Let \(S\subseteq \mathbb{R}^2\) be a subset and \(H_B\) the Landau operator for a give \(B>0\). If there numbers \(B>0\), \(E\ge B\) and \(C>0\) such that for every \(f\in \Ran 𝟙_{(-\infty, E]}(H_B)\) the following inequality holds
\[ \lVert f\rVert_{L^2(\mathbb{R}^2)} \le C \lVert f\rVert_{L^2(S)}, \]then \(S\) is thick . [1, Theorem 11]
Proof
Assume \(S\) is not thick and construct a sequence of \(f\in \Ran 𝟙_{(-\infty ,E]}(H_B)\) such that the inequality fails. See [1] for details.
References Link to heading
- P. Pfeiffer and M. Täufer,
Magnetic Bernstein inequalities and spectral inequality on thick sets for the Landau operator,
2023. doi:10.48550/arXiv.2309.14902