The Beurling and Malliavin Theorem in Several Dimensions
Ioann Vasilyev
TL;DR
The paper develops a multidimensional analogue of the First Beurling–Malliavin Theorem by establishing a radial majorant theorem in $\mathbb{R}^d$ and a general nonradial majorant theorem. For radial $\omega$ with $\log(1/\omega)\in L^1(\mathbb{R}^d,(1+|x|^2)^{-(d+1)/2}\,dx)$ and Lipschitz, and any $\sigma>0$, there exists a nonzero $f\in L^2(\mathbb{R}^d)$ with $\mathrm{supp}(\widehat{f})\subset B(0,\sigma)$ and $|f|\le \omega$ on $\mathbb{R}^d$; the radial case is handled by reducing to a one-dimensional cosine transform problem and constructing a function $g$ with $\mathrm{supp}(Tg)\subset [0,\sigma]$ and then lifting to a radial function $\psi(x)=g(|x|)$, with odd dimensions using derivative/moments and even dimensions via Sonin’s integral formula for Bessel functions. The general nonradial majorant follows from the radial one by a Lipschitz majorant argument, addressing Hörmander’s question. The results have potential applications to fractal uncertainty principles and spectral gaps in hyperbolic geometry, and they introduce a simpler, dimension-sensitive approach to Beurling–Malliavin-type majorants.
Abstract
The present paper is devoted to a new multidimensional generalization of the Beurling and Malliavin Theorem, which is a classical result in the Uncertainty Principle in Fourier Analysis. In more detail, we establish by an elegant but simple new method a sufficient condition for a radial function to be a Beurling and Malliavin majorant in several dimensions (this means that the function in question can be minorized by the modulus of a square integrable function which is not zero identically and which has the support of the Fourier transform included in an arbitrary small ball). As a corollary of the radial case, we also get a new sharp sufficient condition in the nonradial case. The latter result provides an answer to a question posed by L. Hörmander. Our proof is different in the cases of odd and even dimensions. In the even dimensional case we make use of one classical formula from the theory of Bessel functions due to N. Ya. Sonin.