We prove that for any convex polytope $\Omega \subset \mathbb{R}^d$ which is centrally symmetric and whose faces of all dimensions are also symmetric, there exists a Riesz basis exponential functions in the space $L^2(\Omega)$. The result new $d$ greater than one.

Let \(\gamma_{-1}\) be the absolutely continuous measure on \(\mathbf{R}^n\) whose density is reciprocal of a Gaussian function. further \(\mathscr{A}\) natural self-adjoint Laplacian \(L^2(\gamma_{-1})\). In this paper, we prove that Riesz transforms associated with order one or two are weak type \((1,1)\), but those higher not.

It is always attractive and challenging to explore the intricate behavior data and uncover people’s motivations, preference and habits, which can greatly benefit many tasks including link prediction, item recommendation, etc. Traditional work usually studies people’s behaviors without time information in a static or discrete manner, assuming the underlying factors stay invariant in a long perio...

Abstract The aim of this paper is to provide necessary and sufficient conditions on the generator a multiplication operator acting in spaces functions bounded Young Riesz variation so that it is, among other things, invertible, continuous, finite rank, compact, Fredholm or has closed range. Furthermore, we characterize various spectra such operators give some estimates their measure non-compact...

The classical theorems of Riesz [l] on compact operators have been extended by Leray [2] and Williamson [3] to the context of topological linear spaces. Ringrose [4] has shown that if an operator on such a space is compact, the square of its adjoint is also compact, where the topology on the dual space is that of uniform convergence on bounded sets. Thus if an operator is continuous and some po...

We prove L estimates for the maximal Riesz transform in terms of the Riesz transform itself, for 1 < p ≤ ∞. We show that the corresponding weak L1 estimate fails for the maximal Riesz transform, but surprisingly does hold for the maximal Beurling transform.

In this paper, the space-time Riesz fractional partial differential equations with periodic conditions are considered. The equations are obtained from the integral partial differential equation by replacing the time derivative with a Caputo fractional derivative and the space derivative with Riesz potential. The fundamental solutions of the space Riesz fractional partial differential equation (...

In this paper we solve the nonlinear Dirichlet problem (uniquely) for functions with prescribed asymptotic singularities at a finite number of points, and with arbitrary continuous boundary data, on a domain in R. The main results apply, in particular, to subequations with a Riesz characteristic p ≥ 2. It is shown that, without requiring uniform ellipticity, the Dirichlet problem can be solved ...