Eigenfunction and harmonic function estimates in domains with horns and cusps

An interesting example in the paper of Davies and Simon [5] was that of a horn-shaped domain in R. By horn-shaped we mean domains of the form D = {(x, y) : x > 0, ‖y‖ < f(x)} with f : [0,∞) → (0,∞) a function tending to zero as x tends to infinity. Davies and Simon [5] established sufficiently sharp estimates on the first Dirichlet eigenfunction of ∆d (d-dimensional Laplacian) on D to determine when the Dirichlet heat semigroup on D is intrinsically ultracontractive. This last property is important as one can provide bounds in that case for all the eigenfunctions, the heat kernel, and Green function in terms of the first eigenfunction. Thus, if one gets precise bounds on the first eigenfunction and the domain is intrinsically ultracontractive, then one has precise estimates on other important analytic quantities associated to the domain. Several works have appeared providing such bounds, Bañuelos [1],[2], Bañuelos and van den Berg [3], Bañuelos and Davis [4], Lindemann, Pang and Zhao [8]. In this paper we shall obtain pointwise bounds for positive harmonic functions vanishing