# Observability of the heat equation from very small sets

A. Walton Green111Washington University, St. Louis, MO<EMAIL_ADDRESS>A.
W. G. is supported in part by NSF grant DMS-2202813, Kévin Le Balc’h222Inria,
Sorbonne Université, Université de Paris, CNRS, Laboratoire Jacques-Louis
Lions, Paris, France<EMAIL_ADDRESS>Jérémy Martin 333Inria,
Sorbonne Université, Université de Paris, CNRS, Laboratoire Jacques-Louis
Lions, Paris, France<EMAIL_ADDRESS>, Marcu-Antone Orsoni
444University of Toronto Mississauga, Mississauga, ON<EMAIL_ADDRESS>
###### Abstract

We consider the heat equation set on a bounded $C^{1}$ domain of
$\mathbb{R}^{n}$ with Dirichlet boundary conditions. The first purpose of this
paper is to prove that the heat equation is observable from any measurable set
$\omega$ with positive $(n-1+\delta)$-Hausdorff content, for $\delta>0$
arbitrary small. The proof relies on a new spectral estimate for linear
combinations of Laplace eigenfunctions, obtained via a Remez type inequality,
and the use of the so-called Lebeau-Robbiano’s method. Even if this
observability result is sharp with respect to the scale of Hausdorff
dimension, our second goal is to construct families of sets $\omega$ which
have codimension greater than or equal to $1$ for which the heat equation
remains observable.

###### Contents

  1. 1 Introduction
    1. 1.1 Observability and null-controllability of the heat equation
    2. 1.2 Main results
  2. 2 Proof of the main observability estimate, i.e. Theorem 1.3
    1. 2.1 The spectral estimate
    2. 2.2 Proof of the spectral estimate
    3. 2.3 Proof of the Remez inequality
      1. 2.3.1 One-dimensional case
      2. 2.3.2 Higher Dimensions
    4. 2.4 Adaptation of the Lebeau-Robbiano’s method
  3. 3 Observability from measurable sets with lower Hausdorff dimension
    1. 3.1 Observability from a finite subset
    2. 3.2 Spectral inequalities for Cartesian products
    3. 3.3 Observability inequalities for Cartesian products
    4. 3.4 Examples
  4. A Another proof of the spectral estimate

##  1 Introduction

###  1.1 Observability and null-controllability of the heat equation

Given $T>0$, $\Omega$ a bounded open connected set of $\mathbb{R}^{n}$,
$\omega\subset\Omega$ a measurable set, this paper aims at establishing
observability estimates of the heat equation

$\left\\{\begin{array}[]{ll}\partial_{t}u-\Delta u=0&\text{ in
}(0,T)\times\Omega,\\\ u=0&\text{ on }(0,T)\times\partial\Omega,\\\
u(0,\cdot)=u_{0}&\text{ in }\Omega.\end{array}\right.$ (1.1)

First, let us recall some basic well-posedness properties of the heat equation
(1.1). From [Eva10, Sections 7.1], for every $u_{0}\in L^{2}(\Omega)$, there
exists a unique weak solution $u\in L^{2}(0,T;H_{0}^{1}(\Omega))\cap
H^{1}(0,T;H^{-1}(\Omega))$ of the parabolic initial/boundary-value problem
(1.1). By Sobolev embedding on mixed spaces, we deduce that $u\in
C([0,T];L^{2}(\Omega))$. Moreover, by the $L^{2}-L^{\infty}$ regularizing
effect of the heat equation, see for instance [CH98, Proposition 3.5.7], we
have $u(T,\cdot)\in L^{\infty}(\Omega)$. If we assume furthermore that
$u_{0}\in L^{\infty}(\Omega)$, then $u\in L^{\infty}((0,T)\times\Omega)$
according to the weak maximum principle, see for instance [WYW06, Section
4.2].

With this at hand, let us introduce the following notion of observability of
the heat equation (1.1) from $\omega$ at time $T$ as follows.

######  Definition 1.1.

Let $T>0$ and $\omega\subset\Omega$ be a measurable set. The heat equation
(1.1) is said to be observable from $\omega$ at time $T$ if there exists a
positive constant $C=C(\Omega,\omega,T)>0$ such that

$\|u(T,\cdot)\|_{L^{\infty}(\Omega)}\leq
C\int_{0}^{T}\sup_{x\in\omega}|u(t,x)|dt,\qquad\forall u_{0}\in
L^{\infty}(\Omega).$ (1.2)

All the terms in (1.2) are well-defined (and finite) according to the previous
recalled properties of the heat equation. Indeed the left hand side is well-
defined according to the property $u(T,\cdot)\in L^{\infty}(\Omega)$ because
of the regularizing effects while the right hand side is also well-defined
according to the weak maximum principle that implies $u\in
L^{\infty}((0,T)\times\omega)$.

Note that this notion of observability (1.2) differs from the more standard
one, usually called $L^{2}-L^{2}$ observability estimate,

$\|u(T,\cdot)\|_{L^{2}(\Omega)}\leq
C\left(\int_{0}^{T}\int_{\omega}|u(t,x)|^{2}dtdx\right)^{1/2},\qquad\forall
u_{0}\in L^{2}(\Omega).$ (1.3)

First because of the Hilbertian structure of $L^{2}(\Omega)$ and
$L^{2}((0,T)\times\omega)$, (1.3) is well-adapted for employing duality
methods. For instance, let us consider the associated controlled heat equation

$\left\\{\begin{array}[]{ll}\partial_{t}y-\Delta y=h1_{\omega}&\text{ in
}(0,T)\times\Omega,\\\ y=0&\text{ in }(0,T)\times\partial\Omega,\\\
y(0,\cdot)=y_{0}&\text{ in }\Omega.\end{array}\right.$ (1.4)

In the controlled system (1.4), at time $t\in[0,+\infty)$,
$y(t,\cdot):\Omega\to\mathbb{R}$ is the state and
$h(t,\cdot):\omega\to\mathbb{R}$ is the control. The following classical
result, called Hilbert Uniqueness Method [Cor07, Theorem 2.48], holds.

######  Theorem 1.2 (H.U.M.).

Let $T>0$, $\omega\subset\Omega$ be a measurable set and
$C_{\mathrm{Cost}}=C(\Omega,\omega,T)>0$ be a positive constant. The following
two assertions are equivalent.

  1. 1.

The controlled heat equation (1.4) is null-controllable at time $T$ from
$\omega$ with control cost $C=C_{\mathrm{Cost}}>0$, i.e. for every $y_{0}\in
L^{2}(\Omega)$, there exists $h\in L^{2}((0,T)\times\omega)$ with

$\|h\|_{L^{2}((0,T)\times\omega)}\leq
C_{\mathrm{Cost}}\|y_{0}\|_{L^{2}(\Omega)},$ (1.5)

such that the unique weak solution $y\in C([0,T];L^{2}(\Omega))$ of (1.4)
satisfies $y(T,\cdot)=0$.

  2. 2.

The heat equation (1.1) is $L^{2}-L^{2}$-observable at time $T$ from $\omega$
with observability constant $C=C_{\mathrm{Cost}}>0$, i.e. (1.3) holds true
with $C=C_{\mathrm{Cost}}>0$.

Secondly, (1.3) is also well-adapted when considering open sets $\omega$ or
even measurable sets $\omega$ such that $|\omega|>0$ where $|\cdot|$ stands
for the $n$-dimensional Lebesgue measure in $\mathbb{R}^{n}$. But here because
we focus on situations of measurable sets $\omega\subset\Omega$ where possibly
$|\omega|=0$, we will work with the more accurate observability notion (1.2)
of Definition 1.1.

From now and in all the following, we assume that $\Omega$ satisfies the
following hypothesis.

$\text{The set}\ \Omega\ \text{is a bounded Lipschitz open connected set of
$\mathbb{R}^{n}$, locally star-shaped.}$ (H)

See [AEWZ14, Definition 4] for a precise definition of locally star-shaped
domains. Note that from [AEWZ14, Remark 6] such domains include $C^{1}$
bounded domains, Lipschitz polyhedrons in $\mathbb{R}^{n}$ with $n\geq 3$ and
bounded convex domains.

###  1.2 Main results

Our goals in this paper are twofold. Our main theorem provides observability
results for (1.1) when the Hausdorff dimension of the observable set $\omega$
is strictly greater than $n-1$ (i.e. has codimension strictly less than $1$).
Let us recall that for $s\geq 0$, the $s$-Hausdorff content of a set
$E\subset\mathbb{R}^{n}$ is

$\mathcal{C}_{\mathcal{H}}^{s}(E)=\inf\left\\{\sum_{j}r_{j}^{s}\ ;\
E\subset\bigcup_{j}B(x_{j},r_{j})\right\\},$ (1.6)

and the Hausdorff dimension of $E$ is defined as

$\text{dim}_{\mathcal{H}}(E)=\inf\\{s\geq 0\ ;\
\mathcal{C}_{\mathcal{H}}^{s}(E)=0\\},\quad\text{codim}_{\mathcal{H}}(E)=n-\text{dim}_{\mathcal{H}}(E).$

Our main result is as follows.

######  Theorem 1.3.

Let $T>0$ and $\delta>0$. Let $\omega\subset\Omega$ be a measurable subset
satisfying $\mathcal{C}^{n-1+\delta}_{\mathcal{H}}(\omega)>0$, then the heat
equation (1.1) is observable from $\omega$ at time $T$.

More precisely, for measurable subset $\omega\subset\Omega$ satisfying
$\mathcal{C}^{n-1+\delta}_{\mathcal{H}}(\omega)>0$, there exists
$C=C(\Omega,\delta,\omega)>0$ such that

$\|u(T,\cdot)\|_{L^{\infty}(\Omega)}\leq
Ce^{C/T}\int_{0}^{T}\sup_{x\in\omega}|u(t,x)|dt,\qquad\forall u_{0}\in
L^{\infty}(\Omega).$ (1.7)

Some comments with respect to the geometrical assumptions on $\Omega$ and
$\omega$ are in order.

  * •

When $\omega\subset\Omega$ is a nonempty open set, [FR71] proved (1.3) in the
one-dimensional case by moment’s method then more than twenty years later,
[LR95] and [FI96] generalized independently this result to the multi-
dimensional case by using Carleman estimates. When $\Omega$ is
$C^{\infty}$-smooth, [LR95] actually used local elliptic Carleman estimates to
prove a spectral estimate for linear combination of eigenfunctions (see [JL99,
Theorem 4.3] for a precise statement), then the authors introduced the now so-
called Lebeau-Robbiano’s method for passing from a spectral estimate to the
observability of the heat equation (see for instance [Mil10, Theorem 2.2] or
[BPS18, Theorem 2.1]). When $\Omega$ is $C^{2}$, [FI96] directly proved
observability estimates for the heat equation with the help of global
parabolic Carleman estimates. Our proof of Theorem 1.3 follows Lebeau-
Robbiano’s method, so our crucial points would consist in deriving new
spectral estimates.

  * •

When $\omega$ is only assumed to be measurable with positive Lebesgue measure
i.e. $|\omega|>0$ and $\Omega$ is Lipschitz and locally star-shaped, [AEWZ14]
proved (1.3). Their result is actually based on the obtaining of spectral
estimate of Jerison-Lebeau’s type together with the analytic properties of
linear combination of eigenfunctions inside the domain $\Omega$. It is worth
mentioning that as far as we know the obtaining of (1.3) for general bounded
Lipschitz domains $\Omega$ with observation localized in measurable sets
$\omega$ such that $|\omega|>0$ or even for nonempty open sets $\omega$ is
open (see [BP20] for some progress in that direction).

  * •

When $\Omega$ is $W^{2,\infty}$, [BM23] proved (1.2) for measurable subsets
$\omega$ with possibly zero-Lebesgue measure but with positive Hausdorff
content satisfying $\mathcal{C}^{n-1+\delta}_{\mathcal{H}}(\omega)>0$ for a
$\delta\in(0,1)$ that is a priori close to $1$ by using recent quantitative
propagation of smallness results for gradient of solutions to $A$-harmonic
functions from [LM18]. In particular, they went a little bit beyond [AEWZ14]
proving the observability from measurable sets that are of zero
$n$-dimensional Lebesgue measure. Their result is also valid replacing the
flat Laplacian $-\Delta$ by a divergence elliptic operator with Lipschitz
coefficients, i.e. $-\mathrm{div}(A(x)\nabla\cdot)$ with a matrix-valued
Lipschitz and uniformly elliptic diffusion $A$. Note also that it was the
first observability result for the heat equation from a measurable set with
zero-Lebesgue measure in the multi-dimensional case.

  * •

In the one-dimensional case, i.e. $n=1$, [Zhu23] exactly obtained Theorem 1.3
replacing the flat Laplacian $-\partial_{xx}^{2}$ by a rough divergence
elliptic operator, $-\partial_{x}(a(x)\partial_{x}\cdot)$ where $a\in
L^{\infty}(\Omega)$ and uniformly elliptic. Their main result is a [LM18]-like
quantitative propagation of smallness result for gradient of $a$-harmonic
functions in the two-dimensional case for sets with Hausdorff dimensions
$\mathrm{dim}_{\mathcal{H}}(E)=\delta>0$ for arbitrary $\delta>0$. It is
obtained exploiting the very specific relation between $a$-harmonic functions
and holomorphic functions in the complex plane.

  * •

The assumption made in Theorem 1.3 on $\omega$, satisfying
$\mathcal{C}^{n-1+\delta}_{\mathcal{H}}(\omega)>0$ for arbitrary $\delta>0$,
is sharp in the following sense: there exists a subset $\omega$ with
$\mathrm{dim}_{\mathcal{H}}(\omega)=n-1$ such that (1.2) does not hold from
$\omega$. Indeed, assume that $\Omega$ is $C^{1}$ and let
$\omega:=\\{x\in\Omega\ ;\ \varphi_{\lambda}(x)=0\\}$ be a nodal set of a
Laplace eigenfunction $\varphi_{\lambda}$ associated to the eigenvalue
$\lambda>0$ with Dirichlet boundary conditions on $\partial\Omega$ i.e.

$\left\\{\begin{array}[]{ll}-\Delta\varphi_{\lambda}=\lambda\varphi_{\lambda}&\text{
in }\Omega,\\\ \varphi_{\lambda}=0&\text{ on
}\partial\Omega,\end{array}\right.$ (1.8)

such that $\|\varphi_{\lambda}\|_{L^{\infty}(\Omega)}=1$. Then it is a well-
known fact that $\mathcal{C}^{n-1}_{\mathcal{H}}(\omega)>0$, actually one can
prove that $\mathcal{C}^{n-1}_{\mathcal{H}}(\omega)\geq c\sqrt{\lambda}$ for
$\lambda$ sufficiently large, see [LMNN21]. Taking $\varphi_{\lambda}$ as a
initial data, we obtain that the solution $u$ of (1.1) is given by
$u(t,x)=e^{-\lambda t}\varphi_{\lambda}(x)$ so the left hand side of the
observability estimate (1.2) is equal to $e^{-\lambda T}$, while the right
hand side vanishes. Hence, the inequality (1.2) cannot hold for this specific
$\omega$.

  * •

Note that the observability estimate (1.7) is quantitative in function of $T$
only. If one wants to obtain a quantitative estimate in function of the
Hausdorff content of $\omega$, we can establish the following statement in the
spirit of [AEWZ14]. For $x_{0}\in\Omega$, $R\in(0,1]$ be such that
$B(x_{0},4R)\subset\Omega$, for every $m>0$, there exists
$C=C(\Omega,\delta,R,m)>0$ such that for every measurable subset
$\omega\subset B(x_{0},R)$ with
$\mathcal{C}^{n-1+\delta}_{\mathcal{H}}(\omega)\geq m$, one has

$\|u(T,\cdot)\|_{L^{\infty}(\Omega)}\leq
Ce^{C/T}\int_{0}^{T}\sup_{x\in\omega}|u(t,x)|dt,\qquad\forall u_{0}\in
L^{\infty}(\Omega).$ (1.9)

From Theorem 1.3 and a duality argument in the spirit of Theorem 1.2 we can
deduce a null-controllability result for (1.4). In the following, we denote by
$\mathcal{M}(\Omega)$ the space of Borel measures on $\Omega$. Recall that the
norm of $\mathcal{M}(\Omega)$ is defined as

$\|\mu\|_{\mathcal{M}(\Omega)}=\sup_{f\in
C_{c}^{0}(\Omega)}\frac{\left|\int_{\Omega}fd\mu\right|}{\|f\|_{\infty}}.$
(1.10)

######  Theorem 1.4.

Let $T>0$ and $\delta>0$. Let $\omega\subset\Omega$ be a measurable subset
satisfying $\mathcal{C}^{n-1+\delta}_{\mathcal{H}}(\omega)>0$, then (1.4) is
null-controllable at time $T$, that is for every $y_{0}\in L^{2}(\Omega)$,
there exists $h\in L^{\infty}(0,T;\mathcal{M}(\Omega))$ such that the
(very)-weak solution $y$ of (1.4) satisfies $y(T,\cdot)=0$.

More precisely, for measurable subset $\omega\subset\Omega$ satisfying
$\mathcal{C}^{n-1+\delta}_{\mathcal{H}}(\omega)>0$, there exists
$C=C(\Omega,\delta,\omega)>0$, for every $y_{0}\in L^{2}(\Omega)$, there
exists $h\in L^{\infty}(0,T;\mathcal{M}(\Omega))$ satisfying

$\sup_{t\in(0,T)}\|h(t)\|_{\mathcal{M}(\Omega)}\leq
Ce^{\frac{C}{T}}\left\lVert y_{0}\right\rVert_{L^{2}(\Omega)},$ (1.11)

such that the (very)-weak solution $y$ of (1.4) satisfies $y(T,\cdot)=0$.

The proof of Theorem 1.4 will be omitted because it is a standard adaptation
of [BM23, Section 5] starting from Theorem 1.3. We insist on the fact that a
first difficulty in the analysis is to give a meaning of the concept of
(very)-weak solution to (1.4) with a control $h=h(t)$, supported in $\omega$
and taking its values in $\mathcal{M}(\Omega)$.

Notwithstanding the sharpness in the scale of Hausdorff dimension of Theorem
1.3 just demonstrated, our second goal is to exemplify specially constructed
families of sets which have codimension greater than or equal to $1$ for which
the heat equation remains observable. The main examples of interest are first,
the cartesian product of any bounded set
$\omega^{\prime}\subset\mathbb{R}^{n-1}$ with
$\dim_{\mathcal{H}}(\omega^{\prime})>n-2$ and a single, specially chosen point
$x_{0}$; see Corollary 3.5 below. Second, we can in fact construct closed
observation sets of zero dimension, see Corollary 3.8 below. Therein, we show
that the set of points

$\\{(x_{0},k^{-\alpha}):k=1,2,\ldots\\}$

forms an observation set for the heat equation on $(0,1)^{2}$ for any
$\alpha>0$ and a suitably chosen point $x_{0}$.

A few natural natural open questions immediately arise. First, is
$\\{(x_{0},e^{-k}):k=1,2\ldots\\}$ an observation set in $(0,1)^{2}$? Second,
a potential replacement in higher dimensions for a single point with
prescribed algebraic properties would be certain algebraic curves. At this
point it is even open whether the diagonal line
$\\{(x,y)\subset(0,1)^{2}:x=y\\}$ is an observation set for the heat equation
on $(0,1)^{2}$. See the discussion around 3.10 below for one possible path to
solve this problem.

##  2 Proof of the main observability estimate, i.e. Theorem 1.3

In order to prove Theorem 1.3, we will follow the classical Lebeau-Robbiano’s
method, [LR95], consisting in establishing first spectral estimate. The
unbounded operator $-\Delta$ with domain $H^{2}(\Omega)\cap H_{0}^{1}(\Omega)$
is self-adjoint with compact resolvant so it admits an orthonormal basis of
eigenfunctions $(\varphi_{k})_{k\geq 1}$, associated to the sequence of real
eigenvalues $(\lambda_{k})_{k\geq 1}$. Given $\Lambda>0$, we introduce the
spectral projector $\Pi_{\Lambda}$ as follows

$\Pi_{\Lambda}u=\sum_{\lambda_{k}\leq\Lambda}\langle
u,\varphi_{k}\rangle_{L^{2}}\varphi_{k},\qquad\forall u\in L^{2}(\Omega).$
(2.1)

###  2.1 The spectral estimate

The main result of this section states the following spectral estimates.

######  Theorem 2.1.

For all $\delta>0$, for every $\omega\subset\Omega$ such that
$\mathcal{C}_{\mathcal{H}}^{n-1+\delta}(\omega)>0$, there exists
$C=C(\Omega,\delta,\omega)>0$ such that for every $\Lambda>0$, we have

$\left\lVert\Pi_{\Lambda}u\right\rVert_{L^{\infty}(\Omega)}\leq
Ce^{C\sqrt{\Lambda}}\sup_{x\in\omega}\left|(\Pi_{\Lambda}u)(x)\right|\qquad\forall
u\in L^{2}(\Omega).$ (2.2)

On the one hand, Theorem 2.1 improves the spectral estimates from [AEWZ14],
[AE13], [BM23] with respect to the Hausdorff dimension of the observation set
and extends [Zhu23] to the multi-dimensional setting. On the other hand,
Theorem 2.1 is only valid for an elliptic operator with real analytic metric
while the spectral estimate of [BM23] is valid for $W^{1,\infty}$ metric and
the one of [Zhu23] is valid for $L^{\infty}$ metric. We focus our attention on
the case of the pure Laplacian (constant coefficients), but we will point out
the changes needed to handle the case of real-analytic coefficients in the
Appendix when we discuss an alternative proof of Theorem 2.1. Therein, we will
show that Theorem 2.1 actually follows in the now-standard way from a
propagation of smallness result for gradients of harmonic functions due to
Malinnikova [Mal04], and the well-known lifting strategy of Jerison and Lebeau
[JL99]. However, here we give a direct and elementary proof of (2.2) which
allows us to deal with the linear combinations of eigenfunctions directly,
only relying on the Bernstein inequality (2.4), see below, in the vein of e.g.
[Kov01, JM22, NSV04, LM19].

###  2.2 Proof of the spectral estimate

The main step for obtaining Theorem 2.1, which relies on the simple Lagrange
interpolation formula and a slicing/rotation argument for capacity, is the
following proposition which is proved at the end of this part.

######  Proposition 2.2.

Let $r,\delta,m>0$. There exists $C=C(r,\delta,m)>0$ such that for any ball
$B\subset\mathbb{R}^{n}$ of radius $r$, any $E\subset B$ with
$\mathcal{C}^{n-1+\delta}_{\mathcal{H}}(E)\geq m$, any $(M_{\ell})_{\ell\geq
0}\subset(0,\infty)^{\mathbb{N}}$ with

$\frac{M_{\ell}}{\ell!}\to 0\ \text{as}\ \ell\to+\infty,$ (2.3)

any $f\in C^{\infty}(B)$ satisfying

$\sup_{B}|f^{(\ell)}|\leq\frac{M_{\ell}}{r^{\ell}}\sup_{B}|f|,\quad\ell\geq
0,$ (2.4)

then for $\ell_{0}$ defined as

$\ell_{0}:=\inf\\{\ell\in\mathbb{N}\ ;\ 2M_{\ell}\leq\ell!\\}\in\mathbb{N},$
(2.5)

there holds

$\sup_{B}\left|f\right|\leq C^{\ell_{0}}\sup_{E}\left|f\right|.$

For now, let us prove Theorem 2.1 using Proposition 2.2.

###### Proof of Theorem 2.1 using Proposition 2.2.

First, from the assumption on $\omega$, we can find a ball $B\subset\Omega$ of
radius $r>0$ and $c=c(\Omega,\delta,\omega)>0$ such that

$\mathcal{C}_{\mathcal{H}}^{n-1+\delta}(B\cap\omega)\geq c.$

Then, from [AEWZ14, Theorem 3], there exists $C=C(\Omega,\delta,\omega)>0$
such that

$\left\lVert\Pi_{\Lambda}u\right\rVert_{L^{2}(\Omega)}\leq
Ce^{C\sqrt{\Lambda}}\sup_{B}\left|\Pi_{\Lambda}u\right|.$ (2.6)

Henceforth $C$ may change in each instance, yet always remains only dependent
on $\Omega,\delta,\omega$. For simplicity, let us now fix $\Lambda>0$ and set
$v=\Pi_{\Lambda}u$. Then, for any $p\geq 0$,

$(-\Delta)^{p}v=\sum_{\lambda_{k}\leq\Lambda}\lambda_{k}^{p}\langle
v,\varphi_{k}\rangle\varphi_{k}\ \text{in}\ \Omega,\quad v=0\ \text{on}\
\partial\Omega,$ (2.7)

so Sobolev embeddings, orthogonality of $(\varphi_{k})_{k\geq 0}$, and (2.6)
imply that

$\|\nabla^{\ell}v\|_{L^{\infty}(\Omega)}\leq C\|v\|_{H^{\ell+n}(\Omega)}\leq
C\Lambda^{\frac{\ell+n}{2}}\left\lVert
v\right\rVert_{L^{2}(\Omega)}\leq\sqrt{\Lambda}^{\ell}Ce^{C\sqrt{\Lambda}}\sup_{B}\left|v\right|\qquad\forall\ell\geq
0.$ (2.8)

Now, we are in the position to apply Proposition 2.2 with $E=B\cap\omega$,
noticing that $v$ satisfies (2.4) with

$M_{\ell}=C^{\sqrt{\Lambda}}\left(C\sqrt{\Lambda}\right)^{\ell}.$

By Stirling’s formula, (2.3) holds and $M_{\ell}/\ell!<\frac{1}{2}$ once
$\ell$ is larger than $A\sqrt{\Lambda}$ for some $A=A(\Omega,\delta,\omega)$ a
large absolute constant multiple of $C$. Therefore, Proposition 2.2 implies

$\sup_{B}|v|\leq C^{A\sqrt{\Lambda}}\sup_{\omega}|v|,$

which when combined with (2.8) with $\ell=0$ establishes (2.2). ∎

###  2.3 Proof of the Remez inequality

In this part, we prove Proposition 2.2.

####  2.3.1 One-dimensional case

We begin by the one-dimensional case. Our starting point is the following very
simple Bang-type estimate for analytic functions. It is a simplified version
of [NSV04, Estimate (4.1)] or [JM22, Lemma 4.7] with one small modification,
replacing the Lebesgue measure with the following quantity related to
transfinite diameter, see [Pom92].

For any set $E\subset[0,1]$, define

$\gamma_{k}(E)=\sup_{\\{x_{i}\\}_{i=1}^{k}\subset E}\min_{i}\prod_{j\neq
i}|x_{i}-x_{j}|.$ (2.9)

The following one-dimensional Remez estimate holds.

######  Proposition 2.3.

Let $(M_{\ell})_{\ell\geq 0}\subset(0,\infty)^{\mathbb{N}}$ satisfying (2.3),
$f\in C^{\infty}([0,1])$ satisfying

$\sup_{[0,1]}|f^{(l)}|\leq M_{\ell}\sup_{[0,1]}|f|,\qquad\ell\geq 0.$ (2.10)

Then for any $E\subset[0,1]$, for $\ell_{0}$ as in (2.5),

$\sup_{[0,1]}|f|\leq\frac{4\ell_{0}}{\gamma_{\ell_{0}}(E)}\sup_{E}|f|.$ (2.11)

###### Proof.

The proof is quite elementary, relying chiefly on the following Lagrange
interpolation formula (see for instance [Dem16, Chapitre 2, Section 1.2]). For
any $x$, $x_{1}<\cdots<x_{k}\in[0,1]$, there exists $\xi\in[0,1]$ such that

$f(x)=\sum_{i=1}^{k}f(x_{i})\prod_{j\neq
i}\frac{(x-x_{j})}{(x_{i}-x_{j})}+\frac{f^{(k)}(\xi)}{k!}\prod_{j}(x-x_{j}).$
(2.12)

Choosing $\\{x_{i}\\}_{i=1}^{k}\subset E$ such that $2\min_{i}\prod_{j\neq
i}\left|x_{i}-x_{j}\right|\geq\gamma_{k}(E)$, it is immediate that

$\left|\sum_{i=1}^{n}f(x_{i})\prod_{j\neq
i}\frac{(x-x_{j})}{(x_{i}-x_{j})}\right|\leq\frac{2k\cdot\sup_{E}|f|}{\gamma_{k}(E)}$

To estimate the second term in (2.12) we simply apply (2.10) to obtain

$\left|\frac{f^{(k)}(\xi)}{k!}\prod_{j}(x-x_{j})\right|\leq\frac{M_{k}}{k!}\sup_{[0,1]}|f|.$

which is less than $\frac{1}{2}\sup_{[0,1]}\left|f\right|$ for $k=\ell_{0}$
hence (2.11) holds. ∎

The transfinite diameter of $E$ is the limit of the $k$-th root of
$\gamma_{k}(E)$ as $k$ approaches infinity. An equivalent characterization is
also through

$\Delta_{k}(E)=\sup_{\\{x_{i}\\}_{i=1}^{k}\subset
E}\prod_{i=1}^{k}\prod_{j\neq i}\left|x_{i}-x_{j}\right|,$

and the identity

$\lim_{k\to\infty}\Delta_{k}(E)^{\frac{1}{k(k-1)}}=\lim_{k\to\infty}\gamma_{k}(E)^{\frac{1}{k}},$

holds for any compact set $E$, see [Pom92, Proposition 9.6].

To use Proposition 2.3, we recall some basic facts from the theory of capacity
(see e.g. [Mat95, Chapter 8] and [Pom92, Chapters 9 and 10]). For any $0<s\leq
n$ and any probability Radon measure $\mu$ on $\mathbb{R}^{n}$, define then
$s$-energy of $\mu$ by

$I_{s}(\mu)=\iint\left|x-y\right|^{-s}\,d\mu(x)\,d\mu(y),$

with the modification for $s=0$ to define $I_{s}(\mu)$ with
$\left|x-y\right|^{-s}$ replaced by $-\log\left|x-y\right|$. Letting
$\mathcal{P}(E)$ denote the set of all probability Radon measures supported on
$E$, and define the $s$-capacity of $E$ by

$\operatorname{cap}_{s}(E)=\left\\{\begin{array}[]{ll}\sup\left\\{I_{s}(\mu)^{-1}:\mu\in\mathcal{P}(E)\right\\},&s>0;\\\
\sup\left\\{\exp\left(-I_{0}(\mu)\right):\mu\in\mathcal{P}(E)\right\\},&s=0.\end{array}\right.$

The concavity of the logarithm and Jensen’s inequality show that

$\operatorname{cap}_{0}(E)^{s}\geq\operatorname{cap}_{s}(E).$ (2.13)

Furthermore, $\operatorname{cap}_{0}(E)$ coincides with the transfinite
diameter of $E$ for $E$ compact (see [Ahl10, Theorem 2.3, page 28]), which
means

$\operatorname{cap}_{0}(E)=\lim_{k\to\infty}\Delta_{k}(E)^{\frac{2}{k(k-1)}}=\lim_{k\to\infty}\gamma_{k}(E)^{\frac{1}{k}},$

and in fact,

$\gamma_{k}(E)\geq(\operatorname{cap}_{0}E)^{k-1}.$ (2.14)

See [Pom92, Proposition 9.6 and Equation (16)].

Capacities will be more natural to work with than the closely related, and
more familiar Hausdorff content, defined above in (1.6). This relationship
between capacity and Hausdorff content is facilitated by Frostman’s lemma
[Mat95, Theorem 8.8] which implies the following result.

######  Lemma 2.4.

For each $s>t\geq 0$ and $m>0$ there exists $p>0$ such that

$\mathcal{C}_{\mathcal{H}}^{s}(E)\geq m\implies\operatorname{cap}_{t}(E)\geq
p.$ (2.15)

Consequently,

$\dim_{\mathcal{H}}E=\sup\left\\{s>0:\operatorname{cap}_{s}E>0\right\\}.$
(2.16)

With these preliminaries behind us, we are ready to state and prove the
promised corollary for sets of positive Hausdorff dimension.

######  Corollary 2.5.

Let $(M_{\ell})_{\ell\geq 0}\subset(0,\infty)^{\mathbb{N}}$ satisfying (2.3),
$f\in C^{\infty}([0,1])$ satisfying (2.10). Then for any compact set
$E\subset[0,1]$, for $\ell_{0}$ as in (2.5),

$\sup_{[0,1]}\left|f\right|\lesssim\left(\frac{1}{\operatorname{cap}_{0}E}\right)^{\ell_{0}}\sup_{E}\left|f\right|.$
(2.17)

Furthermore, for each $s,m>0$, there exists a constant $C(s,m)>0$ such that if
$\mathcal{C}^{s}_{\mathcal{H}}(E)\geq m$,

$\sup_{[0,1]}\left|f\right|\lesssim\left(\frac{1}{C(s,m)}\right)^{\ell_{0}}\sup_{E}\left|f\right|.$
(2.18)

###### Proof.

(2.17) follows immediately from Proposition 2.3 and (2.14). The second
inequality (2.18) follows from the first statement (2.15) with in Lemma 2.4
with $t=0$ and the application of (2.17). ∎

####  2.3.2 Higher Dimensions

First, we prove the following Lebesgue estimate in higher dimensions, which
well-known on its own for analytic functions [Bru99], but we outline its
standard extension from Proposition 2.3.

######  Proposition 2.6.

For any ball $B\subset\mathbb{R}^{n}$, any $E\subset\frac{1}{2}B$ with
$|E|>0$, any $(M_{\ell})_{\ell\geq 0}\subset(0,\infty)^{\mathbb{N}}$
satisfying (2.3), any $f\in C^{\infty}(B)$ satisfying (2.4) then for
$\ell_{0}$ as in (2.5), the following estimate holds

$\sup_{\frac{1}{2}B}\left|f\right|\leq\left(\frac{c_{n}|B|}{|E|}\right)^{\ell_{0}}\sup_{E}\left|f\right|,$
(2.19)

for some $c_{n}>0$ a positive constant depending on the dimensional constant
$n$.

###### Proof.

Rescaling, we can assume the radius of $B$ is one. Let $x_{0}\in\frac{1}{2}B$
such that $\sup_{\frac{1}{2}B}\left|f\right|\leq 2\left|f(x_{0})\right|$. By
using polar coordinates, there exists a line $L$ through $x_{0}$ such that

$\mathcal{L}^{1}(E\cap L)\geq c_{n}|E|,$

for some dimensional constant $c_{n}>0$, where $\mathcal{L}^{1}$ is the one-
dimensional Lebesgue measure on the line $L$. Restricting $f$ to $L$, the
restriction satisfies the assumptions of Proposition 2.3. Moreover it is not
difficult to see that there is an absolute constant $c>0$ such that

$\gamma_{\ell_{0}}(E\cap L_{\theta})\geq\left(c\mathcal{L}^{1}_{\theta}(E\cap
L_{\theta})\right)^{\ell_{0}}.$ (2.20)

Indeed, we can find $x_{1},\ldots,x_{\ell_{0}}\in\overline{E}\cap L_{\theta}$
such that $|x_{j}-\theta|$ (let $\theta$ be either point on
$L\cap(\mathbb{S}^{n-1}+x_{0})$) is increasing and
$|x_{j}-x_{j+1}|\geq\mathcal{L}^{1}_{\theta}(E\cap L_{\theta})\ell_{0}^{-1}$.
Such points can be constructed as in [AE13, Lemma 1]. Then (2.20) follows from
Stirling’s formula and the binomial theorem as in [NSV04, p. 69]:

$\prod_{j\neq
i}|x_{i}-x_{j}|\geq\frac{(i-1)!(\ell_{0}-i)!}{\ell_{0}^{\ell_{0}-1}}\mathcal{L}^{1}_{\theta}(E\cap
L_{\theta})^{\ell_{0}-1}\geq\frac{i!(\ell_{0}-i)!}{\ell_{0}^{\ell_{0}}}\left(\mathcal{L}^{1}_{\theta}(E\cap
L_{\theta})\right)^{\ell_{0}}\\\
\geq\frac{i!(\ell_{0}-i)!}{\ell_{0}!}\left(\frac{\mathcal{L}^{1}_{\theta}(E\cap
L_{\theta}}{e}\right)^{\ell_{0}}\geq\left(\frac{\mathcal{L}^{1}_{\theta}(E\cap
L_{\theta}}{2e}\right)^{\ell_{0}}.$

Therefore (2.19) follows from the estimate (2.11) in Proposition 2.3. ∎

To obtain the result for sets $E\subset B$ with $\dim_{\mathcal{H}}E>n-1$, we
want to repeat this technique of slicing our set $E$, but for lower
dimensional sets, going to polar coordinates becomes more difficult.
Nonetheless, such coarea-type formulas do persist for capacity (see e.g.
[Mat95, §10.9]). The particular consequence we will need is from [Mal04, Lemma
5.3]. We then obtain the following result leading to Proposition 2.2.

######  Proposition 2.7.

Let $R,s>0$. There exists $C=C(R,s,n)>0$ such that for any ball
$B\subset\mathbb{R}^{n}$ of radius $r\leq R$, any $E\subset\frac{1}{2}B$ with
$\operatorname{cap}_{n-1+s}(E)>0$, any $(M_{\ell})_{\ell\geq
0}\subset(0,\infty)^{\mathbb{N}}$ satisfying (2.3), and any $f\in
C^{\infty}(B)$ satisfying (2.4), there holds, for $\ell_{0}$ as in (2.5), the
following estimate

$\sup_{\frac{1}{2}B}\left|f\right|\leq\left(\frac{Cr^{n-1+s}}{\operatorname{cap}_{n-1+s}(E)}\right)^{\frac{\ell_{0}(s+1)}{s}}\sup_{E}\left|f\right|.$
(2.21)

###### Proof.

Let us assume $B$ is centered at the origin. By [Mal04, Lemma 5.3], there
exist dimensional constants $c_{1}>0,c_{2}>0$ and a line $L$ through the
origin such that

$G:=\left\\{a\in L^{\perp}\ :\ \operatorname{cap}_{s}(E\cap(L+a))\geq
c_{1}r^{1-n}\operatorname{cap}_{n-1+s}(E)\right\\}$

has $\mathcal{L}^{n-1}(G)\geq c_{2}r^{-s}\operatorname{cap}_{n-1+s}(E)$, where
$L^{\perp}$ is the $(n-1)$-dimensional hyperplane perpendicular to $L$ through
the origin and $\mathcal{L}^{n-1}$ is the $(n-1)$-dimensional Lebesgue measure
on $L^{\perp}$. Define now for each $a\in L^{\perp}$ the line segment
$L_{a}^{*}:=\\{x+a:x\in L,|x|\leq\tfrac{1}{4}r\\}$. Then we construct

$F=\bigcup_{\begin{subarray}{c}a\in G,\end{subarray}}L_{a}^{*},$

so that

$|F|\gtrsim r\mathcal{L}^{n-1}(G)\gtrsim
r^{1-s}\operatorname{cap}_{n-1+s}(E).$

On the other hand, for each $x\in F$, there exists a line segment
$L^{\prime}$, of length $\frac{1}{2}r$ passing through $x$, such that
$\operatorname{cap}_{0}(E\cap L^{\prime})^{s}\geq\operatorname{cap}_{s}(E\cap
L^{\prime})\geq c_{1}r^{1-n}\operatorname{cap}_{n-1+s}(E)$ so by (2.17),

$|f(x)|\leq\sup_{L^{\prime}}\left|f\right|\lesssim\left(\frac{r}{\operatorname{cap}_{0}(E\cap
L^{\prime})}\right)^{\ell_{0}}\sup_{E}\left|f\right|\lesssim\left(\frac{Cr^{n-1+s}}{\operatorname{cap}_{n-1+s}(E)}\right)^{\ell_{0}/s}\sup_{E}\left|f\right|.$

Then, the result follows from applying Proposition 2.6 to the set $F$ leading
to

$\sup_{\frac{1}{2}B}\left|f\right|\leq\left(\frac{c_{n}|B|}{|F|}\right)^{\ell_{0}}\sup_{F}\left|f\right|\lesssim\left(\frac{Cr^{n}}{r^{1-s}\operatorname{cap}_{n-1+s}(E)}\right)^{\ell_{0}}\left(\frac{Cr^{n-1+s}}{\operatorname{cap}_{n-1+s}(E)}\right)^{\ell_{0}/s}\sup_{E}\left|f\right|,$

then in particular (2.21). ∎

###  2.4 Adaptation of the Lebeau-Robbiano’s method

With the help of the spectral estimate of Theorem 2.1, one can now prove
Theorem 1.3.

###### Proof of Theorem 1.3.

First, an immediate corollary of Theorem 2.1 is the following one. For all
$\delta>0$, for every $\omega\subset\Omega$ such that
$\mathcal{C}_{\mathcal{H}}^{n-1+\delta}(\omega)>0$, there exists
$C=C(\Omega,\delta,\omega)>0$ such that for every $\Lambda>0$, we have

$\left\lVert\Pi_{\Lambda}u\right\rVert_{L^{2}(\Omega)}\leq
Ce^{C\sqrt{\Lambda}}\sup_{x\in\omega}\left|(\Pi_{\Lambda}u)(x)\right|\qquad\forall
u\in L^{2}(\Omega).$ (2.22)

By using the spectral estimates (2.22), the proof of the observability
estimate (1.7) is by now quite classical and originally comes from the Lebeau-
Robbiano’s method for obtaining the null-controllability of the heat equation
starting from a spectral estimate, see [LR95] and [LRL12, Section 6]. This
strategy was latter extended by Miller in [Mil10]. Note that it has to be
adapted to the particular functional setting of the $L^{2}-L^{\infty}$
spectral estimate (2.22). Here we follow the arguments of [BM23, Proof of
Theorem 3 in Section 4] and [AEWZ14, Remark 2 in Section 2].

From (2.22) and [BM23, Corollary 4.1], we have the following result. There
exists $A>0,\ C>0$ such that that for all $0<t_{1}<t_{2}\leq T$, $u_{0}\in
L^{\infty}(\Omega)$, the solution $u$ of (1.1) satisfies

$e^{-\frac{A}{(t_{2}-t_{1})}}\|u(t_{2},\cdot)\|_{L^{2}(\Omega)}-e^{-\frac{2A}{(t_{2}-t_{1})}}\|u(t_{1},\cdot)\|_{L^{2}(\Omega)}\leq
C\int_{t_{1}}^{t_{2}}\|u(s,\cdot)\|_{L^{\infty}(\omega)}ds.$ (2.23)

We now define $l=T/2$, $l_{1}=3T/4$ and

$l_{m+1}-l=2^{-m}(l_{1}-l)\qquad m\geq 1.$

We apply (2.23) with $t_{1}=l_{m+1}$ and $t_{2}=l_{m}$,

$e^{-\frac{A}{l_{m}-l_{m+1}}}\|u(l_{m},\cdot)\|_{L^{2}(\Omega)}-e^{-\frac{2A}{(l_{m}-l_{m+1})}}\|u(l_{m+1},\cdot)\|_{L^{2}(\Omega)}\leq
C\int_{l_{m+1}}^{l_{m}}\|u(s,\cdot)\|_{L^{\infty}(\omega)}ds.$

Noticing that $\frac{2}{(l_{m}-l_{m+1})}=\frac{1}{(l_{m+1}-l_{m+2})}$, we get

$e^{-\frac{A}{l_{m}-l_{m+1}}}\|u(l_{m},\cdot)\|_{L^{2}(\Omega)}-e^{-\frac{A}{l_{m+1}-l_{m+2}}}\|u(l_{m+1},\cdot)\|_{L^{2}(\Omega)}\leq
C\int_{l_{m+1}}^{l_{m}}\|u(s,\cdot)\|_{L^{\infty}(\omega)}ds,$ (2.24)

then summing the telescopic series (2.24), and using that

$\lim_{m\rightarrow+\infty}e^{-\frac{A}{l_{m+1}-l_{m+1}}}=0,$

we get

$e^{-\frac{A}{l_{1}-l_{2}}}\|u(l_{1},\cdot)\|_{L^{2}(\Omega)}\leq
C\int_{l}^{l_{1}}\|u(s,\cdot)\|_{L^{\infty}(\omega)}ds.$

From the definition of $l,l_{1},l_{2}$ we immediately deduce

$\|u(3T/4,\cdot)\|_{L^{2}(\Omega)}\leq
Ce^{C/T}\int_{0}^{T}\|u(s,\cdot)\|_{L^{\infty}(\omega)}ds,$

then from the $L^{2}-L^{\infty}$ regularizing effect of the heat equation, see
for instance [CH98, Proposition 3.5.7], we obtain

$\|u(T,\cdot)\|_{L^{\infty}(\Omega)}\leq
CT^{-n/4}\|u(3T/4,\cdot)\|_{L^{2}(\Omega)}\leq
Ce^{C/T}\int_{0}^{T}\|u(s,\cdot)\|_{L^{\infty}(\omega)}ds,$

so the quantitative observability estimate (1.7). ∎

##  3 Observability from measurable sets with lower Hausdorff dimension

This section is devoted to study the observability of (1.1) in some particular
cases when the observable sets $\omega$ has a Hausdorff dimension lower than
$n-1$. In fact, we exemplify cases of zero dimension in arbitrarily large
ambient dimension.

###  3.1 Observability from a finite subset

To set the stage, we recall that in the case of the one-dimensional heat
equation posed on $\Omega=(0,1)$, Dolecki fully described in [Dol73] the
situation when $\omega=\\{x_{0}\\}$ in terms of certain algebraic properties
of the number $x_{0}$. Dolecki’s result in fact precisely describes the gap
between nodal sets of the eigenfunctions and observation sets. Namely,
observation points are characterized by their not being too close to a nodal
set, which in Dolecki’s formulation means $x_{0}$ is not well-approximated by
rationals numbers. In [Sam15], Samb precised this statement by showing that,
in the case when $x_{0}$ is an algebraic number of order $d>1$, the
observability inequality holds at any time $T>0$. He also provided an
estimation of the observability cost, which we record for later use.

######  Theorem 3.1 ([Sam15]).

Let $x_{0}\in(0,1)$ be an algebraic number of order $d>1$. There exists a
positive constant $C>0$ such that for any $T>0$,

$\|u(T,\cdot)\|^{2}_{L^{2}(0,1)}\leq
Ce^{\frac{C}{T}}\int_{0}^{T}|u(t,x_{0})|^{2}dt\qquad\forall u_{0}\in
L^{2}(0,1),$

where $u$ is the weak solution of (1.1).

From Samb’s proof, it is clear that the result holds for every diophantine
number (i.e. non Liouville number) since it only involves that an irrational
algebraic number is a diophantine number. We will not use this fact in the
rest of the article, so we will stick to Samb’s formulation of Theorem 3.1.

A straightforward dimension-counting argument shows that such a result cannot
hold on domains for which the dimension of the eigenspaces $E_{r}(-\Delta)$ is
unbounded as $r\to\infty$, in particular when $\Omega=(0,1)^{n}$ for $n\geq
2$.

######  Proposition 3.2.

Let $\Omega=(0,1)^{n}$ with $n\geq 2$ and $T>0$. For any finite subset
$\omega\subset\Omega$, the heat equation (1.1) is not observable from $\omega$
at time $T$.

This proposition shows that the results of [Dol73] and [Sam15] are very
specific to the one-dimensional case. This difference with the multi-
dimensional case comes from the fact that, when $n\geq 2$, the degree of
degeneracy of the energy levels of the operator $-\Delta$ tends to $+\infty$.
Let us recall that in this case, the spectrum of $-\Delta$ is given by

$\text{Sp}(-\Delta)=\left\\{r_{1}^{2}+\cdots+r_{n}^{2};\
(r_{1},\cdots,r_{n})\in(\mathbb{N}^{*})^{n}\right\\}$ (3.1)

and for all $r\in\text{Sp}(-\Delta)$, the eigenspace $E_{r}(-\Delta)$
associated to $r$ is

$E_{r}(-\Delta)=\text{Span}_{\mathbb{C}}\left\\{\varphi_{r_{1}}\otimes\varphi_{r_{2}}\otimes\cdots\otimes\varphi_{r_{n}};\
r_{1}^{2}+\cdots+r_{n}^{2}=r\right\\},$

with dimension equal to the sum of squares counting function

$s_{n}(r)=\\#\left\\{(r_{1},\cdots,r_{n})\in(\mathbb{N}^{*})^{n};\
r_{1}^{2}+\cdots r_{n}^{2}=r\right\\}.$

If $n\geq 2$, then

$\limsup_{r\to\infty}s_{n}(r)=\infty.$ (3.2)

Indeed, for $n=2$ one can explicitly compute $s_{2}(5^{2r-1})=r$ by [Bei64,
pp.140-142]. For $n\geq 2$, of course $s_{n}(r)\geq s_{2}(r)$ hence (3.2)
holds.

###### Proof of Proposition 3.2.

Let $\omega=\\{x_{1},\cdots,x_{N}\\}\subset(0,1)^{n}$. By (3.2), we can find
$r$ such that $s_{n}(r)\geq N+1$. Since $\text{Dim}(E_{r}(-\Delta))=s_{d}(r)$,
it follows that the linear application $T:f\in
E_{r}(-\Delta)\longmapsto(f(x_{1}),...,f(x_{N}))\in\mathbb{R}^{N}$ is not one-
to-one. This provides an eigenfunction $\psi$, associated to the eigenvalue
$r$, such that

$\psi(x_{1})=\cdots=\psi(x_{N})=0.$

Notice that for all $t\in[0,T]$, $e^{t\Delta}\psi=e^{-rt}\psi$. This leads to

$\int_{0}^{T}\sup_{x\in\omega}|e^{t\Delta}\psi(x)|dt=\int_{0}^{T}e^{-rt}\sup\\{|\psi(x_{1})|,...,|\psi(x_{N})|\\}dt=0.$

Since $\|\psi\|_{L^{\infty}(\Omega)}\neq 0$, the observability of (1.1) does
not hold from $\omega$ at time $T>0$. This ends the proof of Proposition 3.2.
∎

We will construct some families on low-dimensional observation sets by taking
the Cartesian product of other sets of low dimension. The first, simple
principle along these lines is we can take the cartesian product of spectral
inequalities; see Proposition 3.3 below. The second, more involved argument is
that we can take cartesian product of an observability estimate (with good
control cost) and a spectral inequality. Naturally, one may wonder if one
could take the cartesian product of two observability inequalities.
Proposition 3.2 above answers this question in the negative in general since
$\\{x_{0}\\}\times\\{x_{0}\\}\subset(0,1)^{2}$ is not an observation set,
while $\\{x_{0}\\}\subset(0,1)$ is.

###  3.2 Spectral inequalities for Cartesian products

To succinctly describe the spectral inequalities of the form (2.2), we say
$\mathrm{spec}(\Omega,\omega,C,\beta)$ holds if

$\forall\Lambda>0,\forall u\in
L^{2}(\Omega),\quad\|\Pi_{\Lambda}u\|_{L^{\infty}(\Omega)}\leq
C\exp(C\Lambda^{\beta})\|\Pi_{\Lambda}u\|_{L^{\infty}(\omega)},$

where $\Pi_{\Lambda}$ is the spectral projection defined in (2.1). Here we
only deal with $-\Delta$ and $L^{\infty}$ estimate but in spirit these
argument works for general $L^{p}$ and general elliptic operators $A$.

######  Proposition 3.3.

Let $\Omega_{j}\subset\mathbb{R}^{n_{j}}$, $j=1,2$ and
$\omega\subset\Omega_{1}\times\Omega_{2}$. Let $\omega_{2}=\pi_{2}(\omega)$ be
the projection of $\omega$ on $\Omega_{2}$. For each $y\in\omega_{2}$, we
denote by

$\omega_{y}:=\pi_{1}(\omega\cap(\Omega_{1}\times\\{y\\}))$

the horizontal slices of $\omega$. If there exists $C,\beta>0$ such that
$\mathrm{spec}(\Omega_{2},\omega_{2},C,\beta)$ holds and for every
$y\in\omega_{2}$, $\mathrm{spec}(\Omega_{1},\omega_{y},C,\beta)$ holds, then
there exists $C^{\prime}$ such that

$\mathrm{spec}{(\Omega_{1}\times\Omega_{2},\omega,C^{\prime},\beta)},\quad\text{holds.}$

When $\omega=\omega_{1}\times\omega_{2}$, notice $\omega_{y}=\omega_{1}$ for
all $y\in\omega_{2}$, thus in this case we have the corollary

$\Big{(}\mathrm{spec}(\Omega_{j},\omega_{j},C,\beta)\quad\text{holds},\qquad
j=1,2\Big{)}\implies\mathrm{spec}(\Omega_{1}\times\Omega_{2},\omega_{1}\times\omega_{2},C^{\prime},\beta)\quad\text{holds}.$

###### Proof.

Let $\Lambda>0$ and $u\in L^{2}(\Omega_{1}\times\Omega_{2})$. Let
$(\varphi_{k}^{1})_{k\in\mathbb{N}}$ (resp.
$(\varphi_{k}^{2})_{k\in\mathbb{N}}$) be an orthonormal basis of
$L^{2}(\Omega_{1})$ (resp. $L^{2}(\Omega_{2})$) formed by eigenfunctions of
$-\Delta_{\Omega_{1}}$ (resp. $-\Delta_{\Omega_{2}}$). Then

$\Pi_{\Lambda}u(x,y)=\sum_{\lambda_{p}^{1}+\lambda_{q}^{2}\leq\lambda}C_{p,q}\varphi^{1}_{p}(x)\varphi^{2}_{q}(y).$

The key observation is that for any $x\in\Omega_{1}$ and $y\in\Omega_{2}$,

$v_{x}=\Pi_{\Lambda}u(x,\cdot),\quad w_{y}=\Pi_{\Lambda}u(\cdot,y)$

are each linear combinations of eigenfunctions of energy level less than
$\Lambda$ to which the spectral inequalities apply. Thus, for any
$x\in\Omega_{1}$, using that $\mathrm{spec}(\Omega_{2},\omega_{2},C,\beta)$
holds,

$\sup_{y\in\Omega_{2}}\left|\Pi_{\Lambda}u(x,y)\right|=\sup_{\Omega_{2}}\left|v_{x}\right|\leq
C\exp(C\Lambda^{\beta})\sup_{\omega_{2}}\left|v_{x}\right|=C\exp(C\Lambda^{\beta})\sup_{y\in\omega_{2}}\left|\Pi_{\Lambda}u(x,y)\right|.$

And for any $y\in\omega_{2}$, $x\in\Omega_{1}$, using
$\mathrm{spec}(\Omega_{1},\omega_{y},C,\beta)$ holds,

$\left|\Pi_{\Lambda}u(x,y)\right|=\left|w_{y}(x)\right|\leq
C\exp(C\Lambda^{\beta})\sup_{z\in\omega_{y}}\left|w_{y}(z)\right|\leq
C\exp(C\Lambda^{\beta})\sup_{\omega}\left|\Pi_{\Lambda}u\right|.$

Combining these two displays and using the fact that $x\in\Omega_{1}$ was
arbitrary concludes the proof. ∎

###  3.3 Observability inequalities for Cartesian products

######  Proposition 3.4.

Let $T>0$, $\omega_{j}\subset\Omega_{j}\subset\mathbb{R}^{n_{j}}$, $j=1,2$.
Let $\omega_{1}\subset\Omega_{1}$ and $\omega_{2}\subset\Omega_{2}$. Suppose

  * (a)

there exists $C_{2}>0$ and $\beta\in(0,1)$ such that
$\mathrm{spec}(\Omega_{2},\omega_{2},C_{2},\beta)$ holds,

  * (b)

there exists $C_{1}>0$ such that for $\alpha=\frac{\beta}{1-\beta}$,

$\forall 0<\tau\leq T,\ \|v(\tau,\cdot)\|^{2}_{L^{2}(\Omega_{1})}\leq
C_{1}\exp\left(\tfrac{C_{1}}{\tau^{\alpha}}\right)\int_{0}^{\tau}\sup_{\omega_{1}}|v(t,x)|^{2}dt$

where $v$ is the solution of (1.1) on $\Omega_{1}$.

Then there exists $C>0$ such that

$\|u(T,\cdot)\|_{L^{2}(\Omega_{1}\times\Omega_{2})}^{2}\leq
C\exp\left(\tfrac{C}{T^{\alpha}}\right)\int_{0}^{T}\sup_{\omega_{1}\times\omega_{2}}|u(t,x)|^{2}dt,$
(3.3)

where $u$ is the solution of (1.1) on $\Omega_{1}\times\Omega_{2}$.

Note that (3.3) differs from the functional spaces of the observability notion
(1.2) in Definition 1.1 but this is because of the assumption (b). By using
the $L^{2}-L^{\infty}$ regularizing effect of the heat equation, at least the
left hand side of (3.3) can be transformed into
$\|u(T,\cdot)\|_{L^{\infty}(\Omega_{1}\times\Omega_{2})}^{2}$.

The following proof is adapted from [Sam15, Theorem 1.1] who was considering
the case of $\\{x_{0}\\}\times\omega\subset(0,1)^{2}$ where $x_{0}$ is an
algebraic number of degree $d>1$ and $\omega$ an open interval of $(0,1)$. A
generalization of this type of result, encompassing the case where the minimal
time of null-controllability $T(x_{0})$ is strictly positive, has been
recently considered in the recent preprint [AKBGB+24].

###### Proof.

The proof consists of applying the observability estimate on $\omega_{1}$
before using spectral estimates and then performing a Lebeau-Robbiano argument
type. Let $0<\tau\leq 1$, $u_{0}\in L^{2}(\Omega_{1}\times\Omega_{2})$ and
$\Lambda>0$ to be chosen later. Let us write the space variable
$z=(x,y)\in\Omega_{1}\times\Omega_{2}$ so that
$\Delta=\Delta_{\Omega_{1}}+\Delta_{\Omega_{2}}$. As before, let
$(\varphi_{k}^{j})_{k\in\mathbb{N}}$ be an orthonormal basis of
$L^{2}(\Omega_{j})$, for $j=1,2$, consisting of eigenfunctions of $-\Delta$
with Dirichlet boundary conditions. If $u$ is the mild solution of (1.1), we
can therefore write

$\Pi_{\Lambda}^{2}u(t,x,y)=\sum_{\lambda^{2}_{k}\leq\lambda}u_{k}(t,x)e^{-t\lambda^{2}_{k}}\varphi^{2}_{k}(y),$

where the $u_{k}$ are solutions of the heat equation on $\Omega_{1}$ and
$\Pi^{j}_{\Lambda}$ is the spectral projection (2.1) on $\Omega_{j}$.

So for $y\in\Omega_{2}$ fixed,
$v_{y}(t,x):=\sum_{\lambda^{2}_{k}\leq\lambda}u_{k}(t,x)e^{-\tau\lambda^{2}_{k}}\varphi^{2}_{k}(y)$
is also a solution of the heat equation on $\Omega_{1}$, whence we have the
observability inequality

$\|v_{y}(\tau)\|^{2}_{L^{2}(\Omega_{1})}\leq
C_{1}\exp\left(\tfrac{C_{1}}{\tau^{\alpha}}\right)\int_{\frac{\tau}{2}}^{\tau}\sup_{x\in\omega_{1}}|v_{y}(t,x)|^{2}dt$
(3.4)

for any $y\in\Omega_{2}$. Furthermore, by the Cauchy-Schwarz inequality, for
$x\in\omega_{1}$ and $t\in\left(\tfrac{\tau}{2},\tau\right]$,

$|v_{y}(t,x)|^{2}\leq\left(\sum_{\lambda_{k}^{2}\leq\Lambda}\left|u_{k}(t,x)\right|^{2}e^{-2t\lambda_{k}^{2}}\right)\left(\sum_{\lambda_{k}^{2}\leq\Lambda}\left|\varphi_{k}^{2}(y)\right|^{2}\right).$

By the Weyl’s law, integrating the second factor over $\Omega_{2}$ contributes
polynomial blow-up in $\Lambda$. This is of no consequence to us since the
first factor, by Parseval’s identity and the spectral inequality, satisfies

$\sup_{x\in\omega_{1}}\sum_{\lambda^{2}_{k}\leq\lambda}\left|u_{k}(t,x)\right|^{2}e^{-2t\lambda^{2}_{k}}=\sup_{x\in\omega_{1}}\int_{\Omega_{2}}\left|\Pi^{2}_{\Lambda}u(t,x,y)\right|^{2}dy$
$\leq
C_{2}\exp\left(C_{2}\Lambda^{\beta}\right)\sup_{(x,y)\in\omega_{1}\times\omega_{2}}\left|\Pi^{2}_{\Lambda}u(t,x,y)\right|^{2}.$

Integrate (3.4) over $y\in\Omega_{2}$ and combine it with the above display
and the identity

$v_{y}(\tau,x)=\Pi^{2}_{\Lambda}u(\tau,x,y)$

to obtain the low-frequency observability inequality

$\left\lVert\Pi^{2}_{\Lambda}u(\tau)\right\rVert_{L^{2}(\Omega_{1}\times\Omega_{2})}^{2}\leq
C_{3}\exp\left(\tfrac{C_{1}}{\tau^{\alpha}}+C_{3}\Lambda^{\beta}\right)\int_{\frac{\tau}{2}}^{\tau}\sup_{(x,y)\in\omega_{1}\times\omega_{2}}\left|\Pi^{2}_{\Lambda}u(t,x,y)\right|^{2}\,dt.$
(3.5)

($C_{3}$ is introduced in order to absorb the polynomial growth in $\Lambda$).

It is classical that since $u^{+}:=(I-\Pi^{2}_{\Lambda})u$ satisfies the heat
equation on $\Omega=\Omega_{1}\times\Omega_{2}$, we have from the
$L^{2}-L^{\infty}$ regularizing effect of the heat equation, see for instance
[CH98, Proposition 3.5.7]

$\forall
t>\frac{\tau}{2},\quad\sup_{\Omega}\left|u^{+}(t,\cdot)\right|\leq\frac{C}{\tau^{\frac{n}{4}}}\left\|u^{+}(\frac{\tau}{4},\cdot)\right\|_{L^{2}(\Omega)}\leq\frac{C}{\tau^{\frac{n}{4}}}e^{-\Lambda\frac{\tau}{4}}\|u_{0}\|_{L^{2}(\Omega)},$
(3.6)

for some positive constant $C>0$. So by applying the triangle inequality,
(3.5), triangle inequality again, followed by (3.6) to absorb the error, we
obtain that for all $0<\tau\leq 1$ and $\lambda>0$

$\displaystyle\|u(\tau)\|^{2}_{L^{2}(\Omega)}$ $\displaystyle\leq
C_{3}\exp\left(\tfrac{C_{1}}{\tau^{\alpha}}+C_{3}\Lambda^{\beta}\right)\left[\int_{\frac{\tau}{2}}^{\tau}\sup_{\omega_{1}\times\omega_{2}}|u(t,\cdot)|^{2}dt+\tau^{-\frac{n}{4}}C\exp\left(-\tfrac{\tau}{4}\Lambda\right)\|u_{0}\|^{2}_{L^{2}(\Omega)}\right].$

One can find a positive parameter $\gamma>0$ such that $\gamma\geq
10\left(C_{1}+C_{3}\gamma^{\beta}\right)$. Let us choose then
$\Lambda=\frac{\gamma}{\tau^{\alpha+1}}$, noting the key relationship
$\beta(\alpha+1)={\alpha}$, so that

$\exp\left(\tfrac{C_{1}}{\tau^{\alpha}}+C_{3}\Lambda^{\beta}\right)\leq\exp\left(\tfrac{\gamma}{10\tau^{\alpha}}\right),\quad\tau^{-\frac{n}{4}}\exp\left(-\tfrac{\tau}{4}\Lambda\right)\leq
C_{5}\exp\left(-\tfrac{\gamma}{5\tau^{\alpha}}\right).$

Therefore,

$f(\tau)\|u(\tau)\|^{2}_{L^{2}(\Omega)}\leq\int_{\frac{\tau}{2}}^{\tau}\sup_{\omega_{1}\times\omega_{2}}|u(t,\cdot)|^{2}dt+f(\eta\tau)\|u_{0}\|^{2}_{L^{2}(\Omega)},$
(3.7)

with $f(\tau)=C_{6}\exp\left(-\frac{\gamma}{10\tau^{\alpha}}\right)$, and
$\eta=2^{-\frac{1}{\alpha}}<1$. Let us assume that $0<T\leq 1$ and define the
following sequence:

$T_{0}=T\quad\text{and}\quad\forall k\in\mathbb{N},\
T_{k+1}=T_{k}-(1-\eta)\eta^{k}T.$

By applying (3.7) with initial data $u_{0}=u(T_{k+1})$ and running until time
$T_{k}$ ($\tau$ then represents the elapsed time
$T_{k}-T_{k+1}=(1-\eta)\eta^{k+1}T$), we obtain

$\displaystyle
f\left((1-\eta)\eta^{k}T\right)\|u(T_{k})\|^{2}_{L^{2}(\Omega)}$
$\displaystyle\leq\int_{T_{k+1}}^{T_{k}}\sup_{\omega_{1}\times\omega_{2}}|u(t,\cdot)|^{2}dt+f\left((1-\eta)\eta^{k+1}T\right)\|u(T_{k+1})\|^{2}_{L^{2}(\Omega)}.$

Since $f\left((1-\eta)\eta^{k}T\right)\to 0$ as $k\to\infty$, we can rearrange
the above display so the LHS is a telescoping series, yielding

$f\left((1-\eta)T\right)\|u(T)\|^{2}_{L^{2}(\Omega)}\leq\int_{0}^{T}\sup_{\omega_{1}\times\omega_{2}}|u(t,\cdot)|^{2}dt,$

leading to (3.3) recalling the form of $f$. ∎

###  3.4 Examples

As demonstrated from the results of [Dol73] and [Sam15], there are zero-
dimensional observation sets in dimension 1. We will now use Propositions 3.3
and 3.4 to supply examples in higher dimensions. Combining Theorem 2.1,
Theorem 3.1, and Proposition 3.4 exhibits observation sets of dimension
$n-2+\delta$.

######  Corollary 3.5.

Let $T>0$, $\delta>0$ and $\Omega=(0,1)\times\Omega^{\prime}$ where
$\Omega^{\prime}\subset\mathbb{R}^{n-1}$. Let $x_{0}\in(0,1)$ be a algebraic
number of order $d>1$ and $\omega\subset\Omega^{\prime}$ be a measurable
subset satisfying $\mathcal{C}^{n-2+\delta}_{\mathcal{H}}(\omega)>0$. The heat
equation (1.1) is observable from $\\{x_{0}\\}\times\omega$ at time $T$.

As we generate sets of even smaller dimension we make the following comment.
Since the eigenfunctions of $-\Delta$ are continuous, $\Pi_{\Lambda}u$ is
always continuous which entails

$\sup_{x\in
E}\left|\Pi_{\Lambda}u(x)\right|=\sup_{x\in\overline{E}}\left|\Pi_{\Lambda}u(x)\right|.$
(3.8)

Therefore, to generate nontrivial examples, we should restrict our attention
to sets whose closure is low-dimensional (or to closed sets). An example to
motivate this restriction is the zero-dimensional set
$\omega=\mathbb{Q}^{d}\cap\Omega$ for which the spectral inequality trivially
holds by (3.8). We now introduce a family of zero-dimensional sets, indexed by
$\alpha>0$,

$\omega_{\alpha}=\left\\{i^{-\alpha}:i=1,2,\ldots\right\\}.$

We have the following crude estimate for $\gamma_{k}(\omega_{\alpha})$ (recall
$\gamma_{k}$ from (2.9)) which suffices for our purposes.

######  Lemma 3.6.

For each $\alpha>0$ there exists $c_{1},c_{2}>0$ such that

$\gamma_{k}(\omega_{\alpha})\geq c_{1}e^{c_{2}k\log k}.$ (3.9)

###### Proof.

By simply selecting the first $k$ terms of the sequence
$(i^{-\alpha})_{i\in\mathbb{N}^{*}}$, we have

$\gamma_{k}(\omega_{\alpha})\geq\inf_{i=1,\ldots,k}\prod_{\begin{subarray}{c}j=1\\\
j\neq i\end{subarray}}^{k}\left|j^{-\alpha}-i^{-\alpha}\right|.$

Rewrite

$\prod_{\begin{subarray}{c}j=1\\\ j\neq
i\end{subarray}}^{k}\left|j^{-\alpha}-i^{-\alpha}\right|=\prod_{\begin{subarray}{c}j=1\\\
j\neq
i\end{subarray}}^{k}j^{-\alpha}i^{-\alpha}\left|j^{\alpha}-i^{\alpha}\right|.$

Obviously $\left|j^{\alpha}-i^{\alpha}\right|$ is minimized when $j$ is close
to $i$ so by the mean value theorem $\left|j^{\alpha}-i^{\alpha}\right|\gtrsim
i^{\alpha-1}$. Therefore,

$\gamma_{k}(\omega_{\alpha})\gtrsim\inf_{i=1,\ldots,k}\prod_{j=1}^{k}j^{-\alpha}i^{-1}=\inf_{i=1,\ldots,k}(k!)^{-\alpha}i^{-k}.$

Clearly this infimum is attained when $i=k$ and then the form (3.9) follows by
Stirling’s formula. ∎

By combining Proposition 2.3 and Lemma 3.6 together with the methodology of
the proof of Theorem 2.1 using Proposition 2.2, we obtain the following
corollary.

######  Corollary 3.7.

For any $\beta\in\left(\tfrac{1}{2},1\right)$ there exists $C>0$ such that
$\mathrm{spec}((0,1),\omega_{\alpha},C,\beta)$ holds.

Finally, from Corollary 3.7 and Proposition 3.4 we conclude that the heat
equation is indeed observable from $\\{x_{0}\\}\times\omega_{\alpha}$, which
is a zero-dimensional set. In fact that the cartesian product
$\omega_{\alpha}\times\ldots\times\omega_{\alpha}$ is countable and therefore
must have Hausdorff dimension zero.

######  Corollary 3.8.

Let $\alpha,T>0$, $n\geq 2$, and $x_{0}$ an algebraic number of degree greater
than one. Then the heat equation (1.1) on $\Omega=(0,1)^{n}$ is observable
from $\\{x_{0}\\}\times\omega_{\alpha}\times\cdots\times\omega_{\alpha}$ at
time $T$.

Another indication of the smallness of this observation set is that when $n=2$
it only has one accumulation point (which is the minimal number in a bounded
domain), namely

$(x_{0},0),$

and the set becomes even more concentrated there as $\alpha\to\infty$. To this
end we pose the following open question.

######  Question 3.9.

Is the heat equation (1.1) on $\Omega=(0,1)^{2}$ observable from

$\\{x_{0}\\}\times\left\\{e^{-k}:k=1,2,\ldots\right\\}$

for some point $x_{0}\in(0,1)$?

Is is relatively straightforward to check that for

$\omega_{\mathrm{exp}}=\left\\{e^{-k}:k=1,2,\ldots\right\\},\quad\gamma_{k}(\omega_{\mathrm{exp}})\sim
e^{-k^{2}},$

which does not yield $\mathrm{spec}((0,1),\omega_{\mathrm{exp}},C,\beta)$
except when $\beta=1$. To conclude observability of the heat equation of
course we need $\beta<1$.

Second, we were unable to prove Proposition 3.4 in the full generality of
Proposition 3.3 using sections and projections of general non-product-type
observation sets. If one were able to do so then a large family of curves in
$(0,1)^{2}$ would be observation sets. Of course, Corollary 3.5 shows easily
that a line parallel to an axis with distance $x_{0}$ from that axis is indeed
an observation set. But for any line not parallel to one of the axes, we do
not know if the heat equation is observable. Our desired extension of
Proposition 3.4 discussed above would prove this, but we do not know if such
an extension is possible.

######  Question 3.10.

Is the heat equation on $(0,1)^{2}$ observable from any line not parallel to
one of the axes?

##  Appendix A Another proof of the spectral estimate

The goal of this part is to present another proof of Theorem 2.1 that is more
in the spirit of [BM23] starting from Malinnikova’s result on the propagation
of smallness for solutions to generalized Cauchy-Riemann systems that are
factor of the Laplacian.

######  Lemma A.1.

Let $\delta>0$, $\mathcal{K}\subset\subset B_{1}$, $\mathcal{E}\subset\subset
B_{1}$ be a subset of an hyperplane such that
$\mathcal{C}_{\mathcal{H}}^{n-2+\delta}(\mathcal{E})>0$. Then, there exist
$C=C(\mathcal{K},\mathcal{E},\delta)>0$ and
$\alpha=\alpha(\mathcal{K},\mathcal{E},\delta)\in(0,1)$ such that for every
$u:B_{1}\to\mathbb{R}$ satisfying $\Delta u=0$ in $B_{1}$ then

$\sup_{x\in\mathcal{K}}|\nabla u(x)|\leq C\sup_{x\in\mathcal{E}}|\nabla
u(x)|^{\alpha}\|\nabla u\|_{L^{\infty}(B_{1})}^{1-\alpha}.$ (A.1)

In other words, Lemma A.1 tells us that we are able to propagate smallness for
gradient of harmonic functions from sets of Hausdorff dimension $d-2+\delta$
for every $\delta>0$. Note that [LM18] obtains the same result for
$A$-harmonic function with $A$ satisfying elliptic and Lipschitz assumptions
and for measurable sets $\mathcal{E}$ satisfying
$\mathcal{C}_{\mathcal{H}}^{n-1-\delta}(\mathcal{E})=m$ for some $\delta>0$ a
priori small. As we will see during the proof of , Lemma A.1 is a
straightforward corollary of [Mal04, Main Result]. Finally, as stated in
[Mal09, Section 3.2], one can certainly replace the standard Laplace operator
$\Delta$ in Lemma A.1 by a second order uniformly elliptic operator $L$ with
analytic coefficients and propagation of smallness of gradient of solutions to
$Lu=0$ still hold. This will explain why Theorem 2.1 still hold for
$L=\mathrm{div}(A\nabla\cdot)$ with $A$, a matrix-valued uniformly elliptic
with real-analytic entries.

###### Proof.

Let us remark that

$F=(F_{1},\dots,F_{n})=\nabla u=\left(\frac{\partial
u}{\partial_{x_{1}}},\dots,\frac{\partial u}{\partial_{x_{n}}}\right),$

is a solution to the following system

$\sum_{j=1}^{n}\frac{\partial F_{j}}{\partial_{x_{j}}}=0,\qquad\frac{\partial
F_{j}}{\partial_{x_{k}}}=\frac{\partial F_{k}}{\partial_{x_{j}}},\ 1\leq
j<k\leq n.$ (A.2)

Indeed, the first equation of (A.2) corresponds to $\mathrm{div}(\nabla u)=0$
while the second corresponds to Schwarz’s theorem on mixed derivatives by
using the fact that because $u$ is harmonic on $B_{1}$ then it is
$C^{\infty}$-smooth on $B_{1}$. By defining the operator $A$ as

$A:\mathbb{R}^{n}\to\mathbb{R}^{m}\qquad AF=\left(\sum_{j=1}^{n}\frac{\partial
F_{j}}{\partial_{x_{j}}},\left(\frac{\partial
F_{j}}{\partial_{x_{k}}}-\frac{\partial F_{k}}{\partial_{x_{j}}}\right)_{1\leq
j<k\leq n}\right),$ (A.3)

with $m=1+(n(n-1))/2$. The last equations of $AF=0$ imply by Poincaré lemma
that $F$ is locally a gradient. Then the first equation of $AF=0$ implies that
$F$ is locally a gradient of a harmonic function so each component of $F$ is
harmonic then the system of equations (A.2) is a generalized Cauchy-Riemann
system in the sense of [Mal04, Section 4.1]. Moreover, the operator $A$
defined in (A.3) is a factor of the Laplacian in the sense of [Mal04,
Definition 4.1] because

$\frac{\partial}{\partial_{x_{i}}}\left(\sum_{j=1}^{n}\frac{\partial
F_{j}}{\partial_{x_{j}}}\right)-\sum_{k<i\leq
n}\frac{\partial}{\partial_{x_{k}}}\left(\frac{\partial
F_{k}}{\partial_{x_{i}}}-\frac{\partial
F_{i}}{\partial_{x_{k}}}\right)+\sum_{i<k\leq
n}\frac{\partial}{\partial_{x_{k}}}\left(\frac{\partial
F_{i}}{\partial_{x_{k}}}-\frac{\partial F_{k}}{\partial_{x_{i}}}\right)\\\
=\Delta F_{i},\ 1\leq i\leq n.$

Moreover, Lemma 2.4 furnishes that
$\mathcal{C}_{\mathcal{H}}^{n-2+\delta}(\mathcal{E})>0\Rightarrow\operatorname{cap}_{n-2+\delta/2}(\mathcal{E})>0$
so one can apply directly the propagation of smallness result from [Mal04],
i.e. [Mal04, Main Result] to $F=\nabla u$ to get (A.1). ∎

With Lemma A.1 at hand, we can prove Theorem 2.1.

###### Proof of Theorem 2.1.

We split the proof into two steps.

Step 1: Observation on a suitable ball. Let $\Lambda\geq 1$ and let us define

$u=\Pi_{\Lambda}u.$ (A.4)

First, let us take $B(x_{0},r)$ be such that $B(4r,x_{0})\subset\Omega$ and

$m:=\mathcal{C}^{d-2+\delta}(\omega\cap B(x_{0},r))>0.$

Such a ball exists because $\mathcal{C}^{d-2+\delta}(\omega)>0$. Then we have
from [AEWZ14, Theorem 3],

$\left\lVert u\right\rVert_{L^{2}(\Omega)}\leq
Ce^{C\sqrt{\Lambda}}\sup_{xB(x_{0},r)}\left|u(x)\right|.$ (A.5)

Note that we have used [AEWZ14] instead of the more classical reference [JL99]
because we are only assuming that $\Omega$ is a bounded Lipschitz domain,
locally star-shaped. Moreover, we have

$-\Delta u=\sum_{\lambda_{k}\leq\Lambda}\lambda_{k}\langle
u,\phi_{k}\rangle\phi\ \text{in}\ \Omega,\quad u=0\ \text{on}\
\partial\Omega,$ (A.6)

so by elliptic regularity and Sobolev embeddings we have

$\|u\|_{L^{\infty}(\Omega)}\leq C\|u\|_{H^{d}(\Omega)}\leq
Ce^{C\sqrt{\Lambda}}\left\lVert u\right\rVert_{L^{2}(\Omega)},$ (A.7)

hence by (2.6),

$\left\lVert u\right\rVert_{L^{\infty}(\Omega)}\leq
Ce^{C\sqrt{\Lambda}}\sup_{x\in B(x_{0},r)}\left|u(x)\right|.$ (A.8)

Step 2: Observation on $\omega$. Let us take

$\hat{u}(x,t)=\sum_{\lambda_{k}\leq\Lambda}u_{k}\frac{\sinh(\sqrt{\lambda_{k}}t)}{\sqrt{\lambda_{k}}}\phi_{k}(x)\qquad(t,x)\in(-1,+1)\times\Omega.$
(A.9)

One can check that

$\Delta_{t,x}\hat{u}=0\qquad(t,x)\in(-1,+1)\times\Omega.$ (A.10)

Up to a translation, let us assume that $x_{0}=0$ and reducing $r>0$ if
necessary, let us consider $B_{1}$ the unit ball in $\mathbb{R}^{n+1}$ and
$\mathcal{K}\subset\mathbb{R}^{n+1}$ such that

$B(x_{0},r)\times(-1/2,+1/2)\subset\subset\mathcal{K}\subset\subset
B_{1}\subset\subset\Omega\times(-1,+1),$

and

$\mathcal{E}=(\omega\cap
B(x_{0},r))\times\\{0\\}\subset\mathbb{R}^{n}\times\\{0\\}.$

Then we can apply the propagation of smallness for gradients of harmonic
functions i.e. (A.1) from Lemma A.1,

$\sup_{(x,t)\in\mathcal{K}}|\nabla\hat{u}|\leq
C\sup_{(x,t)\in\mathcal{E}}|\nabla\hat{u}|^{\alpha}\|\nabla\hat{u}\|_{L^{\infty}(B)}^{1-\alpha}.$
(A.11)

In particular we get

$\sup_{x\in B(x_{0},r)}|\partial_{t}\hat{u}(x,0)|\leq
C\left(\sup_{x\in\omega}|\partial_{t}\hat{u}(x,0)|\right)^{\alpha}\left\|\hat{u}\right\|^{1-\alpha}_{W_{t,x}^{1,\infty}(B)}.$
(A.12)

The left hand side of (A.12) exactly gives

$\sup_{x\in B(x_{0},r)}|\partial_{t}\hat{u}(x,0)|=\sup_{x\in
B(x_{0},r)}|u(x)|.$ (A.13)

The first right hand side term of (A.12) exactly gives

$\sup_{x\in\omega\cap
B(x_{0},r)}|\partial_{t}\hat{u}(x,0)|=\sup_{x\in\omega\cap
B(x_{0},r)}|u(x)|\leq\sup_{x\in\omega}|u(x)|.$ (A.14)

Then by elliptic regularity estimates on (A.10), the second right hand side
term of (A.12) can be estimated as

$\left\|\hat{u}\right\|_{W_{t,x}^{1,\infty}(B)}\leq
C\left\|\hat{u}\right\|_{L^{2}((-1,+1)\times\Omega)}\leq
Ce^{C\sqrt{\lambda}}\left\lVert u\right\rVert_{L^{\infty}(\Omega)}.$ (A.15)

Therefore, we have from (A.12), (A.13), (A.14) and (A.15),

$\sup_{x\in B(x_{0},r)}|u(x)|\leq
Ce^{C\sqrt{\lambda}}(\sup_{x\in\omega}|u(x)|)^{\alpha}\left\lVert
u\right\rVert_{L^{\infty}(\Omega)}^{1-\alpha}.$ (A.16)

We conclude the proof of (2.2) by gathering (A.8) and (A.16). This ends the
proof of Theorem 2.1. ∎

## References

  * [AE13] Jone Apraiz and Luis Escauriaza.  Null-control and measurable sets.  ESAIM Control Optim. Calc. Var., 19(1):239–254, 2013.
  * [AEWZ14] Jon Apraiz, Luis Escauriaza, Gensheng Wang, and Can Zhang.  Observability inequalities and measurable sets.  J. Eur. Math. Soc. (JEMS), 16(11):2433–2475, 2014.
  * [Ahl10] Lars V. Ahlfors.  Conformal invariants.  AMS Chelsea Publishing, Providence, RI, 2010.  Topics in geometric function theory, Reprint of the 1973 original, With a foreword by Peter Duren, F. W. Gehring and Brad Osgood.
  * [AKBGB+24] Farid Ammar Khodja, Assia Benabdallah, Manuel González-Burgos, Morgan Morancey, and Luz de Teresa.  New results on biorthogonal families in cylindrical domains and controllability consequences, 2024.
  * [Bei64] Albert H Beiler.  Recreations in the theory of numbers: The queen of mathematics entertains.  Courier Corporation, 1964.
  * [BM23] Nicolas Burq and Iván Moyano.  Propagation of smallness and control for heat equations.  J. Eur. Math. Soc. (JEMS), 25(4):1349–1377, 2023.
  * [BP20] Jon Asier Bárcena-Petisco.  Null controllability of the heat equation in pseudo-cylinders by an internal control.  ESAIM Control Optim. Calc. Var., 26:Paper No. 122, 34, 2020.
  * [BPS18] Karine Beauchard and Karel Pravda-Starov.  Null-controllability of hypoelliptic quadratic differential equations.  J. Éc. polytech. Math., 5:1–43, 2018.
  * [Bru99] Alexander Brudnyi.  Local inequalities for plurisubharmonic functions.  Ann. Math., 149(2):511–533, 1999.
  * [CH98] Thierry Cazenave and Alain Haraux.  An introduction to semilinear evolution equations, volume 13 of Oxford Lecture Series in Mathematics and its Applications.  The Clarendon Press, Oxford University Press, New York, 1998.  Translated from the 1990 French original by Yvan Martel and revised by the authors.
  * [Cor07] Jean-Michel Coron.  Control and nonlinearity, volume 136 of Mathematical Surveys and Monographs.  American Mathematical Society, Providence, RI, 2007.
  * [Dem16] Jean-Pierre Demailly.  Analyse numérique et équations différentielles.  Grenoble Sciences. EDP Sciences, Les Ulis, fourth edition, 2016.
  * [Dol73] Szymon Dolecki.  Observability for the one-dimensional heat equation.  Studia Mathematica, 48:291–305, 1973.
  * [Eva10] Lawrence C. Evans.  Partial differential equations, volume 19 of Graduate Studies in Mathematics.  American Mathematical Society, Providence, RI, second edition, 2010.
  * [FI96] Andrei V. Fursikov and Oleg Yu. Imanuvilov.  Controllability of evolution equations, volume 34 of Lecture Notes Series.  Seoul National University, Research Institute of Mathematics, Global Analysis Research Center, Seoul, 1996.
  * [FR71] Hector O. Fattorini and David L. Russell.  Exact controllability theorems for linear parabolic equations in one space dimension.  Arch. Rational Mech. Anal., 43:272–292, 1971.
  * [JL99] David Jerison and Gilles Lebeau.  Nodal sets of sums of eigenfunctions.  In Harmonic analysis and partial differential equations (Chicago, IL, 1996), Chicago Lectures in Math., pages 223–239. Univ. Chicago Press, Chicago, IL, 1999.
  * [JM22] Benjamin Jaye and Mishko Mitkovski.  A sufficient condition for mobile sampling in terms of surface density.  Appl. Comput. Harmon. Anal., 61:57–74, 2022.
  * [Kov01] Oleg Kovrijkine.  Some results related to the Logvinenko-Sereda theorem.  Proc. Amer. Math. Soc., 129(10):3037–3047, 2001.
  * [LM18] Alexander Logunov and Eugenia Malinnikova.  Quantitative propagation of smallness for solutions of elliptic equations.  In Proceedings of the International Congress of Mathematicians—Rio de Janeiro 2018. Vol. III. Invited lectures, pages 2391–2411. World Sci. Publ., Hackensack, NJ, 2018.
  * [LM19] Gilles Lebeau and Iván Moyano.  Spectral inequalities for the Schrödinger operator.  arXiv preprint:1901.03513, 2019.
  * [LMNN21] Alexander Logunov, Eugenia Malinnikova, Nikolai Nadirashvili, and Fedor Nazarov.  The sharp upper bound for the area of the nodal sets of Dirichlet Laplace eigenfunctions.  Geom. Funct. Anal., 31(5):1219–1244, 2021.
  * [LR95] Gilles Lebeau and Luc Robbiano.  Contrôle exact de l’équation de la chaleur.  Comm. Partial Differential Equations, 20(1-2):335–356, 1995.
  * [LRL12] Jérôme Le Rousseau and Gilles Lebeau.  On Carleman estimates for elliptic and parabolic operators. Applications to unique continuation and control of parabolic equations.  ESAIM Control Optim. Calc. Var., 18(3):712–747, 2012.
  * [Mal04] Eugenia Malinnikova.  Propagation of smallness for solutions of generalized Cauchy-Riemann systems.  Proc. Edinb. Math. Soc. (2), 47(1):191–204, 2004.
  * [Mal09] Eugenia Malinnikova.  Quantitative transfer of smallness for solutions of elliptic equations with analytic coefficients and their gradients.  In More Progresses In Analysis, pages 1185–1193. World Scientific, 2009.
  * [Mat95] Pertti Mattila.  Geometry of sets and measures in Euclidean spaces, volume 44 of Cambridge Studies in Advanced Mathematics.  Cambridge University Press, Cambridge, 1995.  Fractals and rectifiability.
  * [Mil10] Luc Miller.  A direct Lebeau-Robbiano strategy for the observability of heat-like semigroups.  Discrete Contin. Dyn. Syst. Ser. B, 14(4):1465–1485, 2010.
  * [NSV04] Fedor Nazarov, Mikhail Sodin, and Alexander Volberg.  Lower bounds for quasianalytic functions. I. How to control smooth functions.  Math. Scand., 95(1):59–79, 2004.
  * [Pom92] Christian Pommerenke.  Boundary behaviour of conformal maps, volume 299 of Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences].  Springer-Verlag, Berlin, 1992.
  * [Sam15] El Hadji Samb.  Internal null-controllability of the n-dimensional heat equation in cylindrical domains.  C. R. Math. Acad. Sci. Paris, 353(10):925–930, 2015.
  * [WYW06] Zhuoqun Wu, Jingxue Yin, and Chunpeng Wang.  Elliptic & parabolic equations.  World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2006.
  * [Zhu23] Yuzhe Zhu.  Remarks on propagation of smallness for solutions of elliptic equations in the plane.  arXiv preprint:2304.09800, 2023.