# Magnetic Bernstein inequalities and spectral inequality on thick sets for
the Landau operator

Paul Pfeiffer  FernUniversität in Hagen, Germany,
orcid.org/0000-0001-8538-5314<EMAIL_ADDRESS>and  Matthias
Täufer  FernUniversität in Hagen, Germany, orcid.org/0000-0001-8473-2310
<EMAIL_ADDRESS>
###### Abstract.

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.

###### Key words and phrases:

Landau Hamiltonian, Spectral inequality, Quantitative Unique Continuation,
thick sets, Bernstein Inequalities, Null-controllability, Anderson
localization

###### 2020 Mathematics Subject Classification:

Primary: 35Pxx, 35A23. Secondary: 93B05, 82B44

##  1\. Introduction

The _Landau operator_

$H_{B}:=\left(i\nabla+\frac{B}{2}\begin{pmatrix}-x_{2}\\\
x_{1}\end{pmatrix}\right)^{2}$

occasionally also called _twisted Laplacian_ , describes the motion of a
particle in two dimensions, subject to a constant magnetic field. It is a
self-adjoint operator the spectrum of which consists of infinite degenerate
eigenvalues at the _Landau levels_ $B,3B,5B,\dots$. The Landau operator is
relevant for a host of phenomena in Physics, including explainations for
Landau diamagnetism [Lan30], Hofstadter’s butterfly [Hof76], as well as von
Klitzing’s description of the quantized Hall effect [Kv86].

In this article, we prove optimal _spectral inequalities_ , that are lower
bounds on the mass of functions, sampled on a subdomain
$S\subset\mathbb{R}^{2}$, uniform for all function in the spectral subspace
below a given energy $E$

(1) $\lVert f\rVert_{L^{2}(\mathbb{R}^{2})}^{2}\leq C(E,B,S)\lVert
f\rVert_{L^{2}(S)}^{2}\quad\text{for all}\quad
f\in\operatorname{Ran}\mathbf{1}_{(-\infty,E]}(H_{B}).$

Clearly, not for every $S\subset\mathbb{R}^{2}$ such an inequality can hold.
We identify the necessary and sufficient criterion on $S\subset\mathbb{R}^{2}$
for (1) to hold, namely _thickness_ or _relative density_. Furthermore, we
provide an explicit expression for the constant $C(E,B,S)$, and show that, in
some sense, it is optimal in $E,B$, and parameters determining the thick set
$S$. For details, see the remark below Theorem 3. In particular, for fixed $B$
and $S$, the constant $C(E,B,S)$ grows as $\exp(C\sqrt{E})$ for $E\to\infty$,
which is essential for the applications to control theory.

So far, examples of differential operators on $\mathbb{R}^{d}$ where the class
of all measurable sets $S\subset\mathbb{R}^{d}$ leading to a spectral
inequality has been identified, are rare: One example is the free Laplacian,
where spectral inequalities can be inferred from the Kovrijkine-Logvinenko-
Sereda theorem [Kov00, EV18, WWZZ19], another one is the harmonic Laplacian
where explicit calculations are possible [BJPS21, ES21]. Our main result,
Theorem 3, adds the Landau operator to this exclusive club.

Estimates as in (1), albeit without an explicit depencence of the constant
$E$, have also been used in the context of Anderson localization for random
Schrödinger operators where they are known as _unique continuation
principles_. Even without the quantitative dependence of the constant on $E$
(which might give rise to further developments), our results yield immediate
improvements of existing works since we no longer need to assume that $S$ is
open, an ubiquitous technical assumption so far.

From a technical point, our main contribution are what we call _magnetic
Bernstein inequalities_ (Theorem 7). Indeed, we are aware of two established
strategies for proving spectral inequalities: On the one hand, the Kovrijkine-
Logvinenko-Sereda theorem, on the other hand Carleman inequalities. While the
latter strategy offers more flexibility in terms of the choice of the
operator, it usually requires the sampling set $S$ to be open. The Kovrijkine-
Logvinenko-Sereda theorem on the other hand crucially relies on so-called
Bernstein inequalities which bound (the $L^{2}$-norm of) derivatives of
functions in spectral subspaces to infer analyticity. While almost trivial for
the pure Laplacian, it turns out that in the case of the Landau operator,
(ordinary) Bernstein inequalities no longer hold, see Remark 9. However, our
workaround will be to work with covariant _magnetic derivatives_ and then use
corresponding _magnetic Bernstein inequalities_ in $L^{2}$-norm to infer
(ordinary) _Bernstein-type inequalties in $L^{1}$-norm_.

The paper is organized as follows: Section 2 contains definitions and our main
results, namely the optimal spectral inequality for the Landau operator on
$\mathbb{R}^{2}$ (Theorem 3), as well as its analogon on boxes of finite
volume (Theorem 4). In Section 3, we prove the magnetic Bernstein inequalities
(Theorem 7), and Bernstein-type inequalities for the Landau operator in
$L^{1}$-norm (Theorem 12). Also, Section 3 contains remarks and lemmas on
optimality of our main results. Section 4 uses the Bernstein-type inequalities
to prove Theorem 3. In Section 5, we explain the necessary modifications for
the finite-volume analogon. Finally, Section 6 contains applications:
Subsection 6.1 is about controllability and sharp control cost estimates for
the magnetic heat equation (Theorems 21 and 22) whereas Subsection 6.2
contains applications to random Schrödinger operators, namely Wegner
estimates, regularity of the integrated density of states, and Anderson
localization in the continuum Anderson model in the case where the single-site
potential is no longer assumed to be positive on an open, bute merely on a
measurable set.

##  2\. Definitions and main Results

For $x=(x_{1},x_{2})\in\mathbb{R}^{2}$, we denote by $\lvert
x\rvert=(x_{1}^{2}+x_{2}^{2})^{1/2}$ its Euclidean norm and by $\lvert
x\rvert_{1}=\lvert x_{1}\rvert+\lvert x_{2}\rvert$ its $1$-norm. The
expression $\operatorname{Vol}(S)$ refers to the Lebesgue measure of a
measurable set $S\subset\mathbb{R}^{2}$. We will occasionally also use the
one-dimensional Hausdorff measure of subsets of line segments in
$\mathbb{R}^{2}$ and denote the one-dimensional measure of such a set $T$ by
$\operatorname{Vol}_{1}(T)$ for clarity. For a measurable set $B$,
$\mathbf{1}_{B}$ denotes its indicator function. In particular, given a self-
adjoint operator $A$, and $E\in\mathbb{R}$, we denote by
$\mathbf{1}_{(-\infty,E]}(A)$ the orthogonal projector onto the spectral
subspace up to energy $E$, coresponding to $A$. We write
$C_{0}^{\infty}(\mathbb{R}^{2})$ for the space of smooth functions with
compact support and $\mathcal{S}(\mathbb{R}^{2})$ for the space of Schwarz
functions, that are smooth functions all derivatives of which decay faster at
infinity than any polynomial. We also denote by
$\partial_{i}:=\frac{\mathrm{d}}{\mathrm{dx_{i}}}$ the partial derivative with
respect to the $x_{i}$ coordinate.

######  Definition 1.

Let $\ell=(\ell_{1},\ell_{2})\in(0,\infty)^{2}$, and $\rho\in(0,1]$. A
measurable set $S\subseteq\mathbb{R}^{2}$ is called _$(\ell,\rho)$ -thick_ if
for every rectangle $Q$ with side lengths $(\ell_{1},\ell_{2})$, parallel to
the axes, we have

$\operatorname{Vol}\\{S\cap Q\\}\geq\rho\operatorname{Vol}Q\quad\text{for all
$x\in\mathbb{R}^{2}$}.$

If $S$ is $(\ell,\rho)$-thick for some $\ell,\rho$, it is also simply called
_thick_. In the literature, one also finds the equivalent notion of _relative
dense_ sets. Thick sets seem to have originated in Fourier analysis [Pan61,
Kac73, LS74, Kov00, Kov01] but have attracted interest in the recent years
[EV18, WWZZ19, LM19, EV20, MPS20, BJPS21, GJM22, Täu23, WZ23].

######  Definition 2.

For $B>0$ let

$\tilde{\partial}_{1}=i\partial_{1}-\frac{B}{2}x_{2}\quad\text{and}\quad\tilde{\partial}_{2}=i\partial_{2}+\frac{B}{2}x_{1}$

be the _magnetic derivatives at magnetic field strength $B$_. The Landau
Hamiltonian is

$H_{B}=\tilde{\partial}_{1}^{2}+\tilde{\partial}_{2}^{2}.$

Clearly, $H_{B}$ can be written in the form

$H_{B}=(i\nabla-A)^{2}.$

with the _magnetic potential_ $A=\frac{B}{2}(-x_{2},x_{1})$. Indeed, this is
the so-called _symmetric gauge_ and any $A^{\prime}$ with
$\partial_{1}A^{\prime}_{2}-\partial_{2}A^{\prime}_{1}=B$ will lead to a
unitarily equivalent operator. It is well-known that $H_{B}$ is a self-adjoint
operator in $L^{2}(\mathbb{R}^{2})$, an operator core being
$C_{0}^{\infty}(\mathbb{R}^{2})$, with spectrum
$\sigma(H_{B})=\\{B,3B,5B,\dots\\}$. Our first main result is:

######  Theorem 3.

Let $B>0$ and let $S\subseteq\mathbb{R}^{2}$ be $(\ell,\rho)$-thick. Then,
there are $C_{1},C_{2},C_{3},C_{4}>0$, such that for all $E>0$ we have

$\lVert
f\rVert_{L^{2}(\mathbb{R}^{2})}^{2}\leq\left(\frac{C_{1}}{\rho}\right)^{C_{2}+C_{3}\lvert\ell\rvert_{1}\sqrt{E}+C_{4}(\lvert\ell\rvert_{1}^{2}B)}\lVert
f\rVert_{L^{2}(S)}^{2}\quad\text{for all
$f\in\operatorname{Ran}\mathbf{1}_{(-\infty,E]}(H_{B})$}.$

Let us comment on the expression

$\left(\frac{C_{1}}{\rho}\right)^{C_{2}+C_{3}\lvert\ell\rvert_{1}\sqrt{E}+C_{4}(\lvert\ell\rvert_{1}^{2}B)}$

  1. (1)

In the limit $B\to 0$, the constant converges to the expression for the pure
Laplacian in the Logvinenko-Sereda-Kovrikijne theorem [Kov00]. So, one can
indeed also set $B=0$ in the statement of Theorem 3 and in this sense, the
dependence $\exp(C\sqrt{E})$ is optimal. Indeed, this is the first time that
we aware of any dependence on $E$ in a spectral inequality for the Landau
operator, and it is useful for controllability of the heat equation, see
Section 6.1.

  2. (2)

The relation of $E$ to $\ell$, and $B$ to $\ell$ is optimal. Since $H_{B}$ is
of second order in $\partial_{1}$, $\partial_{2}$ and of the same order in
$B$, simultaneous scaling in $E$ and in $B$ corresponds to the square of the
inverse scaling in space.

  3. (3)

The term $\lvert\ell\rvert_{1}^{2}B$ in the exponent is optimal when
$\lvert\ell\rvert_{1}$ is sent to $\infty$, see Remark 10.

  4. (4)

The dependence on $\lvert\ell\rvert_{1}$ yields a meaningful limit in the
homogenization regime, that is when $\ell\to 0$: In this regime, the maximal
size of holes in the set $S\subseteq\mathbb{R}^{2}$ becomes small. One the one
hand, since $\sqrt{E}\lvert\ell\rvert_{1}\gg B\lvert\ell\rvert_{1}^{2}$ as
$\ell\to 0$, we observe that in the homogenization regime, the influence of
the magnetic field $B$ in the spectral inequality (at fixed $B$ and $E\geq B$)
fades. On the other hand, when sending $\ell\to 0$, the exponent will
disappear and the observation operator $\mathbf{1}_{S}$ strongly converges to
an $E$-independent operator, see also the discussion in [NTTV20a].

  5. (5)

Thickness of $S$ is necessary for any quantitative unique continuation
principle of the form

$\lVert f\rVert_{L^{2}(\mathbb{R}^{2})}^{2}\leq C(E,\ell,B)\lVert
f\rVert_{L^{2}(S)}^{2}\quad\text{for all
$f\in\operatorname{Ran}\mathbf{1}_{(-\infty,E]}(H_{B})$}.$

This is proved in Theorem 11.

We also have the corresponding result for finite-volume restrictions $H_{B,L}$
of $H_{B}$ onto boxes
$\Lambda_{L}=(0,L_{1})\times(0,L_{2})\subseteq\mathbb{R}^{2}$, where
$L=(L_{1},L_{2})\in\mathbb{R}_{>0}^{2}$ satisfies the so-called  _integer flux
condition_ , and $H_{L}$ is defined with appropriate magnetic boundary
conditions, see Section 5 for precise definitions.

######  Theorem 4.

Let $B>0$, and let $S\subseteq\mathbb{R}^{2}$ be $(\ell,\rho)$-thick. Then
there are $C_{1},C_{2},C_{3},C_{4}>0$, such that for all
$L=(L_{1},L_{2})\in(0,\infty)^{2}$ satisfying

$B(L_{2}-L_{1})\in 2\pi\mathbb{Z},\quad\text{and}\quad\ell_{1}\leq L_{1},\
\ell_{2}\leq L_{2},$

we have

$\lVert
f\rVert_{L^{2}(\Lambda_{L})}^{2}\leq\left(\frac{C_{1}}{\rho}\right)^{C_{2}+C_{3}\lvert\ell\rvert_{1}\sqrt{E}+C_{4}(\lvert\ell\rvert_{1}^{2}B)}\lVert
f\rVert_{L^{2}(\Lambda_{L}\cap S)}^{2}\quad\text{for all
$f\in\operatorname{Ran}\mathbf{1}_{(-\infty,E]}(H_{B,L})$}.$

Estimates as in (4) have been commonly used in the context of the spectral
theory of random Schrödinger operators where they are also known as
_Quantitative Unique continuation principle_ , see [CHK03, CHKR04, GKS07,
CHK07, TV16].

However, previous results neither carried the explicit dependence on the
parameters $E,B$ nor were they valid beyond open sets whereas Theorem 4 allows
for any subset $S\subset\Lambda_{L}$ of positive measure. We explain in
Section 6.2 how this leads to improvements of existing results.

##  3\. Magnetic Bernstein inequalities

In this section, we prove magnetic Bernstein inequalities. The first step will
be to express

$\sum_{\alpha\in\\{1,2\\}^{m}}\lVert\tilde{\partial}_{\alpha_{1}}\tilde{\partial}_{\alpha_{2}}\dots\tilde{\partial}_{\alpha_{n}}f\rVert_{L^{2}(\mathbb{R}^{2})}^{2}$

for suitable $f\in\operatorname{Ran}\mathbf{1}_{(-\infty,E]}(H_{B})$ in terms
of $H_{B}$. For this purpose, we need to better understand the algebra
generated by the magnetic derivatives
$\tilde{\partial}_{1},\tilde{\partial}_{2}$.

The magnetic derivatives $\tilde{\partial}_{1}$ and $\tilde{\partial}_{2}$
satisfy the commutator relation

(2) $\displaystyle[\tilde{\partial}_{1},\tilde{\partial}_{2}]$
$\displaystyle=\left[i\partial_{1}-\frac{B}{2}x_{2},i\partial_{2}+\frac{B}{2}x_{1}\right]=\left[i\partial_{1},\frac{B}{2}x_{1}\right]-\left[\frac{B}{2}x_{2},i\partial_{2}\right]=iB.$

Consider the algebra $\mathcal{X}$ of all polynomials in
$\tilde{\partial}_{1},\tilde{\partial}_{2}$ modulo the commutator relation
(2). Clearly, polynomials of $H_{B}$ form a subalgebra of $\mathcal{X}$. We
define a linear operator $R$ mapping $\mathcal{X}$ to itself

$R(P)=\tilde{\partial}_{1}P\tilde{\partial}_{1}+\tilde{\partial}_{2}P\tilde{\partial}_{2}.$

The key idea is now that

(3)
$\displaystyle\sum_{\alpha\in\\{1,2\\}^{m}}\lVert\tilde{\partial}_{\alpha_{1}}\tilde{\partial}_{\alpha_{2}}\dots\tilde{\partial}_{\alpha_{n}}f\rVert_{L^{2}(\mathbb{R}^{2})}^{2}=\left\langle
f,R^{m}(Id)f\right\rangle$

for sufficiently regular $f$, say $f\in\mathcal{S}(\mathbb{R}^{2})$. This can
be verified by integration by parts and is explained in the proof of Theorem
7.

The following Lemma 5 is the first key result of this section, stating that
$R^{m}(\operatorname{Id})$ is not only a polynomial in the variables
$\tilde{\partial}_{1},\tilde{\partial}_{2}$, but actually a polynomial in
$H_{B}$. The subsequent Lemma 6 then provides an explicit bound on this
polynomial, allowing to replace $R^{m}(\operatorname{Id})$ in (3) by a
polynomial in $H_{B}$.

######  Lemma 5.

For all $m\geq 0$, the operator given by $R^{m}(\operatorname{Id})$ is a
polynomial in $H_{B}$, which we denote by $F_{m}$, that is

$F_{m}(H_{B}):=R^{m}(\operatorname{Id}).$

Furthermore,

(4)
$F_{m+1}(H_{B})=R(F_{m}(H_{B}))=\frac{1}{2}\left(\left(H_{B}-B\right)F_{m}(H_{B}-2B)+\left(H_{B}+B\right)F_{m}(H_{B}+2B)\right).$

###### Proof.

Since $R$ is linear, it suffices to consider monomials, and to see (4), it
certainly suffices to show

$2R(H_{B}^{n})=\left(H_{B}-B\right)(H_{B}-2B)^{n}+\left(H_{B}+B\right)(H_{B}+2B)^{n}$

for each $n\geq 0$. We have

$\displaystyle R(H_{B}^{n})$
$\displaystyle=\tilde{\partial}_{1}H_{B}^{n}\tilde{\partial}_{1}+\tilde{\partial}_{2}H_{B}^{n}\tilde{\partial}_{2}.$

Define

$X_{n}:=i\tilde{\partial}_{2}H_{B}^{n-1}\tilde{\partial}_{1}-i\tilde{\partial}_{1}H_{B}^{n-1}\tilde{\partial}_{2},\quad\text{and}\quad
Y_{n}:=\tilde{\partial}_{1}H_{B}^{n-1}\tilde{\partial}_{1}+\tilde{\partial}_{2}H_{B}^{n-1}\tilde{\partial}_{2}.$

The commutator identity (2) leads to

$H_{B}\tilde{\partial}_{1}=\tilde{\partial}_{1}H_{B}-2iB\tilde{\partial}_{2},\quad\text{and}\quad
H_{B}\tilde{\partial}_{2}=\tilde{\partial}_{2}H_{B}+2iB\tilde{\partial}_{1}.$

In particular, this implies $X_{1}=B$, $Y_{1}=H_{B}$, as well as

$\displaystyle\begin{pmatrix}X_{n+1}\\\
Y_{n+1}\end{pmatrix}=\begin{pmatrix}H_{B}&2B\\\ 2B&H_{B}\\\
\end{pmatrix}\begin{pmatrix}X_{n}\\\ Y_{n}\end{pmatrix}$

Diagonalizing

$\begin{pmatrix}H_{B}&2B\\\ 2B&H_{B}\\\
\end{pmatrix}=\frac{1}{2}\begin{pmatrix}-1&1\\\ 1&1\\\
\end{pmatrix}\begin{pmatrix}H_{B}-2B&0\\\ 0&H_{B}+2B\\\
\end{pmatrix}\begin{pmatrix}-1&1\\\ 1&1\\\ \end{pmatrix}$

we obtain

$\begin{pmatrix}X_{n+1}\\\ Y_{n+1}\end{pmatrix}=\begin{pmatrix}H_{B}&2B\\\
2B&H_{B}\\\ \end{pmatrix}^{n}\begin{pmatrix}B\\\
H_{B}\end{pmatrix}=\frac{1}{2}\begin{pmatrix}-1&1\\\ 1&1\\\
\end{pmatrix}\begin{pmatrix}H_{B}-2B&0\\\ 0&H_{B}+2B\\\
\end{pmatrix}^{n}\begin{pmatrix}-1&1\\\ 1&1\\\
\end{pmatrix}\begin{pmatrix}B\\\ H_{B}\end{pmatrix}$

which leads to

$2R(H_{B}^{n})=2Y_{n+1}=\left(H_{B}-B\right)(H_{B}-2B)^{n}+\left(H_{B}+B\right)(H_{B}+2B)^{n}.\qed$

In the next lemma, we use the recursive identity (4) to provide bounds on the
$F_{m}$.

######  Lemma 6.

For every $t\in B(2\mathbb{N}+1)$, we have

$\frac{1}{2^{m}}(t+B)(t+3B)\dots(t+(2m-1)B)\leq
F_{m}(t)\leq(t+B)(t+3B)\dots(t+(2m-1)B).$

In particular,

$\lVert F_{m}(H_{B})\mathbf{1}_{(-\infty,E]}(H_{B})\rVert=\max\left\\{\lvert
F_{m}(t)\rvert\colon t\in\sigma(H_{B})\cap(-\infty,E]\right\\}\leq(E+mB)^{m}.$

###### Proof.

By an iterative application of Lemma 5, $F_{n}(t)$ can be expressed as
$2^{-n}$ times a sum of $2^{n}$ many products of factors of the form $(t-kB)$.
Each summand must have a term $(t\pm B)$, and parameters $k$ in neighbouring
factors differ by $-2,0$ or $+2$. Furthermore, as soon as $t-kB$ is zero, the
summand containing this factor will vanish whence each summand is non-
negative. The lower bound follows by dropping all but one term. The upper
bound follows by replacing all $2^{n}$ many summands by the expression that
maximises such products. ∎

With this, we can prove the magnetic Bernstein inequalities:

######  Theorem 7.

For every $E,B\geq 0$ and $m\in\mathbb{N}$, we have the magnetic Bernstein
inequality

(5)
$\sum_{\alpha\in\\{1,2\\}^{m}}\lVert\tilde{\partial}_{\alpha_{1}}\tilde{\partial}_{\alpha_{2}}\dots\tilde{\partial}_{\alpha_{m}}f\rVert_{L^{2}(\mathbb{R}^{2})}^{2}\leq
C_{B}(m)\lVert f\rVert_{L^{2}(\mathbb{R}^{2})}^{2}\quad\text{for all}\quad
f\in\operatorname{Ran}\mathbf{1}_{(-\infty,E]}(H_{B}),$

where

$C_{B}(m)=(E+Bm)^{m}.$

###### Proof.

Note that $\mathbf{1}_{(-\infty,E]}(H_{B})$ is a finite sum of projectors onto
the Landau Levels up to $E$. These projectors have the integral kernel

$K_{E,B}(x,y)=\frac{B}{2\pi}\sum_{k\in\mathbb{N}_{0}\colon(2k+1)B\leq
E}\exp\left(-\frac{B}{4}\lvert
x-y\rvert^{2}-i\frac{B}{2}(x_{1}y_{2}-x_{2}y_{1})\right)\mathcal{L}_{l}\left(\frac{B}{2}\lvert
x-y\rvert^{2}\right)$

where the $\mathcal{L}_{k}$ are the Legendre polynomials, see [Foc28, Lan30].
This kernel is smooth, exponentially decaying and therefore leaves the Schwarz
space $\mathcal{S}(\mathbb{R}^{2})$ invariant, that is

$\mathbf{1}_{(-\infty,E]}(H_{B})f\in\mathcal{S}(\mathbb{R}^{2})\quad\text{for
all $f\in\mathcal{S}(\mathbb{R}^{2})$}.$

This allows to use integration by parts for the magnetic derivatives, and we
calculate

$\displaystyle\left\langle\mathbf{1}_{(-\infty,E]}(H_{B})f,F_{n}(H_{B})\mathbf{1}_{(-\infty,E]}(H_{B})f\right\rangle=\left\langle\mathbf{1}_{(-\infty,E]}(H_{B})f,R^{n}(\operatorname{Id})\mathbf{1}_{(-\infty,E]}(H_{B})f\right\rangle$
$\displaystyle=$
$\displaystyle\sum_{\alpha\in\\{1,2\\}^{m}}\left\langle\mathbf{1}_{(-\infty,E]}(H_{B})f,\tilde{\partial}_{\alpha_{n}}\tilde{\partial}_{\alpha_{n-1}}\dots\tilde{\partial}_{\alpha_{1}}\tilde{\partial}_{\alpha_{1}}\tilde{\partial}_{\alpha_{2}}\dots\tilde{\partial}_{\alpha_{n}}\mathbf{1}_{(-\infty,E]}(H_{B})f\right\rangle$
$\displaystyle=$
$\displaystyle\sum_{\alpha\in\\{1,2\\}^{m}}\lVert\tilde{\partial}_{\alpha_{1}}\tilde{\partial}_{\alpha_{2}}\dots\tilde{\partial}_{\alpha_{n}}\mathbf{1}_{(-\infty,E]}(H_{B})f\rVert_{L^{2}(\mathbb{R}^{2})}^{2}$

for all $f\in\mathcal{S}(\mathbb{R}^{2})$. By density, this extends to all
$f\in L^{2}(\mathbb{R}^{2})$. Together with Lemma 6, we obtain the claim. ∎

######  Remark 8.

The classic Bernstein inequalities (in two dimensions) are

$\sum_{\alpha\in\\{1,2\\}^{m}}\lVert\partial_{\alpha_{1}}\partial_{\alpha_{2}}\dots\partial_{\alpha_{m}}f\rVert_{L^{2}(\mathbb{R}^{2})}^{2}\leq
E^{m}\lVert f\rVert_{L^{2}(\mathbb{R}^{2})}^{2}\quad\text{for all}\quad
f\in\operatorname{Ran}\mathbf{1}_{(-\infty,E]}(-\Delta).$

They are an immediate consequence of the identity

$\sum_{\alpha\in\\{1,2\\}^{m}}\lVert\partial_{\alpha_{1}}\partial_{\alpha_{2}}\dots\partial_{\alpha_{m}}f\rVert_{L^{2}(\mathbb{R}^{2})}^{2}=\left\langle
f,(-\Delta)^{m}f\right\rangle$

for sufficiently regular $f$, which follows from a repeated application of
integration by parts. Note that, in constrast to the magnetic derivatives
$\tilde{\partial}_{1},\tilde{\partial}_{2}$, the classic derivatives
$\partial_{1},\partial_{2}$ commute. Hence, one usually writes the right hand
side of classic Bernstein inequalities in multiindex notation in the
equivalent form

$\sum_{\lvert\mathfrak{n}\rvert=m}\frac{1}{\mathfrak{n}!}\lVert\partial^{\mathfrak{n}}f\rVert_{L^{2}(\mathbb{R}^{2})}^{2}\leq\frac{E^{m}}{m!}\lVert
f\rVert_{L^{2}(\mathbb{R}^{2})}^{2},$

see [ES21] for an overview. For other operators, Bernstein-type inequalities
are rather rare. One notable exception where Bernstein-type estimates are
known is the Harmonic Oscillator [BJPS21, ES21].

From the proof of Theorem 7 it also follows that
$\operatorname{Ran}\mathbf{1}_{(-\infty,\mu]}(H_{B})\subseteq
C^{\infty}(\mathbb{R}^{2})$.

######  Remark 9.

It is paramount to work with _magnetic derivatives_
$\tilde{\partial}_{1},\tilde{\partial}_{2}$ in Theorem 7, and not with
_ordinary derivatives_ $\partial_{1},\partial_{2}$. Indeed, derivatives of
$f\in\operatorname{Ran}\mathbf{1}_{(-\infty,E]}(H_{B})$ will _not_ be
uniformly bounded in $L^{2}(\mathbb{R}^{2})$ for fixed $E$ and $B$. We
illustrate this with the following example: Let $0<B\leq E$ and consider, for
$y\in\mathbb{R}^{2}$, the eigenfunction to the eigenvalue $B$

(6) $f_{y}(x):=\exp\left(-\frac{B}{4}\lvert
x-y\rvert^{2}-i\frac{B}{2}(x_{1}y_{2}-x_{2}y_{1})\right)\in\operatorname{Ran}\mathbf{1}_{(-\infty,E]}(H_{B}).$

Clearly, $\lVert f_{y}\rVert_{L^{2}(\mathbb{R}^{2})}^{2}=\frac{2\pi}{B}$ is
independent of $y$. However,

$\displaystyle\lVert\partial_{1}f_{y}\rVert_{L^{2}(\mathbb{R}^{2})}^{2}=$
$\displaystyle\frac{B}{2}\int_{\mathbb{R}^{2}}\lvert(-(x_{1}-y_{1})-iy_{2})f_{y}(x)\rvert^{2}\mathrm{d}x\geq\frac{B}{2}\int_{\mathbb{R}^{2}}\left(\lvert
y_{2}\rvert^{2}-\lvert x_{1}-y_{1}\rvert^{2}\right)\lvert
f_{y}(x)\rvert^{2}\mathrm{d}x$ $\displaystyle=\frac{\pi\lvert
y_{2}\rvert^{2}}{2}-\frac{B}{2}\int_{\mathbb{R}^{2}}x_{1}^{2}\exp\left(-\frac{B\lvert
x\rvert^{2}}{4}\right)\mathrm{d}x=\frac{\pi\lvert
y_{2}\rvert^{2}}{2}-\frac{4\pi}{B}.$

This can be made arbitrarily large by choosing $y_{2}$. Consequently, Theorem
7 cannot hold verbatim when replacing
$\tilde{\partial}_{1},\tilde{\partial}_{2}$ by $\partial_{1},\partial_{2}$.

######  Remark 10.

Choosing $y=0$ in (6), the function $f_{0}$ also demonstrates that for fixed
$E,\rho>0$ the constant

$\left(\frac{C_{1}}{\rho}\right)^{C_{2}+C_{3}\lvert\ell\rvert_{1}\sqrt{E}+C_{4}(\lvert\ell\rvert_{1}^{2}B)}\sim\tilde{C}_{1}\exp\left(\tilde{C}_{2}+\tilde{C}_{3}\lvert\ell\rvert_{1}\sqrt{E}+\tilde{C}_{4}(\lvert\ell\rvert_{1}^{2}B)\right)$

in Theorem 3 has the optimal behavior as $\lvert\ell\rvert_{1}^{2}$ tends to
$\infty$. Indeed, let $\ell=(\ell_{1},\ell_{2})\in(0,\infty)^{2}$ and consider
the $(\ell,\rho)$-thick set

$S:=\mathbb{R}^{2}\setminus B_{r}(0)\quad\text{where
$r:=\max(\ell_{1},\ell_{2})(1-\rho)/2$}.$

Then,

$\displaystyle\lVert f_{0}\rVert_{L^{2}(S)}^{2}$
$\displaystyle=\int_{r}^{\infty}2\pi
s\exp(-Bs^{2}/2)\mathrm{d}s=\frac{2\pi}{B}\exp(-Br^{2}/2)$
$\displaystyle=\exp\left(\frac{-B\max(\ell_{1},\ell_{2})^{2}(1-\rho)^{2}}{8}\right)\lVert
f_{0}\rVert_{L^{2}(\mathbb{R}^{d})}^{2}\leq\exp\left(\frac{-B\lvert\ell\rvert_{1}^{2}(1-\rho)^{2}}{2}\right)\lVert
f_{0}\rVert_{L^{2}(\mathbb{R}^{d})}^{2}\,.$

Thus, the constant in Theorem 3 must at least be of order
$\exp(CB\lvert\ell\rvert_{1}^{2})$ as $\lvert\ell\rvert_{1}\to\infty$.

We can furthermore use the functions $f_{y}$, defined in (6), to show that
thickness is necessary for any quantitative unique continuation principle on
spectral subspaces.

######  Theorem 11.

Assume that $S\subset\mathbb{R}^{2}$ is such that for some $B>0$, $E\geq B$,
there is a constant $C>0$ such that

(7) $\lVert f\rVert_{L^{2}(\mathbb{R}^{2})}^{2}\leq C\lVert
f\rVert_{L^{2}(S)}^{2}\quad\text{for all
$f\in\operatorname{Ran}\mathbf{1}_{(-\infty,E]}(H_{B})$}.$

Then $S$ is thick.

###### Proof.

If $S\subset\mathbb{R}^{2}$ was not thick, there would be
$(y^{(n)})_{n\in\mathbb{N}}\subset\mathbb{R}^{2}$ with

$\operatorname{Vol}(B_{n}(y^{(n)})\cap S)\leq\frac{1}{n}\quad\text{for all
$n\in\mathbb{N}$}.$

Defining $f_{y^{(n)}}$ as in (6), we have
$f_{y^{(n)}}\in\operatorname{Ran}\mathbf{1}_{(-\infty,E]}(H_{B})$ with $\lVert
f_{y^{(n)}}\rVert_{L^{2}(\mathbb{R}^{2})}^{2}=\frac{2\pi}{B}$, but

$\displaystyle\lVert f_{y^{(n)}}\rVert_{L^{2}(S)}^{2}$
$\displaystyle\leq\lVert
f_{y^{(n)}}\rVert_{L^{\infty}(\mathbb{R}^{2})}^{2}\cdot\operatorname{Vol}(B_{n}(y^{(n)})\cap
S)+\int_{\lvert x-y_{n}\rvert\geq n}\exp\left(-\frac{B}{2}\lvert
x-y^{(n)}\rvert\right)\mathrm{d}x$
$\displaystyle\leq\frac{1}{n}+\frac{1}{B}\exp\left(-\frac{Bn^{2}}{2}\right).$

This tends to $0$ as $n\to\infty$, so (7) cannot hold. ∎

In the classic strategy of proof of the Kovrijkine-Logvinenko-Sereda theorems,
we would now like to bound the $L^{2}(\mathbb{R}^{2})$-norm of higher order
_ordinary derivatives_ $\partial_{1},\partial_{2}$ of
$f\in\operatorname{Ran}\mathbf{1}_{(-\infty,E]}(H_{B})$ and use this to infer
that $f$ is analytic. Unfortunately, in light of Remark 9, this is impossible.
However, a closer look shows that this lack of a uniform bound is due to the
oscillating phase factor for large $\lvert x\rvert$. This suggests that,
instead of proving $L^{2}(\mathbb{R}^{d})$-bounds on (derivatives of) $f$, we
might be better off proving $L^{1}(\mathbb{R}^{2})$-bounds on derivatives of
$\lvert f\rvert^{2}$. For this, we need some notation. For a finite sequence
$\alpha=(\alpha_{1},\dots,\alpha_{m})\in\\{1,2\\}^{m}$, let

$\displaystyle\partial^{\alpha}:=$
$\displaystyle\partial_{\alpha_{1}}\partial_{\alpha_{2}}\dots\partial_{\alpha_{m}},$
$\displaystyle\tilde{\partial}^{\alpha}:=$
$\displaystyle\tilde{\partial}_{\alpha_{1}}\tilde{\partial}_{\alpha_{2}}\dots\tilde{\partial}_{\alpha_{m}}.$

Furthermore, we write $\beta\leq\alpha$, if $\beta$ is a subsequence of
$\alpha$ and write $\alpha\setminus\beta$ for the complementary subsequence.
We can now formulate our next theorem which are Bernstein-type inequalities
(with ordinary derivatives) on $\lvert f\rvert^{2}$ where
$f\in\operatorname{Ran}\mathbf{1}_{(-\infty,E]}(H_{B})$:

######  Theorem 12.

For every $E,B\geq 0$, $m\in\mathbb{N}$, and
$f\in\operatorname{Ran}\mathbf{1}_{(-\infty,E]}(H_{B})$ we have

(8) $\sum_{\alpha\in\\{1,2\\}^{m}}\lVert\partial^{\alpha}\lvert
f\rvert^{2}\rVert_{L^{1}(\mathbb{R}^{2})}\leq C^{\prime}_{B}(m)\lVert
f\rVert_{L^{2}(\mathbb{R}^{2})}^{2}\quad\text{where}\quad
C^{\prime}_{B}(m)=2^{3m/2}(E+Bm)^{m/2}.$

Furthermore, for all $\alpha\in\\{0,1\\}^{m}$

(9) $\sum_{\alpha\in\\{1,2\\}^{m}}\lVert\partial^{\alpha}\lvert
f\rvert^{2}\rVert_{L^{\infty}(\mathbb{R}^{2})}\leq
C_{\mathrm{sob}}\sum_{m^{\prime}=m}^{m+3}C^{\prime}_{B}(m)\lVert
f\rVert_{L^{2}(\mathbb{R}^{2})}^{2}$

where $C_{\mathrm{sob}}>0$ is a universal constant.

As the notation suggests, $C_{\mathrm{sob}}$ comes from a Sobolev embedding.

###### Proof.

Let $u,v\in\mathcal{C}^{\infty}(\mathbb{R}^{2},\mathbb{C})$ and
$x\in\mathbb{R}^{2}$. We have

$\displaystyle i\partial_{1}(u\bar{v})(x)$
$\displaystyle=\bar{v}(x)\left(\left(i\partial_{1}-\frac{B}{2}x_{2}\right)u\right)(x)-u(x)\overline{\left(\left(i\partial_{1}-\frac{B}{2}x_{2}\right)v\right)}(x)$
$\displaystyle=\bar{v}(x)\tilde{\partial}_{1}u(x)-u(x)\overline{\tilde{\partial}_{1}v}(x).$

Analogously,

$i\partial_{2}(u\bar{v})(x)=\bar{v}(x)\tilde{\partial}_{2}(x)-u(x)\overline{\tilde{\partial}_{2}v}(x).$

By induction, for any $\alpha\in\\{1,2\\}^{n}$, this leads to

$i^{m}\partial^{\alpha}\lvert
u\rvert^{2}(x)=\sum_{\beta\leq\alpha}(-1)^{m-\lvert\beta\rvert}\tilde{\partial}^{\beta}u(x)\overline{\tilde{\partial}^{\alpha\setminus\beta}u(x)}.$

Thus, we can estimate

$\displaystyle\sum_{\lvert\alpha\rvert=m}\lVert\partial^{\alpha}\lvert
f\rvert^{2}\rVert_{L^{1}(\mathbb{R}^{2})}\leq$
$\displaystyle\sum_{\lvert\alpha\rvert=m}\sum_{\beta\leq\alpha}\lVert\tilde{\partial}^{\beta}f\rVert_{L^{2}(\mathbb{R}^{2})}\lVert\tilde{\partial}^{\alpha\setminus\beta}f\rVert_{L^{2}(\mathbb{R}^{2})}$
$\displaystyle=$
$\displaystyle\sum_{k=0}^{m}\binom{m}{k}\sum_{\lvert\beta\rvert=k,\lvert\beta^{\prime}\rvert=m-k}\lVert\tilde{\partial}^{\beta}f\rVert_{L^{2}(\mathbb{R}^{2})}\lVert\tilde{\partial}^{\beta^{\prime}}f\rVert_{L^{2}(\mathbb{R}^{2})}$
$\displaystyle\leq$
$\displaystyle\sum_{k=0}^{m}\binom{m}{k}2^{m/2}\sqrt{\sum_{\lvert\beta\rvert=k,\lvert\beta^{\prime}\rvert=m-k}\lVert\tilde{\partial}^{\beta}f\rVert_{L^{2}(\mathbb{R}^{2})}^{2}\lVert\tilde{\partial}^{\beta^{\prime}}f\rVert_{L^{2}(\mathbb{R}^{2})}^{2}}$
$\displaystyle\leq$
$\displaystyle\sum_{k=0}^{m}\binom{m}{k}2^{m/2}\sqrt{C_{B}(k)C_{B}(m-k)}\lVert
f\rVert_{L^{2}(\mathbb{R}^{2})}^{2}$ $\displaystyle\leq$
$\displaystyle\sum_{k=0}^{m}\binom{m}{k}2^{m/2}(E+Bm)^{m/2}\lVert
f\rVert_{L^{2}(\mathbb{R}^{2})}^{2}=2^{3m/2}(E+Bm)^{m/2}\lVert
f\rVert_{L^{2}(\mathbb{R}^{2})}^{2}.$

Estimate (9) follows from (8) by using the Sobolev estimate $\lVert
g\rVert_{L^{\infty}(\mathbb{R}^{2})}\leq C_{\mathrm{sob}}\lVert
g\rVert_{W^{3,1}(\mathbb{R}^{2})}$ which leads to

$\displaystyle\sum_{\alpha\in\\{1,2\\}^{m}}\lVert\partial^{\alpha}\lvert
f\rvert^{2}\rVert_{L^{\infty}(\mathbb{R}^{2})}$ $\displaystyle\leq
C_{\mathrm{sob}}\sum_{\alpha\in\\{1,2\\}^{m}}\lVert\partial^{\alpha}\lvert
f\rvert^{2}\rVert_{W^{3,1}(\mathbb{R}^{2})}=C_{\mathrm{sob}}\sum_{\alpha\in\\{1,2\\}^{m}}\sum_{\lvert\beta\rvert\leq
3}\lVert\partial^{\beta}\partial^{\alpha}\lvert
f\rvert^{2}\rVert_{L^{1}(\mathbb{R}^{2})}$
$\displaystyle=C_{\mathrm{sob}}\sum_{m^{\prime}=m}^{m+3}\sum_{\lvert\alpha^{\prime}\rvert=m^{\prime}}\lVert\partial^{\alpha^{\prime}}\lvert
f\rvert^{2}\rVert_{L^{1}(\mathbb{R}^{2})}\leq
C_{\mathrm{sob}}\sum_{m^{\prime}=m}^{m+3}C_{B}(m^{\prime})\lVert
f\rVert_{L^{2}(\mathbb{R}^{2})}^{2}.\qed$

Let us emphasize that the the constant $C_{\mathrm{sob}}$ comes from a Sobolev
estimate in $\mathbb{R}^{2}$. When proving the analogous result on _bounded_
domains $\Lambda_{L}$ in Section 5, it is therefore desirable to work with
restrictions onto _one domain_ (or shifted variants thereof). This will be
achieved by possibly extending functions beyond their original domain – using
the magnetic boundary conditions defined in Section 5.

##  4\. Spectral inequality for the Landau operator

In this section, we prove Theorem 3. The strategy of proof roughly follows
Kovrijkine’s proof [Kov00] (for the case of the pure Laplacian) and the more
general argument in [ES21], the latter being however formulated in an
$L^{2}(\mathbb{R}^{d})$ setting instead of the $L^{1}(\mathbb{R}^{d})$ setting
used here.

###  4.1. Analyticity and local estimate

######  Lemma 13.

Let $E\geq 0$, $f\in\operatorname{Ran}\mathbf{1}_{(-\infty,E]}(H_{B})$. Then
$\lvert f\rvert^{2}$ is analytic, i.e. it can be expanded in an absolutely
convergent power series around every $x_{0}\in\mathbb{R}^{2}$. In particular,
it has an analytic extension $\Phi$ to $\mathbb{C}^{2}$.

###### Proof.

Let $m\in\mathbb{N}$. By (9), we have

$\displaystyle\sum_{\alpha\in\\{1,2\\}^{m}}\lVert\partial^{\alpha}\lvert
f\rvert^{2}\rVert_{L^{\infty}(\mathbb{R}^{2})}$ $\displaystyle\leq
C_{\mathrm{sob}}\sum_{m^{\prime}=m}^{m+3}2^{3m^{\prime}/2}(E+Bm^{\prime})^{m/2}\lVert
f\rVert_{L^{2}(\mathbb{R}^{2})}^{2}$ $\displaystyle\leq
4C_{\mathrm{sob}}2^{3(m+3)/2}(E+B(m+3))^{\frac{m+3}{2}}\lVert
f\rVert_{L^{2}(\mathbb{R}^{2})}^{2}.$

Using

$(E+B(m+3))^{\frac{m+3}{2}}\lesssim\sqrt{m!}$

we see that for every point $x_{0}\in\mathbb{R}^{2}$, the series

$\Phi(z):=\sum_{k=0}^{\infty}\sum_{\lvert\alpha\rvert=k}\frac{\partial^{\alpha}\lvert
f\rvert^{2}}{k!}(x-x_{0})^{\alpha},$

where $(x-x_{0})^{\alpha}$ is multi-index notation meaning

$(x-x_{0})^{\alpha}=(x-x_{0})_{1}^{\alpha_{1}}\cdot(x-x_{0})_{2}^{\alpha_{2}},$

converges absolutely and locally uniformly, agrees with $f$ on
$\mathbb{R}^{2}$, and defines an analytic extension $\Phi$ of $f$ to
$\mathbb{C}^{2}$. ∎

Next, we need a local lower bound on such analytic functions. For this
purpose, given $r>0$, we denote by $D_{r}=\\{z\in\mathbb{C}\colon\lvert
z\rvert\leq r\\}$ the complex disc of radius $r$. We also need notation for
two-dimensional complex polydiscs and, given $r_{1},r_{2}>0$, we denote by
$D_{(r_{1},r_{2})}=\\{(z_{1},z_{2})\in\mathbb{C}^{2}\colon\lvert
z_{j}\rvert\leq r_{j}\\}$ the complex polydisc with radii $r_{1}$ and $r_{2}$.

######  Lemma 14.

Let $Q\subseteq\mathbb{R}^{2}$ be a rectangle with sides of lengths
$\ell_{1},\ell_{2}>0$, parallel to the coordinate axes, and let $g\colon
Q\to\mathbb{C}$ be a non-vanishing function admitting an analytic continuation
$G$ to $Q+D_{(4\ell_{1},4\ell_{2})}\subseteq\mathbb{C}^{2}$. Then, for any
measurable $\omega\subseteq Q$ and every linear bijection
$A\colon\mathbb{R}^{2}\to\mathbb{R}^{2}$, we have

$\displaystyle\lVert g\rVert_{L^{1}(Q\cap\omega)}$
$\displaystyle\geq\frac{1}{2}\left(\frac{\operatorname{Vol}A(Q\cap\omega)}{48\pi\operatorname{diam}A(Q)^{2}}\right)^{2\frac{\log
M}{\log 2}}\frac{\operatorname{Vol}(Q\cap\omega)}{\operatorname{Vol}(Q)}\lVert
g\rVert_{L^{1}(Q)}$
$\displaystyle\geq\frac{1}{2}\left(\frac{\operatorname{Vol}A(Q\cap\omega)}{48\pi\operatorname{diam}A(Q)^{2}}\right)^{2\frac{\log
M}{\log 2}+1}\lVert g\rVert_{L^{1}(Q)},$

where

$M:=\frac{\operatorname{Vol}Q}{\lVert g\rVert_{L^{1}(Q)}}\cdot\sup_{z\in
Q+D_{(4\ell_{1},4\ell_{2})}}\lvert G(z)\rvert\geq 1.$

Similar statements as in Lemma 14 can be found at several places in the
literature. The original idea seems to go back to [Kov00]. The proof provided
here is inspired by proof of Lemma 3.5 in [ES21], where a corresponding
statement for $L^{2}$-norms is proved and the linear bijection $A$ was
introduced. The latter will be used in subsequent steps of the proof of
Theorem 3 in order to optimize the constants. Indeed, without $A$, the
_excentricity_ of rectangles with side lengths $(\ell_{1},\ell_{2})$, or more
precisely, the ratio between their diameter and their volume, would enter. The
bijection helps to make the constant independent of the shape such that only
the expression $\lvert\ell\rvert_{1}$ enters the final statement.

The proof of Lemma 14 relies on a dimension reduction argument and the
following one-dimensional estimate, due to [Kov00], which itself relies on the
Remez inequality for polynomials as well as Bleschke products.

######  Lemma 15 (Cf. [Kov00, Lemma 1]).

Let $\varphi\colon D_{4+\epsilon}\to\mathbb{C}$ for some $\epsilon>0$ be an
analytic function with $\lvert\varphi(0)\rvert\geq 1$. Let $E\subseteq[0,1]$
be measurable with positive measure. Then

$\sup_{t\in[0,1]}\lvert\varphi(t)\rvert\leq\left(\frac{12}{\operatorname{Vol}E}\right)^{2\frac{\log
M_{\phi}}{\log 2}}\sup_{t\in E}\lvert\varphi(t)\rvert$

where $M_{\varphi}=\sup_{z\in D_{4}}\lvert\varphi(z)\rvert$.

For convenience of the reader, we provide a proof of Lemma 15 in Appendix A.

###### Proof of Lemma 14.

For all $C>0$, we clearly have

$\displaystyle\lVert g\rVert_{L^{1}(Q\cap\omega)}$
$\displaystyle\geq\lVert\mathbf{1}_{\\{x\in Q\cap\omega\colon\lvert
g(x)\rvert>C\lVert g\rVert_{L^{1}(Q)}\\}}\cdot g\rVert_{L^{1}(Q)}$
$\displaystyle\geq C\lVert
g\rVert_{L^{1}(Q)}\cdot\operatorname{Vol}\left\\{x\in Q\cap\omega\colon\lvert
g(x)\rvert>C\lVert g\rVert_{L^{1}(Q)}\right\\}.$

Using this with

$C=\left(\frac{\operatorname{Vol}(A(Q\cap\omega)}{24\pi\operatorname{diam}(A(Q))^{2}}\right)^{2\frac{\log
M}{\log 2}}\cdot\frac{1}{\operatorname{Vol}Q},$

the first stated inequality follows if we prove

$\operatorname{Vol}\left\\{x\in Q\cap\omega\colon\lvert g(x)\rvert>C\lVert
g\rVert_{L^{1}(Q)}\right\\}\geq\frac{\operatorname{Vol}(Q\cap\omega)}{2}$

which is certainly the case if

(10)
$\operatorname{Vol}(W)\leq\frac{\operatorname{Vol}(Q\cap\omega)}{2},\quad\text{where}\quad
W:=\left\\{x\in Q\colon\lvert g(x)\rvert\leq C\lVert
g\rVert_{L^{1}(Q)}\right\\},$

i.e., the set $W$ where $\lvert g\rvert$ is ”small” has no more than half of
the Lebesgue mass of $Q\cap\omega$. To see (10), we may assume without loss
$W\neq\emptyset$. We will first show that there is a line segment
$I=I(y_{0},W,Q)\subset Q$ of the form

$I=\\{y_{0}+t\xi_{0}\colon t\in[0,t_{\max}]\\}$

such that

(11) $\frac{\operatorname{Vol}_{1}(I\cap
W)}{\operatorname{Vol}_{1}I}\geq\frac{\operatorname{Vol}A(W)}{\pi\operatorname{diam}(A(Q))^{2}}.$

Indeed, there is $y_{0}\in Q$ with $\lvert g(y_{0})\rvert\geq\frac{\lVert
g\rVert_{L^{1}(Q)}}{\operatorname{Vol}(Q)}$. Using spherical coordinates
around $A(y_{0})$,

$\operatorname{Vol}A(W)=\int_{0}^{2\pi}\int_{0}^{\infty}s\cdot\mathbf{1}_{A(W)}\left(A(y_{0})+s\begin{pmatrix}\cos(\theta)\\\
\sin(\theta)\end{pmatrix}\right)\mathrm{d}s\mathrm{d}\theta$

whence there exists $\xi_{0}\in\mathbb{R}^{2}$ with $\lvert\xi_{0}\rvert=1$
such that

$\operatorname{Vol}A(W)\leq\pi\int_{0}^{\infty}s\cdot\mathbf{1}_{A(W)}\left(A(y_{0})+s\xi_{0}\right)\mathrm{d}s.$

Defining

$\eta_{0}:=\frac{A^{-1}(\xi_{0})}{\lvert A^{-1}(\xi_{0})\rvert}$

and denoting by $I\subset Q$ the line segment of maximal length within $Q$,
given by

$I=\\{y_{0}+\operatorname{Vol}_{1}(I)\cdot\eta_{0}t\colon t\in[0,1]\\}$

we have

$\operatorname{Vol}A(W)\leq\pi\operatorname{Vol}_{1}A(I\cap\omega)\operatorname{Vol}_{1}A(I).$

Taking into account $\operatorname{diam}(A(Q))\geq\operatorname{Vol}_{1}(I)$,
the line segment $I\subset Q$ indeed satisfies (11). Since $Q$ is open, there
is $\epsilon>0$ such that $y_{0}+\operatorname{Vol}_{1}I\cdot\eta_{0}z\in
Q+D_{(4\ell_{1},4\ell_{2})}$ for $z\in D_{4+\epsilon}$. Define

$\varphi(z):=\frac{\operatorname{Vol}Q}{\lVert g\rVert_{L^{1}(Q)}}\cdot
G(y_{0}+\operatorname{Vol}_{1}I\cdot\eta_{0}\ z)\in\mathbb{C}.$

By assumption, $\varphi$ is analytic on $D(4+\epsilon)\subset\mathbb{C}$, and
satisfies

$\sup_{t\in[0,1]}\lvert\varphi(t)\rvert\geq\lvert\varphi(0)\rvert=\frac{\operatorname{Vol}Q\cdot\lvert
g(y_{0})\rvert}{\lVert g\rVert_{L^{1}(Q)}}\geq 1$

as well as

$M:=\sup_{z\in
D(4)}\lvert\varphi(z)\rvert\leq\frac{\operatorname{Vol}Q}{\lVert
g\rVert_{L^{1}(Q)}}\sup_{z\in y_{0}+D_{(4\ell_{1},4\ell_{2})}}\lvert
G(z)\rvert\leq M.$

We may assume $M>1$ because if $M=1$, then $g$ would be constant on $Q$ and
the statement would follow immediately. Applying Lemma 15 with
$E:=\\{t\in[0,1]\colon y_{0}+\operatorname{Vol}_{1}I\cdot\eta_{0}\ t\in I\cap
W\\}\subseteq[0,1]$ yields

$\sup_{t\in
E}\lvert\varphi(t)\rvert\geq\left(\frac{\operatorname{Vol}E}{12}\right)^{2\frac{\log
M}{\log
2}}\sup_{t\in[0,1]}\lvert\phi(t)\rvert\geq\left(\frac{\operatorname{Vol}E}{12}\right)^{2\frac{\log
M}{\log 2}}.$

Using the definition of $\varphi$ and recalling that $G\mid_{Q}=g$, this
becomes

$\sup_{t\in E}\lvert
g(y_{0}+\operatorname{Vol}_{1}(I)\cdot\eta_{0}t)\rvert\geq\left(\frac{\operatorname{Vol}E}{12}\right)^{2\frac{\log
M}{\log 2}}\cdot\frac{\lVert g\rVert_{L^{1}(Q)}}{\operatorname{Vol}Q}.$

Since $\operatorname{Vol}E=\frac{\operatorname{Vol}_{1}(I\cap
W)}{\operatorname{Vol}I}\geq\frac{\operatorname{Vol}A(W)}{\pi\operatorname{diam}(A(Q))^{2}}$
by (11), we infer

$\left(\frac{\operatorname{Vol}A(W)}{12\pi\operatorname{diam}A(Q)^{2}}\right)^{2\frac{\log
M}{\log 2}}\cdot\frac{\lVert
g\rVert_{L^{1}(Q)}}{\operatorname{Vol}Q}\leq\sup_{x\in W}\lvert g(x)\rvert.$

Combining this with the definition of $W$, we obtain

$\displaystyle\sup_{x\in W}\lvert g(x)\rvert$
$\displaystyle\leq\left(\frac{\operatorname{Vol}A(Q\cap\omega)}{24\pi\operatorname{diam}A(Q)^{2}}\right)^{2\frac{\log
M}{\log 2}}\cdot\frac{\lVert g\rVert_{L^{1}(\Omega)}}{\operatorname{Vol}Q}$
$\displaystyle=\left(\frac{\operatorname{Vol}A(Q\cap\omega)}{2\operatorname{Vol}A(W)}\cdot\frac{\operatorname{Vol}A(W)}{12\pi\operatorname{diam}A(Q)^{2}}\right)^{2\frac{\log
Mi}{\log 2}}\cdot\frac{\lVert g\rVert_{L^{1}(\Omega)}}{\operatorname{Vol}Q}$
$\displaystyle\leq\left(\frac{\operatorname{Vol}(Q\cap\omega)}{2\operatorname{Vol}W}\right)^{2\frac{\log
M}{\log 2}}\sup_{x\in W}\lvert g(x)\rvert.$

Recalling $M>1$, this implies $\operatorname{Vol}(Q\cap\omega)\geq
2\operatorname{Vol}W$ and concludes the proof of the first stated inequality.
The second inequaliy follows from
$\operatorname{Vol}A(Q)\leq\pi\operatorname{diam}A(Q)^{2}$. ∎

###  4.2. Good and bad rectangles

We cover $\mathbb{R}^{2}$ by a family $(Q_{j})_{j\in\mathbb{N}}$ of open
rectangles of side lengths $\ell_{1}$ and $\ell_{2}$, parallel to the
coordinate axes, such that any two rectangles do not overlap and the
complement of their union is a measure zero set.

######  Remark 16.

To prove Theorem 4, the finite volume analogon of Theorem 3 on rectangles
$\Lambda_{L}$, the side lengths $L_{1},L_{2}$ of the domain might not be
multiples of $\ell_{1},\ell_{2}$ and we will not be able to cover
$\Lambda_{L}$ perfectly by a union of small rectangles of side lengths
$\ell_{1},\ell_{2}$. However, we can obtain a covering such that every point
is contained in at most four elements of the covering. So, the arguments below
will have to be amended with a factor four, see also [ES21], where this
argument is elaborated in a more general setting, using a more general notion
of coverings.

######  Definition 17.

Given $E,B>0$ and $f\in\operatorname{Ran}\mathbf{1}_{(-\infty,E]}(H_{B})$, we
call a rectangle $Q_{j}$ _good_ if

$\lVert\partial^{\alpha}\lvert f\rvert^{2}\rVert_{L^{1}(Q_{j})}\leq
4^{m+1}C_{B}^{\prime}(m)\lVert f\rVert_{L^{2}(Q_{j})}^{2}$

for all $m\in\mathbb{N}$ and $\alpha\in\\{1,2\\}^{m}$, where
$C_{B}^{\prime}(m)$ is defined in (8), and _bad_ otherwise.

######  Lemma 18.

Let $E,B>0$ and $f\in\operatorname{Ran}\mathbf{1}_{(-\infty,E]}(H_{B})$. Then

$\sum_{j\colon Q_{j}\text{good}}\lVert
f\rVert_{L^{2}(Q_{j})}^{2}\geq\frac{1}{2}\lVert
f\rVert_{L^{2}(\mathbb{R}^{2})}^{2}.$

###### Proof.

Using the definition of badness and Theorem 12, we estimate

$\displaystyle\sum_{j\colon Q_{j}\text{bad}}\lVert f\rVert_{L^{2}(Q_{j})}^{2}$
$\displaystyle\leq\frac{1}{4}\sum_{j\colon
Q_{j}\text{bad}}\sum_{m=0}^{\infty}\sum_{\alpha\in\\{1,2\\}^{m}}\frac{1}{4^{m}C_{B}^{\prime}(m)}\lVert\partial^{\alpha}\lvert
f\rvert\rVert_{L^{1}(Q_{j})}^{2}$
$\displaystyle\leq\frac{1}{4}\sum_{m=0}^{\infty}\sum_{\alpha\in\\{1,2\\}^{m}}\frac{1}{4^{m}C_{B}^{\prime}(m)}\lVert\partial^{\alpha}\lvert
f\rvert^{2}\rVert_{L^{1}(\mathbb{R}^{2})}$
$\displaystyle\leq\frac{1}{4}\sum_{m=0}^{\infty}\sum_{\alpha\in\\{1,2\\}^{m}}\frac{1}{4^{m}}\lVert
f\rVert_{L^{2}(\mathbb{R}^{2})}^{2}=\frac{1}{2}\lVert
f\rVert_{L^{2}(\mathbb{R}^{2})}^{2}.\qed$

###  4.3. Proof of Theorem 3

By Lemma 13, $\lvert f\rvert^{2}$ is analytic in $\mathbb{R}^{2}$ with an
analytic extension $\Phi$ to $\mathbb{C}^{2}$. We may assume without loss that
$f$ does not vanish on $\mathbb{R}^{2}$ and thus also, by analyticity, on none
of the $Q_{j}$. Let $Q=Q_{j}$ be a good rectangle. Then, we claim that there
exists a point $x_{0}\in Q$ such that for all $m\in\mathbb{N}$ and all
$\alpha\in\\{1,2\\}^{m}$ one has

(12) $\displaystyle\lvert\partial^{\alpha}\lvert
f\rvert^{2}(x_{0})\rvert\leq\frac{8^{m+1}C_{B}^{\prime}(m)\lVert
f\rVert_{L^{2}(Q)}^{2}}{\operatorname{Vol}Q}.$

Indeed, if there was no such point then for all $x\in Q_{j}$

$\frac{\lVert
f\rVert_{L^{2}(Q)}^{2}}{\operatorname{Vol}Q}<\sum_{m=0}^{\infty}\sum_{\alpha\in\\{1,2\\}^{m}}\frac{1}{8^{m+1}C_{B}^{\prime}(m)}\lvert\partial^{\alpha}\lvert
f\rvert^{2}(x)\rvert.$

But then, integration over $x\in Q_{j}$ and the definition of good rectangles
would imply

$\lVert
f\rVert_{L^{2}(Q)}^{2}<\sum_{m=0}^{\infty}\sum_{\alpha\in\\{1,2\\}^{m}}\frac{1}{8^{m+1}C_{B}^{\prime}(m)}\lVert\partial^{\alpha}\lvert
f\rvert^{2}\rVert_{L^{1}(Q_{j})}\leq\sum_{m=0}^{\infty}\frac{1}{2^{m+1}}\lVert
f\rVert_{L^{2}(Q_{j})}^{2}=\lVert f\rVert_{L^{2}(Q_{j})}^{2},$

a contradiction. This shows the existence of $x_{0}$ as in (12). In
particular, for every $z\in D_{(5\ell_{1},5\ell_{2})}$

$\displaystyle\lvert\Phi(z)\rvert$
$\displaystyle\leq\sum_{m=0}^{\infty}\sum_{\alpha\in\\{1,2\\}^{m}}\frac{\lvert\partial^{\alpha}\lvert
f\rvert^{2}(x_{0})\rvert}{m!}\lvert z-x_{0}\rvert^{\alpha}$
$\displaystyle\leq\sum_{m=0}^{\infty}\sum_{\alpha\in\\{1,2\\}^{m}}\frac{8^{m+1}C^{\prime}_{B}(m)}{m!}(5\ell)^{\alpha}\frac{\lVert
f\rVert_{L^{2}(Q)}^{2}}{\operatorname{Vol}Q}\leq 8\frac{\lVert
f\rVert_{L^{2}(Q)}^{2}}{\operatorname{Vol}Q}\sum_{m\in\mathbb{N}}\frac{(40\lvert\ell\rvert_{1})^{m}C^{\prime}_{B}(m)}{m!}.$

We can therefore apply Lemma 14 with $g=\lvert f\rvert^{2}$, $G=\Phi$, and,
recalling $C_{B}^{\prime}(m)=2^{3m/2}(E+Bm)^{m/2}\leq 3^{m}(E+Bm)^{m/2}$,

$\displaystyle M_{\phi}$
$\displaystyle=8\sum_{m=0}^{\infty}\frac{(40\lvert\ell\rvert_{1})^{m}C^{\prime}_{B}(m)}{m!}\leq
8\sum_{m=0}^{\infty}\frac{(120\lvert\ell\rvert_{1})^{m}(\sqrt{E}+\sqrt{Bm})^{m}}{m!}$
$\displaystyle\leq
8\sum_{m=0}^{\infty}\frac{(240\lvert\ell\rvert_{1}\sqrt{E})^{m}}{m!}+8\sum_{m=0}^{\infty}\frac{(240\lvert\ell\rvert_{1}\sqrt{Bm})^{m}}{m!}.$

where we used $(a+b)^{m}\leq 2^{m}(a^{m}+b^{m})$. Now, note that for all
$s\geq 0$

(13) $\displaystyle\sum_{m=0}^{\infty}\frac{(s\sqrt{m})^{m}}{m!}=$
$\displaystyle\sum_{k=0}^{\infty}s^{2k}\left(\frac{\sqrt{2k}^{2k}}{(2k)!}+s\frac{\sqrt{2k+1}^{2k+1}}{(2k+1)!}\right)\leq(1+s)\sum_{k=0}^{\infty}\frac{(s\sqrt{2k})^{2k}}{(2k)!}$
$\displaystyle=(1+s)\sum_{k=0}^{\infty}\frac{(2s^{2})^{k}k^{k}}{(2k)!}\leq\exp(2s^{2}+s)$

where we used $1+s\leq\exp(s)$, and $(2k)!\geq k^{k}k!$ in the last step.
Using (13) with $s=240\lvert\ell\rvert_{1}\sqrt{B}$, we further estimate

$\displaystyle M_{\phi}$ $\displaystyle\leq
8\exp\left(240\lvert\ell\rvert_{1}\sqrt{E}\right)+8\exp\left(240\lvert\ell\rvert_{1}\sqrt{B}+2\cdot
240^{2}\lvert\ell\rvert_{1}^{2}B\right)$ $\displaystyle\leq 16\exp\left(2\cdot
240^{2}\left(\lvert\ell\rvert_{1}\sqrt{E}+\lvert\ell\rvert_{1}\sqrt{B}+\lvert\ell\rvert_{1}^{2}B\right)\right),$

whence

$\ln M_{\phi}\leq\ln 16+240\lvert\ell\rvert_{1}\sqrt{E}+2\cdot
240^{2}\left(\lvert\ell\rvert_{1}\sqrt{B}+\lvert\ell\rvert_{1}^{2}B\right).$

Therefore, we obtain for every good rectangle $Q_{j}$

$\displaystyle\lVert f\rVert_{L^{2}(Q_{j})}^{2}$ $\displaystyle\leq
2\left(\frac{48\pi\operatorname{diam}A(Q_{j})^{2}}{\operatorname{Vol}A(Q_{j}\cap
S)}\right)^{C_{2}+C_{3}\lvert\ell\rvert_{1}\sqrt{E}+C_{4}\left(\lvert\ell\rvert_{1}\sqrt{B}+\lvert\ell\rvert_{1}^{2}B\right)}\lVert
f\rVert_{L^{2}(Q_{j}\cap S)}^{2}.$

Choose the linear bijection $A$ to map every $Q_{j}$ to a square of unit
length such that

$\frac{48\pi\operatorname{diam}A(Q_{j})^{2}}{\operatorname{Vol}A(Q_{j}\cap
S)}\leq\frac{96\pi}{\rho}.$

Finally, summing over all good rectangles and using Lemma 18 we have

$\displaystyle\lVert f\rVert_{L^{2}(\mathbb{R}^{2})}^{2}$ $\displaystyle\leq
2\sum_{j\colon Q_{j}\text{good}}\lVert
f\rVert_{L^{2}(\mathbb{R}^{2})}^{2}\leq\sum_{j\colon
Q_{j}\text{good}}4\left(\frac{96\pi}{\rho}\right)^{C_{2}+C_{3}\lvert\ell\rvert_{1}\sqrt{E}+C_{4}\left(\lvert\ell\rvert_{1}\sqrt{B}+\lvert\ell\rvert_{1}^{2}B\right)}\lVert
f\rVert_{L^{2}(Q_{j}\cap S)}^{2}$ $\displaystyle\leq
4\left(\frac{C_{1}}{\rho}\right)^{C_{2}+C_{3}\lvert\ell\rvert_{1}\sqrt{E}+C_{4}\left(\lvert\ell\rvert_{1}\sqrt{B}+\lvert\ell\rvert_{1}^{2}B\right)}\lVert
f\rVert_{L^{2}(S)}^{2}.$

Using $\rho\leq 1$ to absorb the prefactor $4$ into the constant $C_{2}$, and
using $B\leq E$ ($H_{B}$ has no spectrum below $B$) to absorb the term
$\lvert\ell\rvert_{1}\sqrt{B}$ into $\lvert\ell\rvert_{1}\sqrt{E}$, we obtain
the statement. ∎

##  5\. Bounded domains

In this section, we define finite-volume restrictions of $H_{B}$ onto
rectangles, and indicate necessary modifications to the proof of Theorem 3 in
order to prove Theorem 4.

Let the _magnetic translations_ be defined by

$\left(\Gamma_{y}\right)_{y\in\mathbb{R}^{2}}\colon L^{2}(\mathbb{R}^{2})\to
L^{2}(\mathbb{R}^{2}),\quad\left(\Gamma_{y}f\right)(x)=\mathrm{e}^{i\frac{B}{2}(y_{2}-y_{1})}f(x-y).$

This is a family of unitary operators, where in contrast to usual translations
on $\mathbb{R}^{2}$, they only form a commutative group if we restrict them to
vectors $y$ satisfying the so-called _integer flux condition_

(14) $B(y_{2}-y_{1})\in 2\pi\mathbb{Z}.$

Following [Sjö91], we define

$\mathcal{H}_{B,\mathrm{loc}}^{m}(\mathbb{R}^{2}):=\\\ =\left\\{f\in
L^{2}_{\mathrm{loc}}(\mathbb{R}^{2})\colon\tilde{\partial}_{\alpha_{1}}\dots\tilde{\partial}_{\alpha_{p}}f\in
L^{2}_{\mathrm{loc}}(\mathbb{R}^{2})\
\forall\alpha=(\alpha_{1},\dots,\alpha_{p})\in\\{1,2\\}^{p},p\leq m\right\\},$

as well as restrictions of these spaces to boxes

$\mathcal{H}_{B}^{m}(\Lambda_{L}):=\left\\{f\mid_{\Lambda_{L}}\colon
f\in\mathcal{H}_{B,\mathrm{loc}}^{m}(\mathbb{R}^{2})\right\\},$

and their ”periodic” versions

$\mathcal{H}_{B,\mathrm{per}}^{m}(\Lambda_{L}):=\left\\{f\mid_{\Lambda_{L}}\colon
f\in\mathcal{H}_{B,\mathrm{loc}}^{m}(\mathbb{R}^{2})\ \text{with
$\Gamma_{y}f=f$ for all $y$ satisfying\leavevmode\nobreak\
\eqref{eq:integer_flux_condition}}\right\\}.$

Functions in $\mathcal{H}_{B,\mathrm{per}}^{m}(\Lambda_{L})$ satisfy
”periodic” boundary conditions where the usual periodicity has been replaced
by invariance under magnetic translations. Then, the local Landau operator
$H_{B,L}$ in the Hilbert space $L^{2}(\Lambda_{L})$ has domain

$\mathcal{D}(H_{B,L})=\mathcal{H}_{B,\mathrm{per}}^{2}(\Lambda_{L}).$

In particular, if (14) is satisfied, then $\sigma(H_{L})$ coincides with
$\sigma(H)=\\{B,3B,\dots\\}$.

Let us now indicate which modifications are necessary for Theorem 4. Large
parts of the proof of Theorem 7 (the magnetic Bernstein inequalities on
$\mathbb{R}^{2}$) are analogous but we need to justify that when performing
integration by parts, boundary terms will disappear. This was obvious for
Schwarz functions on $\mathbb{R}^{2}$, but one needs a reasoning in the finite
volume case. Due to the definition of $\mathcal{D}(H_{L})$, it follows that
one can use integration by parts for the magnetic derivatives
$\tilde{\partial}_{1},\tilde{\partial}_{2}$ of functions
$f\in\mathcal{D}(H_{L})$. However, we also need integration by parts for
higher order magnetic derivatices. For this purpose, the key is to observe
that

$\mathcal{D}(H_{B,L}^{k})\subseteq\mathcal{H}_{B,\mathrm{per}}^{2k}(\Lambda_{L})\quad\text{for
all $k\in\mathbb{N}$}.$

This can for instance be seen by unitarity of the Floquet transform which
pointwise maps $\mathcal{H}^{2k}_{B}(\mathbb{R}^{2})$ to
$\mathcal{D}(H_{B,L}^{k})(\Lambda_{L})$ and which maps the domain of
$H_{B}^{k}$ onto the one of $H_{B,L}^{k}$, cf. [Sjö91]. This justifies to also
use integration by parts for the magnetic derivatives on any function

$f\in\operatorname{Ran}\mathbf{1}_{(-\infty,E]}(H_{B,L})\subset\bigcap_{k\geq
1}\mathcal{D}(H_{B,L}^{k})$

and to prove finite-volume analoga of Theorems 7 and 12.

The remaining steps of the proof of Theorem 4 follow with obvious
modifications:

  * •

Functions in $\operatorname{Ran}\mathbf{1}_{(-\infty,E]}(H_{B},L)$ are a
priori defined on $\Lambda_{L}$, but by magnetic translation they extend to
functions on arbitrarily large boxes.

  * •

Instead of Lemma 13, which shows analyticity of $\lvert f\rvert^{2}$ on
$\mathbb{R}^{2}$, it suffices to prove analyticity of their extension of boxes
of fixed size, depending on $\ell=(\ell_{1},\ell_{2})$ – the parameters from
the definition of thickness. Instead of the global Sobolev estimate, we will
therefore use a local Sobolev estimate in Lemma 13 in order to infer
analyticity, but the constant will remain uniformly bounded.

  * •

We can no longer cover $\Lambda_{L}$ by mutually disjoint rectangles of side
lengths $\ell_{1},\ell_{2}$, but we can bound the overlap, see Remark 16

The rest of the proof of Theorem 4 works verbatim as the proof of Theorem 3.
Note that the corresponding modifications for dealing with domains which are
not $\mathbb{R}^{2}$ itself are treated for example in the general setting in
[ES21], and in a more particular setting for the pure Laplacian in [EV20,
Egi21].

##  6\. Applications

###  6.1. Controllability of the magnetic heat equation

Consider the _controlled heat equation with magnetic generator_

(15) $\begin{cases}\frac{\partial}{\partial
t}u+H_{B}u=\mathbf{1}_{S}f&\quad\text{in $\mathbb{R}^{2}\times(0,T)$},\\\
u(0)=u_{0}&\quad\in L^{2}(\mathbb{R}^{2}).\end{cases}$

System (15) describes the diffusion of a (non-interacting) gas of charged
particles in a plane, subject to a perpendicular magnetic field, and
controlled through an electric potential in $S\subseteq\mathbb{R}^{2}$.

######  Definition 19.

System (15) is called _null-controllable in time $T>0$_ if for every $u_{0}\in
L^{2}(\mathbb{R}^{d})$, there exists $f\in L^{2}((0,T)\times S)$ such that the
solution of (15) satisfies $u(T)=0$.

The reason for restricting to the target state $u(T)=0$ is that by linearity,
this is equivalent to every state $u(T)$ in the range of the semigroup
$(\mathrm{e}^{-H_{B}t})_{t>0}$ being reachable, the best notion of
controllability one can hope for in parabolic systems. By the classic Hilbert
Uniqueness Method (HUM) due to Lions [Lio88], null-controllability is
equivalent to _final-state observability_ , that is the estimate

(16) $\lVert\mathrm{e}^{-H_{B}T}u_{0}\rVert_{L^{2}(\mathbb{R}^{2})}^{2}\leq
C_{\mathrm{obs}}^{2}\int_{0}^{T}\lVert\mathrm{e}^{-H_{B}t}u_{0}\lVert_{L^{2}(S)}^{2}\mathrm{d}t\quad\text{for
all $u_{0}\in L^{2}(\mathbb{R}^{2})$},$

and the least constant $C_{\mathrm{obs}}>0$ in estimate (16) is called
_control cost in time $T>0$_.

Indeed, (15) is an example of a wider class of parabolic systems with lower
semibounded generator. There is a strategy, combining spectral inequalities
with the decay of the semigroup to prove an observability estimate: The so-
called Lebeau-Robbiano-Strategy [LR95, LRL12, TT11]. In recent years,
substantial effort has been devoted to deducing sharp estimates on the control
cost [TT11, Mil04, Mil10, NTTV18, NTTV20a].

######  Proposition 20 (Theorem 2.12 in [NTTV20b]).

Let $A\geq 0$ and let $X$ be a bounded, self-adjoint operator in a Hilbert
space $\mathcal{H}$. Assume that one has the _spectral inequality_

(17) $\lVert u_{0}\rVert_{\mathcal{H}}^{2}\leq
d_{0}\mathrm{e}^{d_{1}\sqrt{E}}\lVert
Xu_{0}\rVert_{\mathcal{H}}^{2}\quad\text{for all
$u_{0}\in\operatorname{Ran}\mathbf{1}_{(-\infty,E]}(A)$}.$

Then, for all $T>0$, the _observability inequality_

$\lVert\mathrm{e}^{-AT}u_{0}\rVert_{\mathcal{H}}^{2}\leq
C_{\mathrm{obs}}^{2}\int_{0}^{T}\lVert
X\mathrm{e}^{-tA}u_{0}\rVert_{\mathcal{H}}^{2}\ \mathrm{d}t\quad\text{for all
$u_{0}\in\mathcal{H}$}$

holds, where

$C_{\mathrm{obs}}^{2}\leq\frac{C_{5}d_{0}}{T}\left(2d_{0}\lVert
X\rVert+1\right)^{C_{6}}\exp\left(\frac{C_{7}d_{1}^{2}}{T}\right)$

for universal constants $C_{5},C_{6},C_{7}>0$.

Combining this with Theorem 3, we obtain:

######  Theorem 21.

Let $B\geq 0$ and let $S\subseteq\mathbb{R}^{d}$ be $(\ell,\rho)$-thick. Then,
System (15) is null-controllable in every time $T>0$ with cost
$C_{\mathrm{obs}}$ satisfying

(18)
$C_{\mathrm{obs}}^{2}\leq\frac{C}{T\rho^{C+C\lvert\ell\rvert_{1}^{2}B}}\exp\left(\frac{\ln\left(\frac{C}{\rho}\right)C\lvert\ell\rvert_{1}^{2}}{T}-BT\right)$

where $C>0$ is a universal constant.

The estimate (18) on the control cost $C_{\mathrm{obs}}$ has an asymptotic
behaviour which is known to be optimal for the free Laplacian, cf. the
discussion in [NTTV20a] and references therein:

  * •

As $T\to 0$, the expression $C_{\mathrm{obs}}$ behaves proportional to
$T^{-1/2}$ if $S\subset\mathbb{R}^{2}$ is dense, and proportionally to
$\exp(C/T)$ otherwise.

  * •

As $T\to\infty$, the cost decays proportionally to $\exp(-CT)$, as necessary
when the generator has a positive of its spectrum.

  * •

Finally, in the homogenization regime, where $\lvert\ell\rvert_{1}$ tends to
zero at fixed $\rho$, and fluctuations within $S$ become small while there is
a uniform lower bound on the relative density, the influence of $S$ and $B$ on
$C_{\mathrm{obs}}$ vanishes.

###### Proof of Theorem 21.

The constant in the spectral inequality of Theorem 3 is of the form

$\displaystyle\left(\frac{C_{1}}{\rho}\right)^{C_{2}+C_{3}\lvert\ell\rvert_{1}\sqrt{E}+C_{4}(\lvert\ell\rvert_{1}^{2}B)}=\underbrace{\left(\frac{C_{1}}{\rho}\right)^{C_{2}+C_{4}\lvert\ell\rvert_{1}^{2}B}}_{:=d_{0}}\cdot\exp\left(\underbrace{\ln\left(\frac{C_{1}}{\rho}\right)C_{3}\lvert\ell\rvert_{1}}_{:=d_{1}}\sqrt{E}\right).$

Applying Proposition 3 with $A:=H_{B}$ and $X:=\mathbf{1}_{S}$ for an
$(\ell,\rho)$-thick $S\subseteq\mathbb{R}^{d}$, we find that (15) is null-
controllable in every time $T>0$ with control cost satisfying

$\displaystyle C_{\mathrm{obs}}^{2}$
$\displaystyle\leq\frac{C_{4}}{T}\left(\frac{2C_{1}+1}{\rho}\right)^{C_{2}(C_{5}+1)+C_{4}(C_{5}+1)\lvert\ell\rvert_{1}^{2}B}\cdot\exp\left(\frac{\ln\left(\frac{C_{1}}{\rho}\right)^{2}C_{3}^{2}\lvert\ell\rvert_{1}^{2}}{T}\right).$
$\displaystyle=\frac{D_{1}}{T\rho^{D_{2}+D_{3}\lvert\ell\rvert_{1}^{2}B}}\exp\left(\frac{\ln\left(\frac{D_{4}}{\rho}\right)^{2}D_{5}^{2}\lvert\ell\rvert_{1}^{2}}{T}\right)$

for universal constants $D_{1}$ to $D_{5}$. This yields the bound

$C_{\mathrm{obs}}^{2}\leq\frac{C}{T\rho^{C+C\lvert\ell\rvert_{1}^{2}B}}\
\exp\left(\frac{\ln\left(\frac{C}{\rho}\right)C\lvert\ell\rvert_{1}^{2}}{T}\right).$

We improve the large time behaviour of $C_{\mathrm{obs}}$ by using
$\inf\sigma(H_{B})=\lvert B\rvert\geq 0$, see for instance [NTTV20a]. Indeed,
instead of controlling in the interval $[0,T]$, one can apply no control in
the interval $[0,T/2]$ and then work with with the new initial state
$\mathrm{e}^{-\frac{T}{2}H_{B}}u_{0}$ satisfying
$\lVert\mathrm{e}^{-\frac{T}{2}H_{B}}u_{0}\rVert_{L^{2}(\mathbb{R}^{d})}^{2}\leq\mathrm{e}^{-TB}\lVert
u_{0}\rVert_{L^{2}(\mathbb{R}^{d})}^{2}$ in the interval $[T/2,T]$. Replacing
$C$ again by $2C$ in order to absorb the factor $\frac{1}{2}$ in $T$, we
obtain the statement. ∎

We next prove that thickness is also a sufficient criterion for observability,
and thus null-controllability of the magnetic heat equation.

######  Theorem 22.

If, for any $B\geq 0$, the observability estimate (16) holds, then
$S\subset\mathbb{R}^{2}$ must be thick.

###### Proof.

In the case of the free Laplacian, that is $B=0$, this was proved
independently in [EV18] and [WWZZ19]. For $B\neq 0$, we argue as in the proof
of Theorem 11. If $S\subset\mathbb{R}^{2}$ was not thick, then for every
$n\in\mathbb{N}$, there would exist
$(y^{(n)})_{n\in\mathbb{N}}\subset\mathbb{R}^{2}$ such that
$\operatorname{Vol}(B_{n}(y^{(n)})\cap S)\leq\frac{1}{n}$ for all
$n\in\mathbb{N}$. Take $f_{y^{(n)}}$ defined as in (6), that is

$f_{y^{(n)}}(x)=\exp\left(-\frac{B}{4}\lvert
x-y^{(n)}\rvert^{2}-i\frac{B}{2}\left(x_{1}y^{(n)}_{2}-x_{2}y^{(n)}_{1}\right)\right).$

This is an eigenfunction to the eigenvalue $B$ satisfying $\lVert
f_{y^{(n)}}\rVert_{L^{2}(\mathbb{R}^{2})}^{2}=\frac{2\pi}{B}$. In particular
$\mathrm{e}^{-H_{B}t}f_{y^{(n)}}=\mathrm{e}^{-Bt}f_{y^{(n)}}$, and
$\lVert\mathrm{e}^{-H_{B}T}f_{y^{(n)}}\rVert_{L^{2}(\mathbb{R}^{2})}^{2}=\frac{2\pi}{B}\mathrm{e}^{-2BT}$.
Hence,

$\int_{0}^{T}\lVert\mathrm{e}^{-H_{B}t}f_{y^{(n)}}\rVert_{L^{2}(S)}^{2}\mathrm{d}t\leq\int_{0}^{T}\mathrm{e}^{-2Bt}\left(\lVert
f_{y^{(n)}}\rVert_{L^{2}(S\cap B_{n}(y^{(n)})}^{2}+\lVert
f_{y^{(n)}}\rVert_{L^{2}(B_{n}(y^{(n)})^{c})}^{2}\right)\mathrm{d}t\\\ \leq\
T\left(\operatorname{Vol}(S\cap
B_{n}(y^{(n)}))+\int_{n}^{\infty}\exp\left(-\frac{B}{2}r^{2}\right)r\mathrm{d}r\right)\leq\frac{T}{n}+\frac{T\exp(-\frac{Bn^{2}}{2})}{2}.$

Since this tends to zero as $n\to\infty$, inequality (16) cannot hold for any
$C_{\mathrm{obs}}>0$. ∎

We conclude that thickness is the _optimal_ , that is necessary and
sufficient, geometric criterion for null-controllability of the magnetic heat
equation – the same as for the classic heat equation.

###  6.2. Random Schrödinger operators

Random Schrödinger operators are families of operators of the form

$H_{\omega}=H_{0}+V_{\omega},\quad\omega\in\Omega$

where $H_{0}$ is a background operator in $L^{2}(\mathbb{R}^{d})$ (usually the
free Laplacian $-\Delta$, the free Laplacian with periodic potential [BCH97,
KSS98, Klo99, Ves02, ST20] or the Landau operator $H_{B}$ [CHK03, CHKR04,
CHK07]), and $(V_{\omega})_{\omega\in\Omega}$ is a random potential drawn from
a probability space $\Omega$ and modeling a disordered solid. The most common
model in this context is the _Alloy-type_ or _continuum Anderson model_

$V_{\omega}(x)=\sum_{j\in\mathbb{Z}^{d}}\omega_{j}u(x-j)$

where $0\leq u\in L^{\infty}(\mathbb{R}^{d})$ is a single-site potential of
compact support, modeling the a single atom, and
$(\omega_{j})_{j\in\mathbb{Z}^{d}}$ is a family of bounded, independent, and
independently distributed random variables.

Physical phenomena of interest in this context are _Anderson localization_ and
_Anderson delocalization_. There are several notions of Anderson localization,
the weakest one being the almost sure emergence of pure point spectrum with
exponentially decaying eigenfunctions at certain energies, and a hierarchy of
stronger notions of _dynamic localization_ , describing decay of correlations
of functions of the operator $H_{\omega}$ in space. Correspondingly, there is
a hierarchy of notions of  _delocalization_ , the strongest one being purely
absolutely continuous spectrum and weaker ones involving dynamical notions and
lower bounds on the decay of correlators in space. We refer to the monographs
[Sto01, Ves08, AW15] for a more comprehensive overview.

Whereas Anderson localization at extremal energies (the bottom of the spectrum
or near band gaps) has been observed in a variety of models, delocalization is
still mostly open and the Landau operator takes a particular role as the only
known ergodic model of random Schrödinger operators on $\mathbb{R}^{d}$ where
– under certain assumptions – a localization-delocalization transition has
been rigorously proved [GKS07]. The latter result crucially relies on the
identification of a strict dichotomy of spectral regions of localization and
delocalization [GK04].

A central ingredient in proofs of localization (and thus, indirectly, of
delocalization) are lower bounds of the form

(19)
$\left\lVert\sum_{j\in\mathbb{Z}^{d}}u(\cdot-j)f\right\rVert_{L^{2}(\Lambda_{L})}\geq
C\lVert f\rVert_{L^{2}(\Lambda_{L})}\quad\text{for all
$f\in\operatorname{Ran}\mathbf{1}_{(-\infty,E]}(H_{0,L})$,}$

and for a family of $L$ of length scales, tending to infinity, where $H_{0,L}$
denotes the restriction of $H_{0}$ onto $L^{2}(\Lambda_{L})$ with self-adjoint
boundary conditions.

Clearly, if $\sum_{j\in\mathbb{Z}^{d}}u(\cdot-j)$ is uniformly positive on a
suitable set $S\subset\mathbb{R}^{d}$, then (19) is a direct consequence of
Theorem 4. Indeed, such estimates have a tradition in the community on random
Schrödinger operators where they are also referred to as _quantitative unique
continuation principles_.

However, so far, for a unique continuation estimate as in (19) to hold, one
has usually had to assume that the function $\sum_{j\in\mathbb{Z}^{d}}u(x-j)$
be uniformly positive on an open set which had to be either periodic [CHK03,
CHK07] or had to have at least some equidistributedness in space [RMV13,
Kle13, TV15, NTTV18, Täu18]. We can now relax this to merely positivity on a
periodic set of positive measure (i.e. a periodic, thick set), which in light
of [TV21] seems to be the minimal assumption possible. Furthermore, in the
recent years, there has been interest in non-ergodic random Schrödinger
operators [RM12, RMV13, Kle13, GMRM15, TT18, MRM22, ST20, TV21], a
generalization which now also becomes accessible since we no longer rely on
periodicity of $\sum_{j\in\mathbb{Z}^{d}}u(x-j)$.

In order to to illustrate that Theorem 4 yields an improvement of existing
results, let us formulate a set of assumptions, inspired by common assumptions
in the alloy-type model, cf. for instance [CHK07, Section 1].

  1. (i)

Let $B>0$ and let the background operator be $H_{B}$. For $L>0$ satisfying the
integer flux condition let $H_{L}$ be the restriction of $H_{B}$ onto
$L^{2}(\Lambda_{L})$ with magnetic boundary conditions as defined in Section
5.

  2. (ii)

Let $(u_{j})_{j\in\mathbb{Z}^{d}}$ be a family of measurable functions
satisfying $0\leq\sum_{j\in\mathbb{Z}^{2}}u_{j}\leq 1$, and
$\sum_{jin\mathbb{Z}^{2}}u_{j}\geq\delta>0$ on a thick set.

  3. (iii)

Let $(\omega_{j})_{j\in\mathbb{Z}^{2}}$ be a family of random variables,
taking values in some interval $[m_{0},M_{0}]$. Call $\mu_{j}$ the conditional
probability measure of $\omega_{j}$, conditioned on all other random variables
$(\omega_{k})_{k\neq j}$

$\mu_{j}([E,E+\epsilon])=\mathbb{P}\left[\omega_{j}\in[E,E+\epsilon]\mid(\omega_{k})_{k\neq
j}\right],$

and define the _conditional modulus of continuity_

$s(\epsilon):=\sup_{j\in\mathbb{Z}^{2}}\mathbb{E}\left[\sup_{E\in\mathbb{R}}\mu_{j}([E,E+\epsilon])\right].$

The novelty is assumption (ii) which no longer requires that
$\sum_{j\in\mathbb{Z}^{2}}u_{j}$ be positive on a periodic, _open_ set. Define
the random Landau Hamiltonian as

$H_{B,\omega}=H_{B}+V_{\omega},\quad
V_{\omega}(x)=\sum_{j\in\mathbb{Z}^{2}}\omega_{j}u_{j}(x),$

and its restriction to boxes $\Lambda_{L}=(-\frac{L}{2},\frac{L}{2})^{2}$ as

$H_{B,\omega,L}:=H_{B,L}+V\mid_{\Lambda_{L}}\quad\text{with boundary
conditions as defined in Section\leavevmode\nobreak\
\ref{sec:finite_volume}}.$

We then obtain a generalization of [CHK07, Theorem 1.3], namely a Wegner
estimate, optimal in energy and volume:

######  Theorem 23.

Assume Hypotheses (i)-(iii) above. Then, there is $L_{0}>0$ such that for all
$E_{0}\in\mathbb{R}$, there exists $C_{W}>0$, such that for all $E\leq E_{0}$,
all $\epsilon\in(0,1]$, and all $L\geq L_{0}$ satisfying the integer flux
condition

$BL\in 2\pi\mathbb{N}$

we have the Wegner estimate

$\displaystyle\mathbb{P}\left[\operatorname{dist}(\sigma(H_{B,\omega,L},E)<\epsilon\right]$
$\displaystyle\leq\mathbb{E}\left[\operatorname{Tr}\mathbf{1}_{[E-\epsilon,E+\epsilon]}(H_{\omega,L}\right]$
$\displaystyle\leq C_{W}s(2\epsilon)L^{2}.$

###### Proof.

The proof is completely analogous to the one in [CHK07], the only difference
being that in our case, the potential

$\tilde{V}(x):=\sum_{j\in\mathbb{Z}^{2}}u(x-j)$

is no longer periodic and not uniformly positive on an open set, but merely on
a thick set. But periodicity and openness were exactly used in [Theorem
4.1][CHK07] to prove

(20) $\Pi_{n,L}\tilde{V}\mid_{\Lambda_{L}}\Pi_{n,L}\geq C\Pi_{n,L}$

in the sense of quadratic forms, where
$\Pi_{n,L}=\mathbf{1}_{\\{(2n+1)B\\}}(H_{B,L})$ is the spectral projector onto
the $n$-th Landau level. But in light of Assumption (ii) above, (20) in our
situation is an immediate consequence of Theorem 4. For more details, we also
refer to [TV21], where the corresponding argument is outlined in the case
where the background operator is the free Laplacian. ∎

If the random family of operator $(H_{\omega})_{\omega\in\Omega}$ is ergodic,
then its integrated density of states

$N(E):=\lim_{L\to\infty}\frac{\operatorname{Tr}\mathbf{1}_{(-\infty,E]}(H_{\omega,L})}{\operatorname{Vol}\Lambda_{L}}$

exists almost surely. As a corollary, we obtain in this case the analogon of
[CHK07, Theorem 1.2], namely regularity of the integrated density of states:

######  Corollary 24.

Assume Hypotheses (i)–(iii) above and assume that the IDS exists almost surely
for the family $(H_{\omega})_{\omega\in\Omega}$. Then, for all
$E_{0}\in\mathbb{R}$, there is $C>0$ such that for all $E\leq E_{0}$ and all
$\epsilon\in(0,1]$, we have

$0\leq N(E+\epsilon)-N(E)\leq Cs(\epsilon).$

In particular, if all $\omega_{j}$ are independent and identically distributed
with bounded density, then the IDS is locally Lipschitz continuous.

Finally, note that Wegner estimates as in Theorem 23 are one important
ingredient in so-called _multiscale analysis_ proofs of localization, the
other central ingredient being _initial length scale estimates_ , see [Sto01,
GK01, GK03]. Initial length scale estimates can for instance be inferred from
exponentially decaying upper bounds on the IDS near its minimum as derived in
[KR06] in the context of Lifshitz tails. Theorem 4.1 (iii) in [KR06] states
such a lower bound under the hypothesis

$u(x)\geq C\ \mathbf{1}_{\lvert x-x_{0}\rvert<\epsilon}(x)\quad\text{for some
$x_{0}\in\mathbb{R}^{2}$, $C,\epsilon>0$}.$

A closer inspection of the proof of said theorem yields that it essentially
relies on lower bounds of the form

$\lVert V_{\omega}\mid_{\Lambda_{L}}\psi\rVert_{L^{2}(\Lambda_{L})}^{2}\geq
C\lVert\psi\rVert_{L^{2}(\Lambda_{L})}^{2}\quad\text{for all
$\psi\in\mathbf{1}_{\\{B\\}}(H_{B,L})$}$

for configurations $\omega$ with sufficiently high probability, cf. [KR06,
Estimate (4.29)]. This can be readily replaced by Theorem 4. In conclusions,
by combining Theorem 23 with an initial scale estimate, derived from Theorem 4
and the method of proof of [KR06] in the bootstrap multiscale analysis, one
infers:

######  Corollary 25.

Let $0\leq u\leq 1$ be measurable with non-empty, compact support. Let
$(\omega_{j})_{j\in\mathbb{Z}^{2}}$ be a family of independent and identically
distributed random variables with bounded support, a bounded density $\rho$,
and $\inf\operatorname{supp}\rho=0$. Then, there is $\epsilon>0$, such that
the family of operators

$H_{B,\omega}:=H_{B}+\sum_{j\in\mathbb{Z}^{2}}\omega_{j}u(\cdot-j)$

exhibits strong dynamical localization in Hilbert Schmidt norm (and thus all
other, weaker forms of Anderson localization) in the interval
$[B,B+\epsilon]$.

The novelty is that the support of $u$ now no longer needs to be open, which
seems to be the minimal assumption necessary.

##  Appendix A Proof of Lemma 15 via Remez inequality

For convenience and the sake of self-containedness, we provide here a proof of
Lemma 15. The version given here is essentially Lemma 1 in [Kov00]. The proof
relies on the following variant of the Remez inequality for polynomials, which
can be inferred from [BE95, Theorem 5.1.1].

######  Lemma 26 (Remez inequality).

Let $P\colon\mathbb{C}\to\mathbb{C}$ be a polynomial of degree
$n\in\mathbb{N}$. Then, for any measurable $E\subset[0,1]$ with positive
measure

(21) $\sup_{t\in[0,1]}\lvert
P(t)\rvert\leq\left(\frac{4}{\operatorname{Vol}E}\right)^{n}\sup_{t\in
E}\lvert P(x)\rvert.$

Recall that $D_{r}\subset\mathbb{C}$ denotes the complex polydisc with radius
$r>0$, centered at $0$.

###### Proof of Lemma 15.

The function $\varphi$ is not the zero function, so it has a finite number of
zeroes in $D_{2}$, which we denote by $w_{1},\dots,w_{n}$ (counting
multiplicities). Define

$g(z):=\varphi(z)\cdot\prod_{k=1}^{n}\frac{4-\overline{w}_{k}z}{2(w_{k}-z)}=\varphi(z)\cdot\frac{Q(z)}{P(z)}.$

We have $\lvert g(0)\rvert\geq 1$ and $\max_{z\in D_{2}}\lvert
g(z)\rvert\leq\max_{z\in D_{2}}\lvert\varphi(z)\rvert\leq M_{\varphi}$ by the
maximum principle since the Blaschke product

$\prod_{k=1}^{n}\frac{2(w_{k}-z)}{4-\overline{w}_{k}z}=\frac{P(z)}{Q(z)}$

has modulus one on the boundary of $D_{2}$. Thus, $g$ is an analytic function
without zeroes in $D_{2}$, and the function $\ln M_{\varphi}-\ln\lvert
g(z)\rvert$ is positive and harmonic in $D_{2}$. By Harnack’s inequality

$\max_{z\in D_{1}}\left(\ln M_{\varphi}-\ln\lvert
g(z)\rvert\right)\leq\frac{1+\frac{1}{2}}{1-\frac{1}{2}}\left(\ln
M_{\varphi}-\ln\lvert g(0)\rvert\right)\leq 3\ln M_{\varphi},$

whence in particular

$\min_{z\in D_{1}}\lvert g(z)\rvert\geq
M_{\varphi}^{-2},\quad\text{and}\quad\frac{\max_{t\in[0,1]}\lvert
g(t)\rvert}{\min_{t\in[0,1]}\lvert g(t)\rvert}\leq M_{\varphi}^{3}.$

Likewise, for every $k\in\\{1,\dots,n\\}$, the function
$z\mapsto(4-\overline{w_{k}}z)$ is analytic in $D_{1}$ without zeroes. By the
maximum principle $z\mapsto\lvert 4-\overline{w_{k}}z\rvert$ takes its maximum
and minimum in $D_{1}$ on the boundary where

$2\leq\lvert 4-\overline{w_{k}}z\rvert\leq 6.$

This implies

$\frac{\max_{t\in[0,1]}\lvert Q(t)\rvert}{\min_{t\in[0,1]}\lvert
Q(t)\rvert}\leq\prod_{k=1}^{n}\frac{\max_{z\in D_{1}}\lvert
4-\overline{w_{k}}z\rvert}{\min_{z\in D_{1}}\lvert
4-\overline{w_{k}}z\rvert}\leq 3^{n}.$

Combining this with Lemma 26, we find

$\displaystyle\sup_{t\in[0,1]}\lvert\varphi(x)\rvert$
$\displaystyle\leq\max_{t\in[0,1]}\lvert
g(x)\rvert\frac{\max_{t\in[0,1]}\lvert P(x)\rvert}{\min_{t\in[0,1]}\lvert
Q(x)\rvert}$ $\displaystyle\leq
M_{\varphi}^{3}\cdot\left(\frac{12}{\operatorname{Vol}E}\right)^{n}\min_{t\in[0,1]}\lvert
g(x)\rvert\frac{\sup_{t\in E}\lvert P(x)\rvert}{\max_{t\in[0,1]}\lvert
Q(x)\rvert}$ $\displaystyle\leq
M_{\varphi}^{3}\cdot\left(\frac{12}{\operatorname{Vol}E}\right)^{n}\sup_{t\in
E}\lvert\varphi(x)\rvert.$

Finally, by Jensen’s formula, the number $n$ of zeroes of $\varphi$ in $D_{2}$
is bounded by $\frac{\ln M_{\varphi}}{\ln 2}$. Thus

$\sup_{t\in[0,1]}\lvert\varphi(x)\rvert\leq
M_{\varphi}^{3}\left(\frac{12}{\operatorname{Vol}E}\right)^{\frac{\ln
M_{\varphi}}{\ln 2}}\sup_{t\in
E}\lvert\varphi(x)\rvert\leq\left(\frac{12}{\operatorname{Vol}E}\right)^{2\frac{\ln
M_{\varphi}}{\ln 2}}\sup_{t\in E}\lvert\varphi(x)\rvert.\qed$

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