Abstract. We consider homogeneous singular kernels, whose angular part is bounded, but need not have any continuity. For the norm of the corresponding singular integral operators on the weighted space L 2 (w), we obtain a bound that is quadratic in the A 2 constant [w] A 2 . We do not know if this is sharp, but it is the best known quantitative result for this class of operators. The proof relies on a classical decomposition of these operators into smooth pieces, for which we use a quantitative elaboration of Lacey's dyadic decomposition of Dini-continuous operators: the dependence of constants on the Dini norm of the kernels is crucial to control the summability of the series expansion of the rough operator. We conclude with applications and conjectures related to weighted bounds for powers of the Beurling transform.
Abstract. It is well-known that the fundamental solution of ut(n, t) = u(n + 1, t) − 2u(n, t) + u(n − 1, t), n ∈ Z, with u(n, 0) = δnm for every fixed m ∈ Z, is given by u(n, t) = e −2t In−m(2t), where I k (t) is the Bessel function of imaginary argument. In other words, the heat semigroup of the discrete Laplacian is described by the formal seriesBy using semigroup theory, this formula allows us to analyze some operators associated with the discrete Laplacian. In particular, we obtain the maximum principle for the discrete fractional Laplacian, weighted ℓ p (Z)-boundedness of conjugate harmonic functions, Riesz transforms and square functions of Littlewood-Paley. Interestingly, it is shown that the Riesz transforms coincide essentially with the so called discrete Hilbert transform defined by D. Hilbert at the beginning of the XX century. We also see that these Riesz transforms are limits of the conjugate harmonic functions.The results rely on a careful use of several properties of Bessel functions.
The analysis of nonlocal discrete equations driven by fractional powers of the discrete Laplacian on a mesh of size h > 0for u, f : Z h → R, 0 < s < 1, is performed. The pointwise nonlocal formula for (−∆ h ) s u and the nonlocal discrete mean value property for discrete s-harmonic functions are obtained. We observe that a characterization of (−∆ h ) s as the Dirichlet-to-Neumann operator for a semidiscrete degenerate elliptic local extension problem is valid. Regularity properties and Schauder estimates in discrete Hölder spaces as well as existence and uniqueness of solutions to the nonlocal Dirichlet problem are shown. For the latter, the fractional discrete Sobolev embedding and the fractional discrete Poincaré inequality are proved, which are of independent interest. We introduce the negative power (fundamental solution)which can be seen as the Neumann-to-Dirichlet map for the semidiscrete extension problem. We then prove the discrete Hardy-Littlewood-Sobolev inequality for (−∆ h ) −s .As applications, the convergence of our fractional discrete Laplacian to the (continuous) fractional Laplacian as h → 0 in Hölder spaces is analyzed. Indeed, uniform estimates for the error of the approximation in terms of h under minimal regularity assumptions are obtained. We finally prove that solutions to the Poisson problem for the fractional Laplacian (−∆) s U = F, in R, can be approximated by solutions to the Dirichlet problem for our fractional discrete Laplacian, with explicit uniform error estimates in terms of h.2010 Mathematics Subject Classification. Primary: 35R11, 49M25. Secondary: 35K05, 65N15. Key words and phrases. Nonlocal discrete diffusion equations, fractional discrete Laplacian, regularity and extension problem, Sobolev and Poincaré inequalities, error of approximation, semidiscrete heat equation.
We study the fractional Laplacian (−∆) σ/2 on the n-dimensional torus T n , n ≥ 1. First, we present a general extension problem that describes any fractional power L γ , γ > 0, where L is a general nonnegative selfadjoint operator defined in an L 2 -space. This generalizes to all γ > 0 and to a large class of operators the previous known results by Caffarelli and Silvestre. In particular it applies to the fractional Laplacian on the torus. The extension problem is used to prove interior and boundary Harnack's inequalities for (−∆) σ/2 , when 0 < σ < 2. We deduce regularity estimates on Hölder, Lipschitz and Zygmund spaces. Finally, we obtain the pointwise integro-differential formula for the operator. Our method is based on the semigroup language approach.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.
hi@scite.ai
10624 S. Eastern Ave., Ste. A-614
Henderson, NV 89052, USA
Copyright © 2024 scite LLC. All rights reserved.
Made with 💙 for researchers
Part of the Research Solutions Family.