Suppose \(B>0\) and \(S\subseteq \mathbb{R}^2\) is \((\gamma,a)\)-thick and \(H_B\) the corresponding Landau operator . Then, there are numerical constants \(C_1,C_2,C_3,C_4>0\) such that for every \(E>0\) and every \(f\in \Ran 𝟙_{(-\infty,E]}(H_B)\) we have
\[ \lVert f\rVert^2_{L^2(\mathbb{R}^2)}\le \biggl(\frac{C_1}{\gamma}\biggr)^{C_2+C_3\lvert a\rvert_1+C_4(\lvert a\rvert^2B)}\lVert f\rVert^2_{L^2(S)}.\][1, Theorem 3]
Proof
The proof is a reformulation of the one of Kovrijkine
.
A main difference is the used Bernstein inequality
.
A delicate step is the Taylor expansion estimate. For details see section 4.3 in [1].
Remarks
- Note, that the result is formulated for \(\mathbb{R}^2\).
- In [1] there is also version where the domain is bounded.
- The given inequality, is sufficient for thickness.
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