Sharp embeddings between weighted Paley-Wiener spaces
Emanuel Carneiro, Cristian González-Riquelme, Lucas Oliveira, Andrea Olivo, Sheldy Ombrosi, Antonio Pedro Ramos, Mateus Sousa
TL;DR
The paper develops a sharp analysis of embeddings between weighted Paley–Wiener spaces, recasting the problem as a Fourier-uncertainty principle and showing that radial symmetrization reduces the multidimensional problem to dimension one. Central to the approach is the de Branges space framework, which yields dimension-shift equalities, existence and symmetry of extremizers, and a robust path to sharp constants through a determinant structure tied to Bessel-type companion functions. In particular, the authors provide precise 1D asymptotics for the sharp constants, derive explicit extremizers, and connect these to Beurling–Selberg-type majorants and sharp higher-order Poincaré inequalities. The results illuminate the role of de Branges spaces in extremal problems for bandlimited functions and offer concrete tools for sharp functional-analytic inequalities with potential applications in harmonic analysis and PDEs. Overall, the work delivers a comprehensive, almost complete sharp-constant theory for multidimensional weighted uncertainty phenomena, with rigorous dimension-reduction principles and a novel extension to multiplication-by-$z^k$ operators in de Branges spaces.
Abstract
In this paper we address the problem of estimating the operator norm of the embeddings between multidimensional weighted Paley-Wiener spaces. These can be equivalently thought as Fourier uncertainty principles for bandlimited functions. By means of radial symmetrization mechanisms, we show that such problems can all be shifted to dimension one. We provide precise asymptotics in the general case and, in some particular situations, we are able to identify the sharp constants and characterize the extremizers. The sharp constant study is actually a consequence of a more general result we prove in the setup of de Branges spaces of entire functions, addressing the operator given by multiplication by $z^k$, $k \in \mathbb{N}$. Applications to sharp higher order Poincaré inequalities and other related extremal problems are discussed.