Poincare inequality for weighted first order Sobolev spaces on loop spaces

	Poincare inequality for weighted first order Sobolev spaces on loop spaces
	Gong, FH ; Rockner, M ; Liming, W
刊名	JOURNAL OF FUNCTIONAL ANALYSIS
	2001-10-01
卷号	185 期号:2 页码:527-563
ISSN号	0022-1236
英文摘要	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.
WOS研究方向	Mathematics
语种	英语
出版者	ACADEMIC PRESS INC
WOS记录号	WOS:000171531300006
内容类型	期刊论文
源URL	[http://ir.amss.ac.cn/handle/2S8OKBNM/15991]
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.