet forms and analysis on Wiener space
cites work: Q4337939
cites work: Topics in Harmonic Analysis Related to the Littlewood-Paley Theory. (AM-63)
cites work: Q3676923
cites work: Stochastic analysis on manifolds
DOI: 10.1007/S10114-005-0538-0
published in: Acta Mathematica Sinica, English Series
description: scientific article