Nazarov's uncertainty principles in higher dimension

Abstract

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