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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Gaussian estimates and regularized groups


Authors: Quan Zheng and Jizhou Zhang
Journal: Proc. Amer. Math. Soc. 127 (1999), 1089-1096
MSC (1991): Primary 47D03, 47F05
MathSciNet review: 1473683
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Abstract: We show that if a bounded analytic semigroup $\{T(z)\}_{ \operatorname{Re}z>0}$ on $L^2({\boldsymbol{\Omega}} )$ $({\boldsymbol{\Omega}} \subset\mathbf{R} ^n)$ satisfies a Gaussian estimate of order $m$ and $A_p$ is the generator of its consistent semigroup on $L^p({\boldsymbol{\Omega}} )$ $(1\le p<\infty)$, then $iA_p$ generates a $(1-A_p)^{-\alpha}$-regularized group on $L^p({\boldsymbol{\Omega}} )$ where $\alpha>2n |\frac{1}{2}-\frac{1}{p}|$. We obtain the estimate of $(\lambda-A_p)^{-1}$ ($|\operatorname{arg}\lambda|<\pi$) and the $p$-independence of $\sigma(A_p)$, and give applications to Schrödinger operators and elliptic operators of higher order.


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Additional Information

Quan Zheng
Affiliation: Department of Mathematics, Huazhong University of Science and Technology, Wuhan 430074, People’s Republic of China
Email: qzheng@hust.edu.cn

Jizhou Zhang
Affiliation: Department of Mathematics, Hubei University, Wuhan 430062, People’s Republic of China
Email: zhangjz@hubu.edu.cn

DOI: http://dx.doi.org/10.1090/S0002-9939-99-04649-3
PII: S 0002-9939(99)04649-3
Keywords: Gaussian estimate, regularized group, analytic semigroup, differential operator
Received by editor(s): February 27, 1997
Received by editor(s) in revised form: July 14, 1997, and July 22, 1997
Additional Notes: This project was supported by the National Science Foundation of China
Communicated by: Palle E. T. Jorgensen
Article copyright: © Copyright 1999 American Mathematical Society