Riesz transform, Gaussian bounds and the method of wave equation

Adam Sikora*

*Corresponding author for this work

Research output: Contribution to journalArticle

96 Citations (Scopus)

Abstract

For an abstract self-adjoint operator L and a local operator A we study the boundedness of the Riesz transform AL on Lp for some α > 0. A very simple proof of the obtained result is based on the finite speed propagation property for the solution of the corresponding wave equation. We also discuss the relation between the Gaussian bounds and the finite speed propagation property. Using the wave equation methods we obtain a new natural form of the Gaussian bounds for the heat kernels for a large class of the generating operators. We describe a surprisingly elementary proof of the finite speed propagation property in a more general setting than it is usually considered in the literature. As an application of the obtained results we prove boundedness of the Riesz transform on Lp for all p ∈ (1, 2] for Schrödinger operators with positive potentials and electromagnetic fields. In another application we discuss the Gaussian bounds for the Hodge Laplacian and boundedness of the Riesz transform on Lp of the Laplace-Beltrami operator on Riemannian manifolds for p > 2.

Original languageEnglish
Pages (from-to)643-662
Number of pages20
JournalMathematische Zeitschrift
Volume247
Issue number3
Publication statusPublished - Jul 2004
Externally publishedYes

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