Riesz transform, Gaussian bounds and the method of wave equation

2003/07/22 by Adam Sikora, Sikora, Adam
Mathematics · #42B15 #42B25 #Advanced Harmonic Analysis Research #Analysis of PDEs (math.AP) #FOS: Mathematics #Mathematical Analysis and Transform Methods #advanced mathematical theories #math.AP #msc:42B15 #msc:42B25

paper · pdf · doi:10.48550/arxiv.math/0307291

17 pages

arxiv created 2003/07/22 · openalex publication_date 2003/07/22 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

For an abstract self-adjoint operator L and a local operator A we study the boundedness of the Riesz transform AL^-α on L^p 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 L^p 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 L^p of the Laplace-Beltrami operator on Riemannian manifolds for p>2 .

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