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Note on Gradient Estimate of Heat Kernel for Schrödinger Operators

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DOI: 10.4236/am.2010.15056    5,912 Downloads   10,150 Views  
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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.

Conflicts of Interest

The authors declare no conflicts of interest.

Cite this paper

S. Zheng, "Note on Gradient Estimate of Heat Kernel for Schrödinger Operators," Applied Mathematics, Vol. 1 No. 5, 2010, pp. 425-430. doi: 10.4236/am.2010.15056.

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