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 $\R^d$ of finite measure, then for every function $f\in L^2(\R^d)$, $$\int_{\R^d}|f(x)|^2 dx \leq C e^{C \min(|S||\Sigma|, |S|^{1/d}w(\Sigma), w(S)|\Sigma|^{1/d})} (\int_{\R^d\setminus S}|f(x)|^2 dx + \int_{\R^d\setminus\Sigma}|\hat{f}(x)|^2 dx) $$ where $\hat{f}$ is the Fourier transform of $f$ and $w(\Sigma)$ is the mean width of $\Sigma$. This extends to dimension $d\geq 1$ a result of Nazarov \cite{pp.Na} in dimension $d=1$

Dynamics Characteristics of Groundwater Exploitation in Tianjin, P.R. China

“…where F stands for the Fourier transform. Recall that the uncertainty principle built up in [26] says: for any S , Σ ⊂ R n with |S | < ∞ and |Σ| < ∞, there is a positive constant C(n, S , Σ) := Ce C min{|S ||Σ|, |S | 1/n ω(Σ), |Σ| 1/n ω(S )} , (5.14)…”

“…with C = C(n), so that for any g ∈ L 2 (R n ), (Here ω(S ) denotes the mean width of S , we refer the readers to [26] for its detailed definition. In particular, when S is a ball in R n , ω(S ) is the diameter of the ball.…”

“…Then there is L > 0 and c > 0 so that [26]) and is independently interesting. With regard to the observability inequality at two points in time for Schrödinger equations, we mention papers [47] and [24].…”

On the Fourier transform of the symmetric decreasing rearrangements

“…Now |ϕ| * is supported in S * , so that the particular case of Conjecture 5.4 that has already been proved in [13] (and |S…”

“…This conjecture has been proved in dimension d = 1 by F. Nazarov [23] and for d 2 and either S or Σ convex by the author in [13]. (The result was stated with a constant of the form [23] or [12]) and easy to prove, it is equivalent to show that…”