Riesz transform, Gaussian bounds and the method of wave equation

Abstract

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 . IntroductionLet ∆ = − n i=1 ∂ 2 i be the standard Laplace operator acting on R n . Then the corresponding Riesz transform is defined by the formula ∂ j ∆ −1/2 . The L p continuity of the Riesz transform for all p ∈ (1, ∞) is one of the most important and celebrated results in analysis. Papers devoted to the study of the Riesz transform and its generalizations are too numerous to list here. Hence we would like to mention only a few most relevant works [1,3,9,10,17,18,21,26,27,33,34,35,38,39,42].The operator ∇L −1/2 , where ∇ is the gradient and L is the Laplace-Beltrami operator on a Riemannian manifold M, is a natural generalization of the classical Riesz transform. L 2 boundedness of the Riesz transform ∇L −1/2 is a consequence of the equality ∇f L 2 = L 1/2 f L 2 , which is actually the definition of the operator L. In [42] Strichartz asked whether one could extend L p continuity of the classical Riesz transform to the setting of Laplace-Beltrami operators described above. An answer to this question was given in [9] for 1 ≤ p ≤ 2. In [9] Coulhon and Duong proved that if M is a complete Riemannian manifold which satisfies the doubling volume property (see Assumption 1), L is the Laplace-Beltrami operator on M and the heat kernel corresponding to L satisfies the Gaussian bounds then the Riesz transform ∇L −1/2 is of weak type (1, 1) and so bounded on L p for all p ∈ (1, 2]. Note that (∂ j ∆ −1/2 ) * = −∂ j ∆ −1/2 so the boundedness of the standard Riesz transform ∂ j ∆ −1/2 for p ∈ (1, 2] implies continuity of the standard Riesz transform for all p ∈ (1, ∞). Surprisingly in general the Riesz transform corresponding to the Laplace-Beltrami operator 1991 Mathematics Subject Classification. 42B20.

Riesz transform on manifolds and heat kernel regularity

“…In the first direction, some results already exist in the range 1 < p < ∞ in the group case, see [33], [55]. It should also be possible to cover the general setting of [88].…”

Hardy spaces associated to non-negative self-adjoint operators satisfying Davies-Gaffney estimates

“…Examples of families of operators for which condition (3.2) holds includes semigroups generated by second order elliptic self-adjoint operators in divergence form, Schrödinger operators with real potential and magnetic field (see, for example [Si2]). Condition (3.2) is well-known to hold for LaplaceBeltrami operators on all complete Riemannian manifolds (see [Da2], [Ga]).…”

Sobolev spaces adapted to the Schrödinger operator with inverse-square potential

“…The wave equation with inverse-square potential u tt + L a u = 0 obeys finite speed of propagation; see, for example, [22] or [11, §3]. Noting that ϕ(τ /R) is supported on the set {τ : |τ | ≤ R 2 }, we therefore obtain…”