Riesz transform, Gaussian bounds and the method of wave equation
By Adam S. Sikora. 2003
230 citations recorded by the source corpus.
Abstract.For an abstract self-adjoint operator L and a local operator A we study the boundedness of the Riesz transform AL−α 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.
Subfields: Applied Mathematics, Mathematical Physics
View Paper: https://www.semanticscholar.org/paper/23979b22631e9a73bab41b1118fb86ca5b625100
How researchers see this paper
The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.
In Applied Mathematics
In Mathematical Physics
Citations in the corpus
- A Quadrature Formula for Diffusion Polynomials Corresponding to a Generalized Heat Kernel (2010, methodology): “While Sikora's formulation of the statement is in terms of certain operators in a very general setting, the formulation we have chosen here is merely a property of complex numbers.”
- Maximal inequalities and Riesz transform estimates on $L^p$ spaces for Schr\ (2006, unknown): “ is controlled pointwise by the one of −∆ and a theorem in [DMc]. See [DOY] where the needed estimates are proved and [CD] where a similar argument is done for Riesz transforms on manifolds. See also [Sik] for a different proof using finite speed of propagation for the wave equation. We are interested”
- Riesz transform on manifolds and heat kernel regularity (2004, methodology): “ty argument would not help us either as we would have to make assumptions on the semigroup acting on 1-forms as explained in Section 1.4; this would bring us back to the state of the art in [20] (see [88] for this approach to the results in [20]). But recently, it was shown in [72] that this regular”
- Endpoint estimates for Riesz transforms of magnetic Schrödinger operators (2006, unknown): “It was recently proved in [20] (Theorem 11) that by using the finite speed propagation property, for each k=1, .”
- Orlicz-Hardy Spaces Associated with Operators Satisfying Davies-Gaffney Estimates (2009, background): “By Assumptions (A) and (B), we have the following results which were established in [17].”
- Restriction and spectral multiplier theorems on asymptotically conic manifolds (2010, unknown): “Our implementation of this principle in the proof of Theorem 1.1 is more geometrical and explicit than in other similar approaches because it is based on finite speed propagation of the wave equations, see [5, 9, 32] for comparison.”
- Gaussian heat kernel upper bounds via the Phragmén–Lindelöf theorem (2006, unknown): “Next, as in [27, 36, 55], we consider the integral”
- Hardy spaces associated to operators satisfying Davies–Gaffney estimates and bounded holomorphic functional calculus☆ (2013, background): “For example, a non-negative self-adjoint operator L with Davies–Gaffney estimates would satisfy the finite speed propagation property for solutions of the corresponding wave equation, see [32] (see also [20]), and allows us to construct the Hardy space via (ω, M)-atoms, see [20], [25], [26].”
- Hardy Spaces of Differential Forms on Riemannian Manifolds (2006, unknown): “Part (b) is proved in [44].”