Note on Gradient Estimate of Heat Kernel for Schr&amp;#246;dinger Operators

Shijun Zheng

doi:10.4236/am.2010.15056

Abstract

Let be a Schrödinger operator on . We show that gradient estimates for the heat kernel of with upper Gaussian bounds imply polynomial decay for the kernels of certain smooth dyadic spectral operators. The latter decay property has been known to play an important role in the Littlewood-Paley theory for and Sobolev spaces. We are able to establish the result by modifying Hebisch and the author’s recent proofs. We give a counterexample in one dimension to show that there exists in the Schwartz class such that the long time gradient heat kernel estimate fails.

Highlights

Consider a Schrödinger operator H = V on n, where V is a real-valued potential in L1loc ( n )
We show that gradient estimates for the heat kernel of H with upper Gaussian bounds imply polynomial decay for the kernels of certain smooth dyadic spectral operators
We give a counterexample in one dimension to show that there exists V in the Schwartz class such that the long time gradient heat kernel estimate fails

Summary

Introduction

Consider a Schrödinger operator H = V on n , where V is a real-valued potential in L1loc ( n ) It is noted in [1,2] that for positive V , if H admits the following gradient estimates for its heat kernel pt (x, y) = e tH (x, y) : for all x, y n and t > 0 ,. We are able to overcome this difficulty by establishing Lemma 6, a scaling version of the weighted L1 inequality for j (H )(x, y) with H s , for which we directly use the scaling information indicated by the time variable appearing in Lemma 2 This leads to the proof of the main theorem by combining methods of Hebisch and the author’s in [7,2]. Holds whenever V 0 is locally integrable on n , n 1

Proof of Theorem 1

A Counterexample to the Gradient Heat Kernel Estimate