Some results related to the Logvinenko-Sereda theorem (Q2723503)

The Logvinenko-Sereda theorem states that if \(J\) is an interval of length \(b\) supporting the Fourier transform of \(f\in L^p\) (\(p\in [1,\infty]\)) and if a measurable set \(E\) intersects every interval of length \(a\) in a set of length at least \(\gamma a\) for some fixed \(\gamma>0\) and \(a>0\), then \(\|f\chi_E\|_{L^p}/\|f\|_{L^p}\) is at least \(1/e^{c(ab+1)/\gamma}\) in which \(c\) does not depend on \(f\). In other words, \(f\) cannot be too well-localized along the set \(E\). The author sharpens this estimate to polynomial dependence \(\gamma\), that is, \(\|f\chi_E\|_{L^p}/\|f\|_{L^p}\) is at least \((\gamma/c)^{c(ab+1)}\). The author also generalizes the Logvinenko-Sereda theorem to the case in which \(f\) is bandlimited to a finite collection of disjoint intervals each having length \(b\). In this case, the lower bound naturally depends on the number of intervals \(n\) because the total bandwidth is \(nb\); however it also depends marginally on \(p\). The first step is to prove that one can essentially restrict \(E\) to a subset of \(\mathbb R\) on which a local Bernstein inequality applies. On this set one can then apply a Taylor estimate and get improved bounds on constants by using a pointwise estimate for a function analytic on a disc.