We characterize geometric properties of Banach spaces in terms of boundedness of square functions associated to general Schrödinger operators of the form ℒ = -Δ + V, where the nonnegative potential V satisfies a reverse Hölder inequality. The main idea is to sharpen the well known localization method introduced by Z. Shen. Our results can be regarded as alternative proofs of the boundedness in H¹, and BMO of classical ℒ-square functions.
We establish L2 and Lp bounds for a class of square functions which arises in the study of singular integrals and boundary value problems in non-smooth domains. As an application, we present a simplified treatment of a class of parabolic smoothing operators which includes the caloric single layer potential on the boundary of certain minimally smooth, non-cylindrical domains.
We study the bases and frames of reproducing kernels in the model subspaces of the Hardy class in the upper half-plane. The main problem under consideration is the stability of a basis of reproducing kernels under “small” perturbations of the points . We propose an approach to this problem based on the recently obtained estimates of derivatives in the spaces and produce estimates of admissible perturbations generalizing certain results of W.S. Cohn and E. Fricain.
We study the relation between standard ideals of the convolution Sobolev algebra and the convolution Beurling algebra L¹((1+t)ⁿ) on the half-line (0,∞). In particular it is proved that all closed ideals in with compact and countable hull are standard.
The paper considers stationary vector subdivision schemes, that is, subdivision schemes acting on vector valued sequences by using a matrix valued mask, and derives the analog of the well-known "zero condition" for an arbitrary number of variables as well as arbitrary expanding dilation matrices.
We prove Strichartz's conjecture regarding a characterization of Hardy-Sobolev spaces.
The notion of “strong boundary values” was introduced by the authors in the local theory of hyperfunction boundary values (boundary values of functions with unrestricted growth, not necessarily solutions of a PDE). In this paper two points are clarified, at least in the global setting (compact boundaries): independence with respect to the defining function that defines the boundary, and the spaces of test functions to be used. The proofs rely crucially on simple results in spectral asymptotics.
The martingale Hardy space and the classical Hardy space are introduced. We prove that certain means of the partial sums of the two-parameter Walsh-Fourier and trigonometric-Fourier series are uniformly bounded operators from to (0 < p ≤ 1). As a consequence we obtain strong convergence theorems for the partial sums. The classical Hardy-Littlewood inequality is extended to two-parameter Walsh-Fourier and trigonometric-Fourier coefficients. The dual inequalities are also verified and a...