\[ \DeclareMathOperator{\supp}{supp} \]

Let \(J\) be an interval with \(|J|=b\). If \(f\in L^p(\mathbb{R})\), \(p \in [1,\infty)\), and \(\supp \hat{f} \subset J\), then a family of disjoint intervals \((I_i)_{i \in \mathbb{N}}\) of length \(a>0\) and a numerical constant \(C>0\) exists, such that

\begin{equation*} \lVert f^{(k)}\rVert_{L^p(I_i)}\le (Cb)^k \lVert f\rVert_{L^p(I_i)}, \end{equation*}

for every \(k\ge 1\) and \(i \in \mathbb{N}\) and

\begin{equation*} \lVert f\rVert_{L^p(\bigcup_{i \in \mathbb{N}} I_i)} \ge 2^{-\frac{1}{p}} \lVert f\rVert_{L^p(\mathbb{R})}. \end{equation*}

We call such intervals good [1].

Implications Link to heading

Remarks
  • Phillipe Jaming once mentioned, that this Lemma is the genius trick Kovrijkine discovered. To be more precise, he discovered that a local Bernstein estimate holds, with a constant which is bit worse than the original one.

See also Link to heading

References Link to heading

  1. 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