Suppose \(B>0\) and \(H_B\) is the corresponding Landau operator . For every \(m\in \mathbb{N}\) and every \(f\in \Ran 𝟙_{(-\infty,E]}(H_B)\) the following Bernstein inequality in terms of ordinary derivatives hold
\[ \sum_{\alpha \in \{1,2\}^m} \lVert \partial^\alpha \lvert f\rvert^2 \rVert^2_{L^1(\mathbb{R}^n)} \le 2^{3m/2}(E+mB)^{m/2} \lVert f\rVert^2_{L^2(\mathbb{R}^n)}. \][1, Theorem 12]
Proof
Using the relation between ordinary and magnetic derivatives we obtain by induction
\[ i^m \partial^{\alpha} = \sum_{\beta \le \alpha} (-1)^{m-\lvert \beta \rvert} \widetilde{\partial }{}^\beta u \overline{\widetilde{\partial }{}^{\alpha \setminus \beta} u}. \]The result is than a consequence of the magnetic Bernstein inequality with respect to magnetic derivatives .
Remarks
- This result is used to proof (0x67d96c9b) .
- Using Sobolev embeddings gives a similar Bernstein inequality for the \(L^{\infty}\) of ordinary derivatives of \(\lvert f\rvert^2\) (see [1]).
- As mentioned here , a Bernstein inequality for the \(L^2\)-norm of ordinary derivatives of \(f\) does not hold in general. The authors of [1] provide a counter example in Remark 9.
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