Nazarov's uncertainty principles in higher dimension

Abstract

In this paper we prove that there exists a constant $C$ such that, if $S,\Sigma$ are subsets of ${\mathbb{R}}^d$ of finite measure, then for every function $f\in L^2({\mathbb{R}}^d)$, $${\int }_{{\mathbb{R}}^d}|f(x){|}^{2}dx\le C{e}^{C\min (|S||\Sigma |,|S{|}^{1/d}w(\Sigma ),w(S)|\Sigma {|}^{1/d})}({\int }_{{\mathbb{R}}^d\setminus S}|f(x){|}^{2}dx+{\int }_{{\mathbb{R}}^d\setminus \Sigma }|\hat{f}(x){|}^{2}dx)$$ where $\hat{f}$ is the Fourier transform of $f$ and $w(\Sigma )$ is the mean width of $\Sigma$. This extends to dimension $d\ge 1$ a result of Nazarov \cite{pp.Na} in dimension $d=1$.

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