Gradient Estimates and Intrinsic Ultracontractivity of Neumann and Shrödinger semigroups

Abstract

We show the estimation of the logarithmic Sobolev constant and give the gradient estimates of heat semigroups. We study Wiener’s lemma for localized integral operators on a Hilbert space We consider the stability of localized operators including infinite matrices. We derived an explicit gradient estimates and show the first Neumann eigenvalue on the manifolds with boundary. Also a second fundamental form and gradient of Neumann semigroups are considered .The positvity and negativity with compactness of the ground state energy for the Schrödinger operator on a Hilbert space are shown. We investigate the intrinsic ultracontractivity for Schrödinger operators based on fractional Laplacian and semigroups on a Hilbert space.