Gradient estimates for fundamental solutions of a Schrödinger operator on stratified Lie groups

Abstract

Let L = − Δ G + Υ \mathcal L=-\Delta _{\mathbb {G}}+\Upsilon be a Schrödinger operator with a nonnegative potential Υ \Upsilon belonging to the reverse Hölder class B Q / 2 B_{Q/2} , where Q Q is the homogeneous dimension of the stratified Lie group G \mathbb {G} . Inspired by Shen’s pioneer work and Li’s work, we study fundamental solutions of the Schrödinger operator L \mathcal L on the stratified Lie group G \mathbb {G} in this paper. By proving an exponential decreasing variant of mean value inequality, we obtain the exponential decreasing upper estimates, the local Hölder estimates and the gradient estimates of the fundamental solutions of the Schrödinger operator L \mathcal L on the stratified Lie group. As two applications, we obtain the De Giorgi-Nash-Moser theory on the improved Hölder estimate for the weak solutions of the Schrödinger equation and a Liouville-type lemma for L \mathcal {L} -harmonic functions on G \mathbb {G} .