Intrinsic ultracontractivity via capacitary width

Abstract

The semigroup associated with the Dirichlet heat kernel is said to be intrinsic ultracontractive if

(i) the Dirichlet realization of the associated self adjoint operator has the first positive eigenvalue with positive L2 eigenfunction;

(ii) the heat kernel is bounded above and below by the product of the eigenfunctions with positive multiplicative constants depending on time.

We give an upper and lower estimate of the first eigenvalue in terms of capacitary width, which yields a sharp sufficient condition for (i). Our parabolic argument also yields an exponential decay property of a certain caloric measure. Then, the caloric measure is controlled by the elliptic Green function with the aid of a parabolic box argument. This is a crucial step for (ii). We give a sharp sufficient integral condition for intrinsic ultracontractivity in terms of capacitary width. A similar integral condition for the boundary Harnack principle is also obtained. Under geometric specifications, these integral conditions generalize known results and give more precise conditions. Sharpness is examined by an infinite funnel, for which we obtain a complete characterization of intrinsic ultracontractivity. Our method is purely analytic and elementary; it enables us to dispense with logarithmic Sobolev inequalities.