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, \(z_0\in I\), \(S\subseteq I\) is a measurable subset with nonzero measure and \(p\ge 1\). If \(\lvert f(z_0)\rvert\ge 1\) and \(M=\max_{|z-z_0|\le 4} |f(z)|\), then

\begin{equation*} \lVert f\rVert_{L^p(I)} \le \biggl(\frac{24}{|S|}\biggr)^{2 \frac{\log M}{\log 2} + \frac{1}{p}} \lVert f\rVert_{L^p(S)}. \end{equation*}

Proof

According to (0x6698f5c7) , we have

\begin{equation*} \biggl(\frac{\lvert S\rvert}{12}\biggr)^{2 \frac{\log M}{\log 2}}\sup_{x\in I} \lvert f(x)\rvert \le \sup_{x\in S} \lvert f(x)\rvert, \end{equation*}

since \(M>1\) (why?) the exponent on the left hand side is positive and we may apply (0x6698f127) . This yields

\begin{equation*} \sup_{x\in I}|f(x)|\le 12^{2\frac{\log M}{\log 2}}\biggl(\frac{2}{|S|}\biggr)^{2 \frac{\log M}{\log 2} + \frac{1}{p}} \lVert f\rVert_{L^p(S)} \le \biggl(\frac{24}{|S|}\biggr)^{2 \frac{\log M}{\log 2} + \frac{1}{p}} \lVert f\rVert_{L^p(S)}. \end{equation*}

Using Hölder’s inequality we estimate the left-hand side from below with

\begin{equation*} \lVert f\rVert_{L^p(I)}\le \sup_{x\in I}|f(x)|. \end{equation*}

Note, since \(|I|=1\) no constant appears.

Remarks

The proof is given in [1] in the proof of Theorem 1.
Regarding [2] this lemma is inspired by [3].
Indeed, the factor in the exponential is in [4] \(\frac{\log M}{\log 2}\). However, this could not be reproduced by [2] and the authors suggest \(2 \frac{\log M}{\log 2}\) instead.
The assumption \(|f(z_0)|\ge 1\) ensures \(\log M> 0\) if \(f\) is not constant. (This is stated in [2]. Why this is true? - I think it has to do with the assumption that \(f\) is analytic.)

References Link to heading

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
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
F. Nazarov, Local estimates for exponential polynomials and their applications to inequalities of the uncertainty principle type, Rossiĭskaya Akademiya Nauk. Algebra i Analiz, vol. 5, no. 4, p. 3–66, 1993.
O. Kovrijkine, Some estimates of Fourier transforms, Ph.D. thesis, United States -- California: California Institute of Technology, 2000.

0x6698f0dc

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