Poincaré 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 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 Schrödinger operator −LV≔−LD+V always has a spectral gap at the bottom λ0(V) of its spectrum and thus has its ground state transformed operator φ−1(−LV−λ0(V)) φ, where φ is the unique ground state of −LV. In particular, our result proves L. Grossʹs conjecture about the existence of a spectral gap for the ground state transform of the Schrödinger 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 φ2, thus establishing a Poincaré 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.