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