Poincare inequality for weighted first order Sobolev spaces on loop spaces

Let E be the loop space over a compact connected Riemannian manifold with a torsion skew symmetric connection. Let LD be the Ornstein Uhlenbeck operat or 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 ineq uality 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 tile SChrodinger o perator - 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 ph i (-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 existen ce of a spectral gap for the ground state transform of the Schrodinger oper ator 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 st ate transforms as weighted First order Sobolev spaces,vith weight given by phi (2), thus establishing a Poincare inequality for them. All these result s are consequences from some new results in this paper on Dirichlet forms c haracterizing certain classes with spectral gaps and from results by S. Aid a and M. Hino. (C) 2001 Academic Press.