Intrinsic Ultracontractivity on Riemannian Manifolds with Infinite Volume Measures

TL;DR

This paper establishes explicit conditions under which diffusion semigroups on non-compact Riemannian manifolds with infinite volume are intrinsically ultracontractive, providing sharp bounds on heat kernels.

Contribution

It introduces explicit criteria for intrinsic ultracontractivity on manifolds with infinite volume, along with sharp heat kernel bounds, advancing understanding of diffusion processes in geometric analysis.

Findings

Explicit conditions for intrinsic ultracontractivity are derived.

Uniform upper bounds on intrinsic heat kernels are established.

Conditions are shown to be sharp for specific examples.

Abstract

By establishing the intrinsic super-Poincar\'e inequality, some explicit conditions are presented for diffusion semigroups on a non-compact complete Riemannian manifold to be intrinsically ultracontractive. These conditions, as well as the resulting uniform upper bounds on the intrinsic heat kernels, are sharp for some concrete examples.