Poincare inequality for weighted first order Sobolev spaces on loop spaces

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	Poincare inequality for weighted first order Sobolev spaces on loop spaces
	Gong, FH ; Rockner, M ; Liming, W
	2001-10-01
发表期刊	JOURNAL OF FUNCTIONAL ANALYSIS
ISSN	0022-1236
卷号	185 期号:2 页码:527-563
摘要	Let E be the loop space over a compact connected Riemannian manifold with a torsion skew symmetric connection. Let LD be the Ornstein Uhlenbeck operator on a nonempty connected component D of the loop space E and let V: D --> R be the restriction on D of the potential in the logarithmic Sobolev inequality found by L. Gross on the loop group by S. Aida and by F. Z. Gong and Z. M. Ma on the loop space, respectively. We prove that the SChrodinger operator - L-V := - L-D + V always has a spectral gap at the bottom lambda (0)(V) of its spectrum and thus has its ground state transformed operator phi (-1)(-L-V-lambda (0)(V))phi. where phi is the unique ground state of - L-V. In particular, our result proves L. Gross's conjecture about the existence of a spectral gap for the ground state transform of the Schrodinger operator studied by him on the loop group. In addition, in all the above cases we identify the domain of the Dirichlet forms associated with the ground state transforms as weighted First order Sobolev spaces with weight given by phi (2), thus establishing a Poincare inequality for them. All these results are consequences from some new results in this paper on Dirichlet forms characterizing certain classes with spectral gaps and from results by S. Aida and M. Hino. (C) 2001 Academic Press.
语种	英语
WOS研究方向	Mathematics
WOS类目	Mathematics
WOS记录号	WOS:000171531300006
出版者	ACADEMIC PRESS INC
引用统计
文献类型	期刊论文
条目标识符	http://ir.amss.ac.cn/handle/2S8OKBNM/15992
专题	中国科学院数学与系统科学研究院
通讯作者	Gong, FH
作者单位	1.Chinese Acad Sci, Inst Appl Math, Acad Math & Syst Sci, Beijing 100080, Peoples R China 2.Univ Bielefeld, Fak Math, D-33615 Bielefeld, Germany 3.Univ Clermont Ferrand, Lab Math Appl, F-63177 Aubiere, France
推荐引用方式 GB/T 7714	Gong, FH,Rockner, M,Liming, W. Poincare inequality for weighted first order Sobolev spaces on loop spaces[J]. JOURNAL OF FUNCTIONAL ANALYSIS,2001,185(2):527-563.
APA	Gong, FH,Rockner, M,&Liming, W.(2001).Poincare inequality for weighted first order Sobolev spaces on loop spaces.JOURNAL OF FUNCTIONAL ANALYSIS,185(2),527-563.
MLA	Gong, FH,et al."Poincare inequality for weighted first order Sobolev spaces on loop spaces".JOURNAL OF FUNCTIONAL ANALYSIS 185.2(2001):527-563.