Intrinsic Ultracontractivity for Domains in Negatively Curved Manifolds

Hiroaki Aikawa,Michiel Van Den Berg,Jun Masamune

doi:10.1007/s40315-021-00402-8

Abstract

Let M be a complete, non-compact, connected Riemannian manifold with Ricci curvature bounded from below by a negative constant. A sufficient condition is obtained for open and connected sets D in M for which the corresponding Dirichlet heat semigroup is intrinsically ultracontractive. That condition is formulated in terms of capacitary width. It is shown that both the reciprocal of the bottom of the spectrum of the Dirichlet Laplacian acting in L^2(D), and the supremum of the torsion function for D are comparable with the square of the capacitary width for D if the latter is sufficiently small. The technical key ingredients are the volume doubling property, the Poincaré inequality and the Li-Yau Gaussian estimate for the Dirichlet heat kernel at finite scale.

Highlights

For a domain D ⊂ M we denote by pD(t, x, y), t > 0, x, y ∈ D, the Dirichlet heat kernel for ∂/∂t − in D, i.e., the fundamental solution to (∂/∂t − )u = 0 subject to the Dirichlet boundary condition u(t, x) = 0 for x ∈ ∂ D and t > 0
Davies and Simon [12] introduced the notion of intrinsic ultracontractivity
The following is in terms of the heat kernel estimate

Summary

Main Results

Let M be a complete, non-compact, n-dimensional connected Riemannian manifold, without boundary, and with Ricci curvature bounded below by a negative constant, i.e., Ric ≥ −K with non-negative constant K. Lierl and Saloff-Coste [15] studied a general framework including Riemannian manifolds In that paper, they gave a precise heat kernel estimate for a bounded inner uniform domain, which implies IU ([15, Thm. 7.9]). The condition wη(D) < R0 in Theorem 1.6 is more convenient since the capacitary width wη(D) can be more estimated than the bottom of the spectrum λmin(D). 2 we summarize the key technical ingredients of the proofs: the volume doubling property, the Poincaré inequality and the Li-Yau Gaussian estimate for the Dirichlet heat kernel at finite scale Observe that these fundamental tools are available for manifolds with Ricci curvature bounded below by a negative constant and for unimodular Lie groups and homogeneous spaces. The constant C is referred to as the constant of comparison

Preliminaries

Torsion Function and the Bottom of Spectrum

Capacitary Width and Harmonic Measure

Remarks