Suppose \(z_0\in \mathbb{R}\), \(f(z)\) is holomorphic on \(D(z_0,5)\), \(I\subseteq \mathbb{R}\) is an interval with length 1, \(z_0\in I\) and \(S\subseteq I\) is a measurable subset with nonzero measure. If \(\lvert f(z_0)\rvert\ge 1\) and \(M=\max_{|z-z_0|\le 4} |f(z)|\), then

\begin{equation*} \sup_{x\in I} \lvert f(x)\rvert \le \biggl(\frac{12}{\lvert S\rvert}\biggr)^{2 \frac{\log M}{\log 2}} \sup_{x\in S} \lvert f(x)\rvert. \end{equation*}
Proof
This theorem is proven in [1, Lemma 1] and in [2, Lemma 1]. In both proves the 12 in the constant as well as the factor 2 in the exponent does not appear as discussed in [3, Remark 4.2].

See also Link to heading

References Link to heading

  1. O. Kovrijkine, Some estimates of Fourier transforms, Ph.D. thesis, United States -- California: California Institute of Technology, 2000.
  2. O. Kovrijkine, Some results related to the Logvinenko-Sereda theorem, Proceedings of the American Mathematical Society, vol. 129, no. 10, pp. 3037–3047, 2001. doi:10.1090/S0002-9939-01-05926-3
  3. M. Egidi and I. Veselić, Scale-free Unique Continuation Estimates and Logvinenko–Sereda Theorems on the Torus, Annales Henri Poincaré, vol. 21, no. 12, pp. 3757–3790, 2020. doi:10.1007/s00023-020-00957-7