Nazarov's uncertainty principles in higher dimension

In this paper we prove that there exists a constant C   such that, if S,ΣS,Σ are subsets of RdRd of finite measure, then for every function f∈L2(Rd)f∈L2(Rd),∫Rd|f(x)|2dx⩽CeCmin(|S||Σ|,|S|1/dw(Σ),w(S)|Σ|1/d)∫Rd⧹S|f(x)|2dx+∫Rd⧹Σ|f^(x)|2dx,where f^ is the Fourier transform of f   and w(Σ)w(Σ) is the mean width of ΣΣ. This extends to dimension d⩾1d⩾1 a result of Nazarov [Local estimates for exponential polynomials and their applications to inequalities of the uncertainty principle type, Algebra i Analiz 5 (1993) 3–66 (in Russian); translation in St. Petersburg Math. J. 5 (1994) 663–717] in dimension d=1d=1.