Local Poincaré inequalities on loop spaces

Let M be a compact connected Riemannian manifold, and fix x,y∈M. For a sufficiently small constant R>0, Poincaré inequalities w.r.t. pinned Wiener measure with time parameter T>0 are proven on the sets $\text{Ω}{}^{\mathrm{R,N}}{}_{\mathrm{x,y}}$, $\text{N∈}\text{N}$, consisting of all continuous paths ω:[0,1]→M such that ω(0)=x, ω(1)=y, and d(ω(s),ω(t))<R if s,t∈[(i−1)/N,i/N] for some integer i. Moreover, the asymptotic behaviour of the best constants in the Poincaré inequalities as T goes to 0 is studied. It turns out that the asymptotic depends crucially on the Riemannian metric on M and, in particular, on the geodesics contained in $\text{Ω}{}^{\mathrm{R,N}}{}_{\mathrm{x,y}}$. Key ingredients in the proofs are a bisection argument, estimates for finite-dimensional spectral gaps, and a crucial variance estimate by Malliavin and Stroock.