朗道算子的磁伯恩斯坦不等式与厚集上的谱不等式
Magnetic Bernstein inequalities and spectral inequality on thick sets for the Landau operator
AI总结：
本文证明朗道算子在厚集上的谱不等式，确定满足该不等式的厚集类别并推导常数的渐近最优表达式，其证明依赖磁伯恩斯坦不等式，还首次证明磁热方程零能控性并放宽安德森局域化证明假设。
摘要：
我们证明了朗道算子的谱不等式。这意味着对于谱子空间中所有能量不超过E的函数f，其在合适的S⊂ℝ²上的L²积分可由一个显式常数乘以f自身的L²范数下界估计。我们确定了所有满足该不等式的可测集S⊂ℝ²的类别，即所谓的厚集或相对稠密集，并根据能量、磁场强度以及决定厚集S的参数，推导出常数的渐近最优表达式。我们的证明依赖于所谓的磁伯恩斯坦不等式。作为结果，我们首次证明了磁热方程的零能控性（控制成本有精确界），并可放宽连续合金型模型中现有安德森局域化证明的假设。
英文摘要：
We prove a spectral inequality for the Landau operator. This means that for all $f$ in the spectral subspace corresponding to energies up to $E$, the $L^2$-integral over suitable $S \subset \mathbb{R}^2$ can be lower bounded by an explicit constant times the $L^2$-norm of $f$ itself. We identify the class of all measurable sets $S \subset \mathbb{R}^2$ for which such an inequality can hold, namely so-called thick or relatively dense sets, and deduce an asymptotically optimal expression for the constant in terms of the energy, the magnetic field strength and in terms of parameters determining the thick set $S$. Our proofs rely on so-called magnetic Bernstein inequalities. As a consequence, we obtain the first proof of null-controllability for the magnetic heat equation (with sharp bound on the control cost), and can relax assumptions in existing proofs of Anderson localization in the continuum alloy-type model.
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