Boundary value problems and
Heisenberg uniqueness pairs

S. Rigat and F. Wielonsky

We describe a general method for constructing Heisenberg uniqueness pairs $(\Gamma,\Lambda)$ in the euclidean space $\R^{n}$ based on the study of boundary value problems for partial differential equations. As a result, we show, for instance, that any pair made of the boundary $\Gamma$ of a bounded convex set $\Omega$ and a sphere $\Lambda$ is an Heisenberg uniqueness pair if and only if the square of the radius of $\Lambda$ is not an eigenvalue of the Laplacian on $\Omega$. The main ingredients for the proofs are the Paley-Wiener theorem, the uniqueness of a solution to a homogeneous Dirichlet or initial boundary value problem, the continuity of single layer potentials, and some complex analysis in $\C^{n}$. Denjoy's theorem on topological conjugacy of circle diffeomorphisms with irrational rotation numbers is also useful.

§ INTRODUCTION

The interpretation of the uncertainty principle in mathematics is a classical theme, especially in harmonic analysis, see e.g. [17] for a thorough presentation of the subject.
Heisenberg uniqueness pair is a related notion which was introduced in [18]. It consists in the following.

A pair
$(\Gamma,\Lambda)$, where $\Gamma$ is a hypersurface in $\R^{n}$, $n\geq2$, and $\Lambda$ is a set in $\R^{n}$, is an Heisenberg uniqueness pair (sometimes abbreviated HUP) if for any complex-valued function $g$ integrable on $\Gamma$ with respect to $d\sigma$, the surface measure on $\Gamma$, the following implication holds true,

$$\hat g(\lambda)=\int_{\Gamma}e^{-i\lambda\cdot x}g(x)d\sigma(x)=0\text{  on }\Lambda\quad\implies
\quad g=0~\text{ a.e.\ on }\Gamma.$$

The function $\hat g(\lambda)$ is the Fourier transform, in the sense of distribution, of the measure $gd\sigma$, also called the Fourier-Stieltjes transform of $g$. Note that, by the Hahn-Banach theorem, the above condition is equivalent to the fact that the vector space spanned by the family of exponentials $e^{i\lambda\xi}$, $\lambda\in\Lambda$, is weak-* dense in $L^{\infty}(\Gamma)$.
Also, a similar notion is that of a mutually annihilating pair $(S,\Sigma)$ of Borel subsets of $\R$ of positive measure, satisfying

\forall\vphi\in L^{2}(S),~\supp(\hat\vphi)\subset\Sigma\quad\implies\quad\vphi=0,

see [17].

It is easy to check that the notion of an Heisenberg uniqueness pair satisfies two elementary properties, namely,

1) it is invariant by translation: if $(\Gamma,\Lambda)$ is an HUP then $(\Gamma+x_{0},\Lambda+\lambda_{0})$ with $x_{0},\lambda_{0}\in\R^{n}$, is also an HUP,

2) if $T$ is an invertible linear transformation with adjoint $T^{*}$, then

\begin{equation}\label{rem-T}
(\Gamma,\Lambda)\text{ is an HUP}\implies(T(\Gamma),(T^{-1})^{*}(\Lambda))\text{ is also an HUP.}
\end{equation}

In [18], it was proven that the pair made of the hyperbola and a cross lattice,

\Gamma=\left\{\left(x_{1}, x_{2}\right) \in \mathbb{R}^{2} : x_{1} x_{2}=1\right\},\quad
\Lambda_{\alpha, \beta} :=(\alpha \mathbb{Z} \times\{0\}) \cup(\{0\} \times \beta \mathbb{Z}),\quad
\alpha,\beta>0,

is a HUP if and only if $\alpha\beta\leq1$.
Since the publication of [18], many other examples have appeared in the literature. For instance, in [30, 22, 31, 4, 14, 2, 20], one may find examples of HUP's in the plane $\R^{2}$ involving sets like circles, union of lines, ellipses, parabola, polygons or very specific curves. In [15, 33], one may also find examples of HUP's in $\R^{n}$, $n>2$, again involving spheres, paraboloids or certain cones.

The aim of the present paper is to obtain Heisenberg uniqueness pairs in $\R^{n}$, $n\geq2$, under rather general assumptions on the set $\Gamma$ since, in some of our examples, it is only assumed to be the boundary of a bounded convex set.
Our method is inspired by the study in [1], which was concerned with providing rigorous foundations of the Fokas method for boundary value problems, showing, for linear elliptic pde's on bounded convex domains, that a solution to the global relation between the Dirichlet and Neumann boundary data implies the existence of a solution to the Dirichlet problem, and that the solution to the global relation is unique. The link between [1] and the problem of determining Heisenberg uniqueness pairs does not seem to have been previously noticed.

When convenient, we will denote by $X$ a $n$-tuple of variables $X_{1},\ldots,X_{n}$ and by $D$ the tuple of partial derivations $\p_{X_{1}},\ldots,\p_{X_{n}}$.
For a polynomial $P(X)=\sum_{|\alpha|\leq m}a_{\alpha}X^{\alpha}\in\C[X]$, the differential operator $P(-iD)$ is defined by

P(-iD)=\sum_{|\alpha|\leq m}a_{\alpha}(-i)^{|\alpha|}\p_{X_{1}}^{\alpha_{1}}\dots\p_{X_{n}}^{\alpha_{n}}.

We will also denote by $Z(P)$ the subset of $\C^{n}$ of zeros of $P$, and by $Z_{\R}(P)=Z(P)\cap\R^{n}$ the subset of its real zeros.

Let $E_{P}(x)$ be a fundamental solution of the differential operator $P(-iD)$, that is
$P(-iD)E_{P}=\delta$ in the sense of distribution, where $\delta$ denotes the Dirac delta distribution.
We will make use of potentials of a single layer,

\begin{equation}\label{def-Phi}
\Phi g(x):=\int_{\Gamma}g(y)E_{P}(x-y)d\sigma(y),\qquad x\in\R^{n},
\end{equation}

where $g$ is some function defined on $\Gamma$, see Section <ref> for more details.

Let us now state our main result.

Let $\Omega$ be a bounded convex domain in $\R^{n}$ with a piecewise $C^{1}$ boundary $\Gamma$,
and let $d\sigma$ be the surface measure on $\Gamma$ (the arc measure when $n=2$).
Let $P(X_{1},\ldots,X_{n})$ be a polynomial with real coefficients such that

(i) $P$ is a square-free polynomial in $\C[X]$, i.e. each irreducible factor of $P$ is of multiplicity 1, and for at least one variable, e.g. $X_{1}$, one has $P(X)=a_{1}X_{1}^{d}+\cdots$, where the nonzero leading coefficient $a_{1}$ is a constant (i.e. independent from the other variables).

each irreducible component $V_{i}$ of the variety $Z(P)$ is defined by a real polynomial, and contains a point in $V_{i,\R}=V_{i}\cap\R^{n}$, which is smooth as a point of $V_{i}$ (i.e. the Jacobian has rank 1 at this point).

(iii) For some space $\DD_{P}(\bar\Omega)$ of functions defined on $\bar\Omega$,
the homogeneous Dirichlet problem

\begin{align*}
P(-iD)u(x) & =0,\qquad x\in\Omega, %\label{DP1}
\\[5pt]
u(x) & = 0,\qquad x\in\Gamma, %\label{DP2}
\end{align*}

has the unique solution $u=0$ in $\DD_{P}(\bar\Omega)$.

(iv) For some space $\GG(\Gamma)\subset L^{1}(\Gamma)$, of complex-valued functions $g$ defined on $\Gamma$, the potential $\Phi g$, associated to $g\in\GG(\Gamma)$,
defined in (<ref>), is a continuous function in $\R^{n}$, and

\forall g\in\GG(\Gamma),\quad\Phi g|_{\bar\Omega}\in\DD_{P}(\bar\Omega).

$(\Gamma,Z_{\R}(P))$ is a Heisenberg uniqueness pair for $\GG(\Gamma)$, that is,

\forall g\in \GG(\Gamma),\quad\hat g=0\text{ on }Z_{\R}(P)\quad\implies\quad g=0\text{ a.e.\ on }\Gamma.

The spaces of functions $\DD_{P}(\bar\Omega)$ and $\GG(\Gamma)$ in the above statement will be chosen in accordance with the domain $\Omega$ and the polynomial $P$, see Section <ref> for details.

Applying the Theorem <ref> in specific cases, we derive general Heisenberg uniqueness pairs as described next.

The following holds true:

i) Let $\Gamma$ be the boundary of any bounded convex domain in $\R^{n}$ of class $C^{1}$. Let $S_{n-1}(c_{1})$ be the sphere of dimension $n-1$ and radius $c_{1}$. Then $(\Gamma,S_{n-1}(c_{1}))$ is a HUP for $L^{p}(\Gamma)$, $p>n-1$, if and only if $c_{1}^{2}$ is not an eigenvalue of $-\Delta$ on $\Omega$.

ii) Let $\Gamma$ be the rectangle with vertices $(0,0), (T,0), (T,L), (0,L)$, $T,L>0$, and let $\PP$ be the parabola $X_{1}+X_{2}^{2}=0$ in $\R^{2}$.
The pair $(\Gamma, \PP)$ is a HUP

for $C^{2}(\Gamma)$.

iii) Let $\Gamma$ be the rectangle as in ii), and assume that $T/L\notin\Q$. Let $\Delta_{+}$ and $\Delta_{-}$ be the two lines
$\{X_{1}= X_{2}\}$, $\{X_{1}=-X_{2}\}$ in $\R^{2}$.
Then the pair
$(\Gamma, \Delta_{+}\cup\Delta_{-})$ is a HUP

for $C^{1}(\Gamma)$.

iv) Let $\T$ denote the unit circle, and let $\Delta_{1}, \Delta_{2}$ be two lines
through the origin.
Then the pair $(\T,\Delta_{1}\cup\Delta_{2})$
is a HUP for $L^{1}(\T)$ if and only if $\Delta_{1}$ and $\Delta_{2}$ make an angle of $\pi\rho$, with $\rho$ an irrational number.

We note that the HUP in item iv) was obtained previously, in [30], by a different method. Also, making use of the property (<ref>), item iii) may be rephrased as the fact that the pair $(\CC,\Delta_{\rho}\cup\Delta_{-\rho})$ is a HUP, where $\CC$ is the rectangle of sides $\pi$ and 1, and
$\Delta_{\pm\rho}=\{X_{2}=\pm\pi\rho X_{1}\}$, with $\rho\not\in\Q$. This shows some relation between iii) and iv).

Acually, item iv) can be extended in the following way.

Let $\Gamma$ be any strictly convex planar curve of class $C^{2}$. There exists an angle $\rho$ between two lines $\Delta_{1}, \Delta_{2}$ through the origin, such that the pair
$(\Gamma,\Delta_{1}\cup\Delta_{2})$ is a HUP for $L^{1}(\Gamma)$.

Unfortunately, we were unable to characterize the angles $\rho$, in terms of $\Gamma$, for which the above pair is a HUP.
It may possibly be an interesting problem.

In Section <ref> we recall or state several preliminary results that will be useful for our study. In Section <ref>, we give the proof of Theorem <ref>. In Section <ref>, we apply our result in the cases of an elliptic equation (the Helmholtz equation), the Schrödinger equation, a hyperbolic equation (the wave equation), and finally, transport equations. For each of these equations, we derive the corresponding Heisenberg uniqueness pairs.

§ PRELIMINARIES

We start with recalling a few details about the notion of a potential of a single layer, as defined in (<ref>). The following can be found in <cit.>, in the case of the Newton potential.
In the sequel, $\vphi$ denotes a test function. For $g\in L^{1}(\Gamma)$, the Radon measure $gd\sigma$ is the distribution, supported on $\Gamma$,
defined by

The potential of a single layer $\Phi g$, associated to $g$, is then defined as the convolution of distributions,

\Phi g=E*gd\sigma,

where $E$ denotes a fundamental solution of the differential operator $\PP=P(-iD)$. Note that the convolution is well defined as the measure $gd\sigma$ has compact support.

By the classical result of Malgrange and Ehrenpreis,
a fundamental solution $E$ of the linear differential operator $P(-iD)$ with constant coefficients, always exists, satisfying

\begin{equation}\label{sol-fund}
\end{equation}

With $\check E$ denoting the distribution symmetric to $E$, defined by $<\check E,\vphi>=<E,\check\vphi>$, $\check\vphi(x)=\vphi(-x)$, and from basic properties of the convolution, we get

\begin{equation}\label{eq-Tg}
<\Phi g,\vphi>=<gd\sigma,\check E*\vphi>=\int_{\Gamma}(\check E*\vphi)gd\sigma
=\int_{\Gamma}<\check E(y),\vphi(x-y)>g(x)d\sigma(x).
\end{equation}

Now, with $\PP^{*}$, the adjoint of the differential operator $\PP$, we have

\begin{align*}
<\PP(\Phi g),\vphi> & =<\Phi g,\PP^{*}\vphi>=\int_{\Gamma}(\check E*\PP^{*}\vphi)gd\sigma
=\int_{\Gamma}(\PP^{*}\check E*\vphi)gd\sigma
\\[5pt]
%=\int_{\Sigma}(P^{*}\check G*\vphi)gd\sigma
& =\int_{\Gamma}(\check{\PP E}*\vphi)gd\sigma=\int_{\Gamma}(\delta*\vphi)gd\sigma
=\int_{\Gamma}\vphi gd\sigma=<gd\sigma,\vphi>,
\end{align*}

and thus $\PP(\Phi g)=gd\sigma$, supported on $\Gamma$. Hence

\begin{equation}\label{P-Phi=0}
\PP(\Phi g)|_{\R^{n}\setminus\Gamma}=0.
\end{equation}

In the case when the fundamental solution $E$ is a locally integrable function,
one gets, from (<ref>),

<\Phi g,\vphi>=\int_{x\in\Gamma}\int_{y\in\R^{n}}\check E(x-y)\vphi(y)dyg(x)d\sigma(x)

i.e. $\Phi g$ is just the function

\Phi g(y)=\int_{\Gamma}E(y-x)g(x)d\sigma(x).

Next, we recall a version of the Paley-Wiener theorem that will be useful to us, see <cit.>.

Let $K$ be a convex compact subset of $\R^{n}$ with supporting function $H$. Every entire function $u$ in $\C^{n}$ satisfying, for some positive integer $N$,  an estimate

|u(\zeta)| \leqq C(1+|\zeta|)^{N} e^{H(\operatorname{lm} \zeta)}, \quad \zeta \in \mathbb{C}^{n},

is the Fourier transform of a distribution with support contained in $K$.

We will also make use of a result about analytic functions in $\C^{n}$ vanishing on the real points of a variety, see e.g. <cit.>. Recall that a point of an
affine variety $V \subset \mathbb{C}^n$ of dimension $d$, defined by a family of polynomials, is smooth if the Jacobian of the defining family has rank $n-d$ at this point. We first consider the case of an irreducible variety $V$.

Let $V \subset \mathbb{C}^n$ be an irreducible
affine variety of dimension $d$, defined by a family of $n-d$ polynomials $P_{1},\ldots,P_{n-d}\in\R[X]$.
Assume $V$ has a smooth real point $a\in V_{\R}$.

Then $V_{\mathbb{R}}$ is Zariski dense in $V$, that is any function $f$ analytic in
$V$, which vanishes on $V_{\R}$, must vanish on $V$.

Consider the smooth point $a\in V$, and assume, for simplicity, that the first minor $(\p_{z_{j}} P_{i}(a))_{i,j=1,\ldots,n-d}$ of the Jacobian  is non singular.
By the implicit function theorem, there exists a small ball $\BB_{a}$ in $\C^{d}$, centered at $(a_{n-d+1},\ldots,a_{n})$, a neighborhood $W$ of $(a_{1},\ldots,a_{n})$, and $n-d$ analytic functions

defined in $\BB_{a}$ such that,

\begin{align}
(z_{1},\ldots,z_{n})\in W\quad & \text{ and } \quad P_{i}(z_{1},\ldots,z_{n})=0,~ i=1,\ldots,n-d
\quad\iff \notag
\\[5pt]
\label{impl1}
(z_{n-d+1},\ldots,z_{n})\in\BB_{a}\quad & \text{ and }\quad  z_{j}=h_{j}(z_{n-d+1},\ldots,z_{n}),~
\end{align}

Since the $P_{i}$'s have real coefficients, $a$ is also a regular point of $V_{\R}$.

Applying the implicit function theorem, this time in $\R^{n}$, we get,

\begin{align}
(x_{1},\ldots,x_{n})\in W\cap\R^{n}\quad & \text{ and } \quad P_{i}(x_{1},\ldots,x_{n})=0,~ i=1,\ldots,n-d
\quad\iff \notag
\\[5pt]
\label{impl2}
(x_{n-d+1},\ldots,x_{n})\in\BB_{a}\cap\R^{d}\quad & \text{ and }\quad  x_{j}=h_{j}(x_{n-d+1},\ldots,x_{n}),~
\end{align}

where the functions $h_{j}$ in (<ref>) and in (<ref>) are actually the same (and are real-valued when restricted to $\R^{d}$).
Now, for $f$ analytic on $V$ and vanishing on $V_{\R}$, the function

is analytic in $\BB_{a}$ and, because of (<ref>), vanishes on $\BB_{a,\R}$. Hence, $g$ vanishes on the whole of $\BB_{a}$. Equivalently, $f$ vanishes in a neighborhood (in the topology of $V$) of $a$. Finally, applying the identity principle for holomorphic functions on the irreducible algebraic set $V$, see <cit.>, $f$ must vanish on all of $V$.

The above result extends easily to the case of a non irreducible variety.

Let $V\subset\C^{n}$ be an affine variety of dimension $d$, and let $V=\cup V_{i}$ its decomposition into a finite number of irreducible components $V_{i}$. Assume each $V_{i}$ is defined by real polynomials, and contains a smooth real point $a_{i}\in V_{i,\R}$. Then, as in Proposition <ref>, $V_{\R}$ is Zariski dense in $V$.

It suffices to notice that $V_{\R}=\cup V_{i,\R}$ and to apply Proposition <ref> to each of the components $V_{i}$.

Another result about analytic functions in $\C^{n}$, useful to us, is the following lemma.

Let $\Omega$ be an open connected subset of $\mathbb{C}^n$ and let $f$ be analytic in $\Omega$. Assume $P\in\mathbb{C}[z_1,\ldots,z_n]$ is a square-free polynomial.

If $f$ vanishes on the zero set $Z(P)$ of $P$, then $f/P$ is analytic in $\Omega$.

By the local analytic nullstellensatz, see <cit.>, for each $z\in\Omega$, there exists a neighborhood $U_{z}$ of $z$ and a function $g_{z}$ analytic in $U_{z}$ such that $f=g_{z}P$. By the identity principle applied in $\Omega$, we may glue together the functions $g_{z}$ to get a function $g$ analytic in $\Omega$ such that $f=gP$ there.

Finally, for one of our applications, we will need some results about diffeomorphisms of the unit circle $\T=\R/2\pi\Z$.  The rotation number of an orientation-preserving homeomorphism $f:\T\to\T$ is defined as the  limit

\tau(f)=\lim_{n\to\infty}\frac{\tilde f^{n}(x)-x}{2\pi n},\qquad\text{mod }1,$$

independent of $x$, where $\tilde f^{n}$ denotes the $n$-th iterate of a lift $\tilde f$ to $\R$ of $f$,
satisfying $\Pi\circ\tilde  f=f\circ\Pi$, with $\Pi$ the projection $\Pi:\R\to\T$. Intuitively, the rotation number represents the
average portion of the circle that a point is moved by $f$.

The map $f\mapsto \tau(f)$ is continuous in the uniform topology of $C(\T)$, the space of continuous maps from $\T$ to $\T$.

A basic result concerning the diffeomorphisms of the circle is the theorem of Denjoy.

A $C^{1}$ diffeomorphism $f:\T\to\T$ with irrational rotation number $\tau(f)$ and derivative of bounded variation (e.g. a $C^{2}$ diffeomorphism) is topologically conjugate to the rotation $R_{\tau(f)}$ of angle $2\pi\tau(f)$. Namely, there exists a homeomorphism $h$ of the circle such that

f=h^{-1}\circ R_{\tau(f)}\circ h.

§ PROOF OF THEOREM <REF>

Let $g\in\GG(\Gamma)$, the function space introduced in assumption (iv), such that its Fourier-Stieltjes transform $\hat g$ satisfies

$$\hat g(\lambda)=\int_{\Gamma}e^{-i\lambda\cdot x}g(x)d\sigma(x)=0,\quad\lambda\in Z_{\R}(P),$$

with $d\sigma$ the surface measure on $\Gamma$.
Taking the Fourier transform $\FF$ in $\R^{n}$ on both sides of (<ref>), we get

Moreover, The function $\hat g$ is an entire function in $\C^{n}$ which, by assumption, vanishes on $Z_{\R}(P)$. Hence, by Proposition <ref> and its Corollary <ref>, it vanishes on $Z(P)$,
which implies, together with Lemma <ref>, that the quotient $\hat g/P$ is also
an entire function
(recall assumption (i) that $P$ is a square-free polynomial in $\C[X]$).
Hence, for the Fourier transform of $\Phi g$, we have

\begin{equation}\label{Four-trans}
\FF(\Phi g)=\FF(E* gd\sigma)=\FF(E)\FF(gd\sigma)=\hat g\FF(E)=(\hat g/P)P\FF(E)=\hat g/P.
\end{equation}

Next, we use the elementary fact that, if $h(z)$ is an analytic function in a neighborhood of the closed unit disk $\bar\D\subset\C$ and $p(z)$ is a polynomial with leading coefficient $\alpha$, one has

|\alpha h(0)|\leq\sup_{|z|=1}|h(z)p(z)|.

Thus, applying this inequality with the variable $z_{1}$, $z=(z_{1},0,\ldots,0)$, $p(z_{1})=P(z+\lambda)$,
$h(z_{1})=\FF(\Phi g)(z+\lambda)$, and recalling the second part of assumption (i), we get

\begin{align*}
|\FF(\Phi g)(\lambda)| & \leq a_{1}^{-1}\sup_{|z_{1}|=1}|\FF(\Phi g)(z+\lambda)P(z+\lambda)|
\leq a_{1}^{-1}\sup_{|z_{1}|=1}\left|\int_{\Gamma}e^{-i(z+\lambda)\cdot y}g(y)d\sigma(y)\right|
\\[10pt]
& \leq C\sup_{y\in\Gamma}e^{\Im(\lambda)\cdot y}=Ce^{H_{\bar\Omega}(\Im(\lambda))},
\end{align*}

where $C$ is some constant independent of $\lambda$, and $H_{\bar\Omega}$ is the supporting function of $\bar\Omega$, the closure of $\Omega$. Now, applying
the Paley-Wiener theorem, see Theorem <ref>, one derives that $\Phi g$ vanishes outside $\bar\Omega$. On the other hand, from the continuity of $\Phi g$ in $\R^{n}$, recall assumption (iv), $\Phi g$ also vanishes on $\Gamma$.

From assumption (iii), the homogeneous Dirichlet problem for the operator $P(-iD)$ has the unique zero solution in $\DD_{P}(\bar\Omega)$.  Hence $\Phi g$, which lies in $\DD_{P}(\bar\Omega)$ by assumption (iv), and satisfies, see (<ref>),

$$P(-iD)(\Phi g)(x)=0,\quad x\in\R^{n}\setminus\Gamma,$$

must vanish inside $\Omega$ and thus everywhere in $\R^{n}$. Consequently,

$\FF(\Phi g)=0$ i.e. $\hat g=\FF(gd\sigma)=0$. The measure $gd\sigma$ is a distribution with compact support, hence a tempered distribution. From the fact that the Fourier transform is an isomorphism from the space of tempered distributions to itself, we thus obtain that $g=0$.

We have thus proved that $(\Gamma,Z_{\R}(P))$ is a Heisenberg pair for the space $\GG(\Gamma)$.

§ APPLICATIONS

We shall restrict ourselves to quadratic polynomial
$P(X_{1},\ldots,X_{n})\in\R[X]$ of the form

\begin{equation}\label{def-P}
P(X_{1},\ldots,X_{n})=-\sum_{i,j=1}^{n} a_{ij}X_{i}X_{j}+\sum_{i=1}^{n}b_{i}X_{i}+c,
\qquad a_{i,j}, b_{i}, c\in\R,
\end{equation}

with $(a_{ij})_{i,j}$, a symmetric matrix.

§.§ Elliptic equations

In this section, we consider elliptic operators, that is, we assume that the matrix $(a_{ij})_{i,j}$ in (<ref>) is positive.
By a linear change of variables with real coefficients (so that real points of $Z(P)$ correspond by the change of variables), we may restrict ourselves to operators of the form

P(-iD)=\Delta-i\sum_{i=1}^{n}b_{i}\p_{x_{i}}+c\quad\text{where}\quad P(X)= -\sum_{i=1}^{n} X_{i}^{2}+\sum_{i=1}^{n}b_{i}X_{i}+c,\quad b_{i},c\in\R.

Assume that the $b_{i}$'s are zero. Then, $Z(P)$ is defined by the equation

\begin{equation}\label{def-Z-ell}
\sum_{i=1}^{n} X_{i}^{2}-c=0.%\qquad\sum_{i=1}^{n}b_{i}X_{i}=0.
\end{equation}

– If $c<0$, $Z_{\R}(P)$ is the empty set so that Theorem <ref> does not apply.

– If $c=0$ then $Z_{\R}(P)$ reduces to the singleton $0$, which is a singular point of $Z(P)$.

– If $c=c_{1}^{2}>0$, with $c_{1}>0$, then $Z_{\R}(P)$ is the sphere $S_{n-1}(c_{1})$ of dimension $n-1$ and radius $c_{1}$.

Before we elaborate on the third case, and prove the first item in Theorem <ref>, let us mention a result about continuity of

potentials of a single layer of the form

\begin{equation}\label{def-single}
(\Phi g)(x)=\int_{\Gamma}g(y)\vphi(x-y)d\sigma(y),\qquad x\in\R^{n}.
\end{equation}

Assume the following holds :

1) the hypersurface $\Gamma$ is of class $C^{1}$,

2) The function $\vphi:\R^{n}\setminus\{0\}\to\R$ is continuous,

3) The function $\vphi$ is weakly singular at 0, namely there exists constants $0<\nu<n-1$ and $C>0$ such that

\begin{equation*}%\label{weak-sing}
|\vphi(x)|\leq\frac{C}{|x|^{\nu}}\quad\text{as }x\to0.
\end{equation*}

4) $g\in L^{p}_{d\sigma}(\Gamma)$ with $p>1+\nu/(n-1-\nu)$.

Then the potential $\Phi g$ in (<ref>) is continuous on $\R^{n}$. In particular, its limits on $\Gamma$ from the inside and the outside coincide,

(\Phi g)_{+}(x)=(\Phi g)_{-}(x),\qquad x\in\Gamma.

Since we were unable to find a convenient reference for this result, we provide a proof of Theorem <ref> in the Appendix.

We assume $c=c_{1}^{2}>0$.
When $n\geq2$ and $c\neq0$, the polynomial in (<ref>) is irreducible in $\C[X]$, and all points of $Z(P)$ are non singular. Moreover, the homogeneous Dirichlet problem for the Helmholtz operator $\Delta+c_{1}^{2}$ on $\Omega$ has non trivial solutions in $C^{0}(\bar\Omega)\cap C^{2}(\Omega)$ only when $c_{1}^{2}$ is an
an eigenvalue
of the positive operator $-\Delta$ on $\Omega$. Finally, a fundamental solution of the Helmholtz operator is given (see e.g. <cit.>), up to a multiplicative constant, by

%{-\frac{i}{4} H_{0}^{(1)}(\omega|x|),} & {d=2} \\[5pt]
%{\displaystyle -\frac{e^{i \omega|x|}}{4 \pi|x|},} & {d=3}

for \(x \neq 0,\) where $Y_{\alpha}$ is the Bessel function of the second kind of order $\alpha$.
Near 0, the following estimates hold,

\begin{cases}
(2/\pi)\log(z),\quad& \alpha=0,
\\[5pt]
-(\Gamma(\alpha)/\pi)(2/z)^{\alpha},\quad & \alpha>0.
\end{cases}

and thus

\begin{equation}\label{sol-fonda-Helm}
\begin{cases}
(2/\pi)\log(|x|),\quad& n=2,
\\[5pt]
-(\Gamma(n/2-1)/\pi)(2/c_{1})^{n/2-1}|x|^{2-n},\quad & n>2.
\end{cases}
\end{equation}

In view of (<ref>), Theorem <ref> applies with $\vphi=E_{n}$ and $p>n-1$. Together with Theorem <ref>, we get that $(\Gamma,S_{n-1}(c_{1}))$ is a HUP for $L^{p}(\Gamma)$, $p>n-1$.

Assume now that $c_{1}^{2}$ is an eigenvalue of $-\Delta$ on $\Omega$. Then, there exists some nonzero solution $u\in C^{2}(\Omega)\cap C(\bar\Omega)$ of the homogeneous Dirichlet problem in $\Omega$. Moreover, it is known, see e.g. Theorems 3.27 and 3.1 of [8] for the case $n=3$, that $u$ actually lies in $C^{1}(\bar\Omega)$, and that

\int_{\Gamma}\frac{\partial u}{\partial \nu}(y) E_{n}(x- y)d \sigma(y)=
\begin{cases}
u(x), \quad  & x \in \Omega,
\\[5pt]
~0,  \quad & x \in \R^{n} \setminus \bar{\Omega},
\end{cases}

where $\nu$ denotes the outward normal to the surface $\Gamma$. Hence, with $v=\p u/\p \nu\in C^{1}(\Gamma)$, one obtains $u=\Phi v$. First, notice that $v\neq0$ since $u\neq0$. Second, similarly to (<ref>), one has

P(\lambda)\FF(u)(\lambda)=\hat v(\lambda),

and $\FF(u)$ is an entire function since $u$ is continuous, with compact support, hence in $L^{1}(\R^{n})$.
Hence, $\hat v$ vanishes on $Z_{\R}(P)$ and $(\Gamma,S_{n-1}(c_{1}))$ is not a HUP. This finishes the proof of
item i) of Theorem <ref>.

When $\Omega$ is a ball, it is known that the eigenvalues of $\Delta$ are the zeros of Bessel functions of the first kind.

By the remark in (<ref>) and the fact that convexity is preserved by linear transformation, one can make use of a linear diagonal transformation

T:~(x_{1},\ldots,x_{n})\mapsto (a_{1}x_{1},\ldots,a_{n}x_{n}),\quad a_{i}>0,\quad i=1,\ldots,n,$$

normalized by $\sum_{i=1}^{n}a_{i}^{2}=1$,
so that the statement i) in Theorem <ref> becomes the following one. Let $\EE_{n-1}(c_{1})$ be the ellipsoid

\sum_{i=1}^{n}\left(\frac{X_{i}}{a_{i}}\right)^{2}=c_{1}^{2},

whose vector of semi-axes $c_{1}(a_{1},\ldots,a_{n})$ has norm $c_{1}$.

Let $\Gamma$ be the boundary of a bounded convex domain in $\R^{n}$.
Then, $(\Gamma,\EE_{n-1}(c_{1}))$ is a HUP for $L^{p}(\Gamma)$, $p>n-1$ if $c_{1}^{2}$ is different from an eigenvalue of the operator

-\sum_{i=1}^{n}\frac{1}{a_{i}^{2}}\frac{\p^{2}}{\p x_{i}^{2}}\quad\text{on }\Omega.

§.§ The Schrödinger equation

In this section, we prove item ii) of Theorem <ref> by considering the 1-dimensional Schrödinger equation, corresponding to the operator

P(-iD)=i\p_{t}+\p_{x}^{2}\quad\text{where}\quad P(X)=-X_{1}-X_{2}^{2}.

Then $Z(P)=\{(X_{1},X_{2})\in\C^{2},~X_{1}+X_{2}^{2}=0\}$ and $Z_{\R}(P)$ is a parabola in $\R^{2}$.

In the literature, a fundamental solution of the Schrödinger equation is usually given by

H(t)t^{-1/2}\exp(ix^{2}/(4t)),\qquad t\geq0,

see e.g. <cit.>, where $H(t)$ denotes the Heaviside function

H(t)=0\text{ for }t\leq0\text{ and }H(t)=1\text{ for }t>0.

Actually, one can also take as a fundamental solution,

E_{1}(t,x)=t^{-1/2}\exp(ix^{2}/(4t)),\qquad t\neq0,

without the factor $H(t)$, which allows one to reconstruct solutions to the Schrödinger equation for $t\in(-\infty,\infty)$ from an initial data at $t=0$, see <cit.>.

For some $T,L>0$, and $I=(0,L)$, we consider the rectangular domain

\begin{equation}\label{rect}
\Omega:=(0,T)\times I\subset\R^{2},
\end{equation}

with vertices

$$V_{1}=(0,0),\quad V_{2}=(T,0), \quad V_{3}=(T,L), \quad V_{4}=(0,L).$$

\begin{equation}\label{Phi-Schr}
\Phi g(t,x)=\int_{\Gamma}g(u,y)\frac{e^{i(x-y)^{2}/(4(t-u))}}{\sqrt{t-u}}d\sigma(u,y),
\end{equation}

where $g$ is a function defined on the boundary $\Gamma$ of $\Omega$, and $d\sigma$ is the arc measure on $\Gamma$, which we assume to be positively oriented.
Also, we assume that the square root in (<ref>) satisfies

\sqrt{t-u}=-i\sqrt{|t-u|}\quad\text{when }t<u.

Our aim is to show the following result.
Assume the function $g$ to be of class $C^{2}$ on $\Gamma_{1}$ and $\Gamma_{3}$.
Then the function $\Phi g(t,x)$ is of class $C^{2}$ on $\R^{2}\setminus(\Delta_{0}\cup\Delta_{T})$, where $\Delta_{0}, \Delta_{T}$ are the vertical lines $t=0$ and $t=L$ respectively. Moreover, $\Phi g(t,x)$ admits left and right limits at each point of $\Delta_{0}$, resp. $\Delta_{L}$, and these limits coincide.

We decompose the integral according to the four sides $\Gamma_{1}$, $\Gamma_{2}$, $\Gamma_{3}$, $\Gamma_{4}$, of the rectangle, starting with the horizontal side $\Gamma_{1}$ from $(0,0)$ to $(T,0)$, and get

\Phi g(t,x)=\Phi_{1}g(t,x)+\Phi_{2}g(t,x)-\Phi_{3}g(t,x)-\Phi_{4}g(t,x),

Φ_1 g(t,x)     :=  ∫_u=0^Tg(u,0)e^ix^2/(4(t-u))/√(t-u)du,
Φ_2 g(t,x)        :=  ∫_y=0^Lg(T,y)e^i(x-y)^2/(4(t-T))/√(t-T)dy,

Φ_3 g(t,x)     :=  ∫_u=0^Tg(u,L)e^i(x-L)^2/(4(t-u))/√(t-u)du,
Φ_4 g(t,x)        :=  ∫_y=0^Lg(0,y)e^i(x-y)^2/(4t)/√(t)dy.

It is clear that the first and third integrals

are well-defined for any $(t,x)\in\R^{2}$ since the integrands are absolutely integrable. The second one is well-defined when $t\neq T$, and the fourth one is well defined when $t\neq0$. Moreover, the four integrals define
$C^{\infty}$ functions with respect to the variable $x\in\R$. The second and fourth ones define $C^{\infty}$ functions with respect to the variable $t$ when $t\notin\{0,T\}$. The first and third ones are also $C^{\infty}$ functions of $t$ when $t\notin[0,T]$. When $0\leq t\leq T$, we may write, with a change of variable $v=t-u$,

\Phi_{1} g(t,x)  =  \int_{v=t-T}^{t}g(t-v,0)\frac{e^{ix^{2}/(4v)}}{\sqrt{v}}dv,
\quad \Phi_{3} g(t,x)  =  \int_{v=t}^{t-T}g(t-v,L)\frac{e^{i(x-L)^{2}/(4v)}}{\sqrt{v}}dv,

which shows, together with the Leibniz integral rule, that the two integrals are $C^{2}$ functions of $t$ (recall that we assumed $g(u,y)$ to be a $C^{2}$ function on $\Gamma_{1}$ and $\Gamma_{3}$).

Finally, it remains to consider
$\Phi_{2}g(t,x)$ and $\Phi_{4}g(t,x)$, respectively when $t=T$ and $t=0$. Obviously, the integral expressions for $\Phi_{2}g(T,x)$ and $\Phi_{4}g(0,x)$ are not defined, we thus compute the left and right limits as $t\to T_{\pm}$ or $t\to 0_{\pm}$. Recall from the method of stationary phase that, for $\alpha<0<\beta$,

\frac{1}{\sqrt{\pi \epsilon}} \int_{\alpha}^{\beta} e^{iy^{2}/\eps} a(y) d y= e^{i {\pi}/{4}}a(0)+O(\epsilon) \quad \text { as } \epsilon \rightarrow 0_{+},

where $a(y)$ is a $C^{1}$ function, see <cit.> or <cit.>.
Hence, for $0<x<L$,

\lim_{t\to T_{+}}\Phi_{2}g(t,x)=\lim_{t\to T_{+}}\int_{y=0}^{L}g(T,y)
\frac{e^{i(x-y)^{2}/(4(t-T))}}{\sqrt{t-T}}dy=\frac{\sqrt{\pi}}{2}e^{i\pi/4}g(T,x),

\begin{align*}
\lim_{t\to T_{-}} \Phi_{2}g(t,x)
& = \lim_{t\to T_{-}}\int_{y=0}^{L}g(T,y)
\frac{e^{-i(x-y)^{2}/(4(T-t))}}{-i\sqrt{T-t}}dy%=\frac{\sqrt{\pi}}{2}e^{i\pi/4}g(T,x)
\\[10pt]
& = i\lim_{t\to T_{-}}\int_{y=0}^{L}g(T,y)
\frac{\bar{e^{i(x-y)^{2}/(4(T-t))}}}{\sqrt{T-t}}dy
=i\frac{\sqrt{\pi}}{2}e^{-i\pi/4}g(T,x)=\lim_{t\to T_{+}}\Phi_{2}g(t,x),
\end{align*}

which shows, together with the assumption that $x\in[0,L]\mapsto g(T,x)$ is continuous, that $\Phi_{2}g$ can be extended to the right vertical side $\Gamma_{2}$ as a continuous function. With help of <cit.>, one could also check that, for $x\not\in(0,L)$, the left and right limits of $\Phi_{2}g(t,x)$ coincide at $T$, and are equal to 0 when $x\not\in[0,L]$. Though this will be not important for our analysis, we note that $\Phi_{2}g$ has a discontinuity at the right vertices $(T,0)$ and $(T,L)$ of the rectangle if the function $g(t,x)$ does not vanish at these points.

One may check that the function $\Phi_{4}g$ exhibits similar behavior on the line $t=0$, in particular, it can be continuously extended on the left vertical side $\Gamma_{4}$ of the rectangle.

Next, in order to apply Theorem <ref>, we need an uniqueness result for the initial boundary value problem related to the Schrödinger equation, see <cit.>,

The initial boundary value problem

\begin{cases}
iu_{t}(t,x)+\Delta u(t,x)=0,\quad x\in I,~t>0,
\\[5pt]
u(0,x)=\vphi(x)\in H_{0}^{1}(I),\quad x\in \bar I=[0,L],
\\[5pt]
u(t,0)=0,~u(t,L)=0,\quad t\geq0,
\end{cases}

has a unique solution $u$ in the space $C([0,\infty),L^{2}(I))$
of continuous functions from $[0,\infty)$ to $L^{2}(I)$
(actually, the solution is in

$$C([0,\infty),H_{0}^{1}(I))\cap C^{1}([0,\infty),H^{-1}(I)),$$

but this result will not be needed to us). Here, $H_{0}^{1}(I)$ and $H^{-1}(I)$
denote the usual Sobolev spaces of the segment $I$.

Since $\Phi g(t,x)$ is a $C^{2}$ function when $t>0$, that is on the right of the vertical line $\Delta_{0}$, in order to prove that $t\mapsto\Phi g(t,x)$ is a continuous function from  $[0,T)$ to $L^{2}(I)$, it is sufficient to check that

\|\Phi g(t,x)-\Phi g(0,x)\|_{L^{2}(I)}\to0\quad\text{as }t\to0_{+},

and, more precisely, that

\begin{equation}\label{est-phi4}
\|\Phi_{4} g(t,x)-\Phi_{4} g(0,x)\|_{L^{2}(I)}\to0\quad\text{as }t\to0_{+},
\end{equation}

since $\Phi_{i}g$, $i=1,2,3$, are of class $C^{2}$ in a neighborhood of the segment $\{0\}\times I$. The fact that (<ref>) holds true can be derived from uniform error estimates for the stationary phase approximations, see [26]. We let the details to the reader.

Putting together Proposition <ref> and Theorem <ref> applied with the initial data $\vphi(x)=0$ on $\bar I=[0,L]$, we see that the method from Theorem <ref> applies, leading to item ii) of Theorem <ref>.

§.§ Hyperbolic equations

Here we prove item iii) of Theorem <ref> by considering the 1-dimensional wave equation, that is, the operator

P(-iD)=\p_{t}^{2}-\p_{x}^{2}\quad\text{where}\quad P(X)=-X_{1}^{2}+X_{2}^{2}=(X_{2}-X_{1})(X_{2}+X_{1}).

Then $Z(P)$ decomposes into the union of two irreducible varieties, namely the two lines $X_{1}=\pm X_{2}$ in $\C^{2}$, and

$$Z_{\R}(P)=\Delta_{+}\cup\Delta_{-}\subset\R^{2},\qquad \Delta_{+}:X_{1}= X_{2},\quad\Delta_{-}: X_{1}=-X_{2}.$$

A fundamental (distributional) solution of the wave equation is known to be, up to a multiplicative constant,

see <cit.>, where $H$ still denotes the Heaviside function.

As in the case of the Schrödinger equation, we consider the rectangular domain $\Omega\subset\R^{2}$ with vertices $(0,0)$, $(T,0)$, $(T,L)$, $(0,L)$, with $T,L>0$. Let $g$ be a $C^{1}$ function defined on the boundary $\Gamma$ of $\Omega$. Then,

\Phi g(t,x)=\int_{\Gamma}g(u,y)H(t-u)H((t-u)-(x-y))H((t-u)+(x-y))d\sigma(u,y),

where $d\sigma$ is the arc measure on $\Gamma$, which we assume to be positively oriented. In view of the definition of $H$, the function $\Phi g$ rewrites as

\begin{equation}\label{Phi-g-wave}
\Phi g(t,x)=\int_{\Gamma_{t,x}}g(u,y)d\sigma(u,y),
\end{equation}

where $\Gamma_{t,x}$ is the subarc (possibly empty) of $\Gamma$, which intersects the sector

\{(u,y)\in\R^{2},~u< t,\quad u-y< t-x,\quad u+y<t+x\}.

Depending on the location of the point $(t,x)\in\R^{2}$, it is readily checked that the endpoints of $\Gamma_{t,x}$ are two of the points

Because $g$ has been assumed to be a $C^{1}$ function, and all coordinates of the above points are just affine functions of $t$ and $x$, it follows from (<ref>) that the function $\Phi g$ is of class $C^{2}$ in $\R^{2}$.

Assuming $\hat g=0$ on the two lines $X_{1}=\pm X_{2}$ in $\R^{2}$, we know from the proof of Theorem <ref> that the function $\Phi g$ vanishes outside of $\Omega$, and by continuity also on $\Gamma$. Hence, $\Phi g$, which belongs to $C^{2}(\Omega)\cap C^{1}(\bar\Omega)$ is a solution of the wave equation, and solves the homogeneous Dirichlet problem in $\bar\Omega$.
Let us now recall the following result, see <cit.>,

The homogeneous Dirichlet problem

$$(\p_{t}^{2}-\p_{x}^{2})f=0\text{ in }\Omega,\quad f=0\text{ on }\Gamma,$$

has the unique solution $f=0$ in $C^{2}(\Omega)\cap C^{1}(\bar\Omega)$ if and only if $T/L$ is an irrational number.

Actually, [10] considers the $(n+1)$-dimensional wave equation in a rectangular domain, and [11] considers more general hyperbolic equations in cylindrical domains.

From what precedes, we see that Theorem <ref> applies, leading to item iii) of Theorem <ref>.

More generally, making use of (<ref>), one gets that, for $c>0$, $(\Gamma, \Delta_{c,+}\cup\Delta_{c,-})$ is an Heisenberg uniqueness pair  for $C^{1}(\Gamma)$ if $T/(cL)\notin\Q$, where
$\Delta_{c,+}$ and $\Delta_{c,-}$ are the lines with slopes $c$ and $-c$.

§.§ A simple first order equation

The goal of this section is to prove item iv) of Theorem <ref> and Theorem <ref>.
We consider the simple differential equation of first order $u_{y}=0$ in $\R^{2}$, associated to the operator

P(-iD)=i\p_{y}\quad\text{where}\quad P(X)=-X_{2},

in a convex domain $\Sigma$ with boundary $\Gamma$. The set $\Lambda$ is the horizontal line $\Delta_{0}:=\{X_{2}=0\}$.

As is easily checked, a fundamental solution is given by the distribution

In view of (<ref>), we have, for $g$ a function on $\Gamma$, integrable with respect to $d\sigma$, $\vphi$ a test function, and $x=(x_{1},x_{2})$,

<\Phi g,\vphi>=\int_{\Gamma}<E(y),\vphi(x+y)>g(x)d\sigma(x)

We choose as a domain the unit disk $\Sigma=\D$ with boundary the unit circle $\Gamma=\T$. We set

\T_{+}=\{(x_{1},x_{2})\in\T,~x_{2}\geq0\},\quad\T_{-}=\{(x_{1},x_{2})\in\T,~x_{2}<0\},

and we parameterize these two arcs by the functions $s\in[-1,1]\to(s,\pm\sqrt{1-s^{2}})$, so that

Setting $g(x)=g_{+}(s)$ for $x\in\T_{+}$ and $g(x)=g_{-}(s)$ for $x\in\T_{-}$, we get,

\begin{align*}
<\Phi g,\vphi> & =\int_{s=-1}^{1}\int_{y=0}^{\infty}\vphi(s,\sqrt{1-s^{2}}+y)|s|g_{+}(s)\frac{dyds}{\sqrt{1-s^{2}}}
\\[5pt]
& + \int_{s=-1}^{1}\int_{y=0}^{\infty}\vphi(s,-\sqrt{1-s^{2}}+y)|s|g_{-}(s)\frac{dyds}{\sqrt{1-s^{2}}}.
\end{align*}

Letting $u=y+\sqrt{1-s^{2}}$ in the first double integral and $u=y-\sqrt{1-s^{2}}$ in the second one, the above expression becomes

\begin{align*}
<\Phi g,\vphi> =\int_{(s,u)\in D_{+}}\vphi(s,u)|s|g_{+}(s)\frac{d\lambda(s,u)}{\sqrt{1-s^{2}}}
+ \int_{(s,u)\in D_{-}}\vphi(s,u)|s|g_{-}(s)\frac{d\lambda(s,u)}{\sqrt{1-s^{2}}},
\end{align*}

where $D_{+}$ (resp. $D_{-}$) denotes the subset of $\R^{2}$ which lie above $\T_{+}$ (resp. $\T_{-}$), and $d\lambda$ denotes the planar Lebesgue measure. Thus, the distribution $\Phi g$ is actually a locally integrable function in $L^{1}_{loc}(\R^{2})$, namely

\begin{align*}
\Phi g(s,u) & =\chi_{D_{+}}(s,u)\frac{g_{+}(s)|s|}{\sqrt{1-s^{2}}}
\\[5pt]
& =\chi_{\D}(s,u)\frac{g_{-}(s)|s|}{\sqrt{1-s^{2}}}+
\chi_{D_{+}}(s,u)(g_{+}(s)+g_{-}(s))\frac{|s|}{\sqrt{1-s^{2}}},
\end{align*}

where $\chi_{A}$ denotes the characteristic function of a set $A\subset\R^{2}$.

Now, assume that $\hat g=0$ on $\Lambda$.
Here we can not apply directly Theorem <ref> since, a priori, $\Phi g$ is not a continuous function. Nevertheless, the first part of the proof of Theorem <ref> tells us that $\Phi g=0$ on $D_{+}$, that is

\begin{equation}\label{rel-g1}
g_{+}(s)=-g_{-}(s),\quad a.e.~s\in[-1,1]\quad\iff\quad
g(e^{i\theta})=-g(e^{-i\theta}),\quad a.e.~\theta\in[0,2\pi].
\end{equation}

Now, if we assume that, for some given angle $\rho\in(0,\pi)$, one has $\hat g=0$ on the line $\Delta_{\rho}:=\{X_{1}=(\cot\rho) X_{2}\}$, and consider, instead of $u_{y}=0$, the differential equation

$$u_{x}-(\cot\rho) u_{y}=0,$$

we derive, in a completely similar fashion, that

\begin{equation}\label{rel-g2}
g(e^{i(\rho+\theta)})+g(e^{i(\rho-\theta)})=0,\quad a.e.~ \theta\in[0,2\pi].
\end{equation}

Combining (<ref>) and (<ref>), we obtain

g(e^{i(\theta+\rho)})=g(e^{i(\theta-\rho)}),\quad a.e.~\theta\in[0,2\pi],

that is, $2\rho$ is a period of the function $\theta\mapsto g(e^{i\theta})$. By uniqueness of the Fourier coefficients of a function in $L^{1}(\T)$,

we get

a_{n}=a_{n}e^{i2n\rho},\quad n\in\Z.

Assume $a_{n}\neq0$ for some $n\neq0$. Then $2n\rho=2k\pi$ for some $k\in\Z$ which implies that $\rho/\pi$ is a rational number. Hence, if $\rho/\pi\not\in\Q$, $g$ is a constant function, and by (<ref>), it has to vanish almost everywhere. Finally, making use of the property (<ref>) with $T$ a rotation, we obtain the HUP stated in item iv) of Theorem <ref>.

Conversely, assume that $\rho/\pi=k/n\in\Q$. Then, the nonzero function $g(e^{i\theta})=e^{in\theta}-e^{-in\theta}$ satisfies (<ref>) and (<ref>). Hence, with $\Phi_{1}g$ and $\Phi_{2}g$ the potentials associated to the fundamental solutions of $\p_{y}$ and $\p_{x}-(\cot\rho)\p_{y}$, one gets that $\Phi_{1}g$ and $\Phi_{2}g$ have compact support since they vanish outside $\D$. Consequently, their Fourier transforms are entire functions in $\C^{n}$, which, in view of (<ref>), implies that $\hat g$ vanishes on $\Delta_{0}\cup\Delta_{\rho}$. Hence, $(\T,\Delta_{0}\cup\Delta_{\rho})$ cannot be a HUP, and item iv) of Theorem <ref> is completely proved.

The above result for the circle $\T$ can be extended, in some sense, to any strictly convex curve $\Gamma$ of class $C^{2}$, see Theorem <ref>. A proof of this theorem is the goal of the remaining part of this section.
Thus we consider a function $g\in L^{1}(\Gamma)$ such that $\hat g$ vanishes on the union $\Delta_{0}\cup\Delta_{\rho}$ of two lines. We introduce two diffeomorphisms $s_{0}$ and $s_{\rho}$ on $\Gamma$ defined by the condition that, for $x\in\Gamma$, the segment $(x,s_{0}(x))$ is vertical (resp. $(x,s_{\rho}(x))$ makes an angle $\rho+\pi/2$ with the oriented horizontal axis). Note that, by the assumption of strict convexity of $\Gamma$, $s_{0}$ and $s_{\rho}$ are well-defined. They are involutive maps, and $s_{0}$
has exactly two fixed points $M_{1},M_{2}$ which are the points where the tangeants to $\Gamma$ are vertical.
Setting $t_{\rho}=s_{\rho}\circ s_{0}$, a reasoning similar to the one we just made for the circle leads to the conclusion that

\begin{equation}\label{inv-g-gam}
g(s_{0}(x))=-g(x),\qquad g(s_{\rho}(x))=-g(x),\quad a.e.\ x\in\Gamma,
\end{equation}

and thus,

\begin{equation}\label{inv-g-gamma}
g(t_{\rho}(x))=g(x),\quad a.e.\ x\in\Gamma.
\end{equation}

Next, pick any point inside $\Gamma$. By translation, we may assume it is the origin $O$ of the plane, and by dilation, we may also assune that the unit circle $\T$ lies inside $\Gamma$. We denote by $p$ the diffeomorphism from $\Gamma$ to $\T$,

p:\Gamma\to\T,\qquad x\mapsto x/\|x\|,

where $\|x\|$ denotes the euclidean norm in $\R^{2}$.

(a) at (.5,0);
(b) at (2,0);
(c) at (1.12,1.4);
(d) at (1.5,1.5);
(e) at (-.5,1.5);

(f) at (-.56,1.2);
(g) at (2.15,.4);
(h) at (.73,1.3);

(i) at (1.5,-.26);
[draw,use Hobby shortcut,closed=true]

(c) node$\bullet$;
(1.33,1.35) node $p(M)$;
(d) node$\bullet$;
(1.6,1.6) node $M$;

(f) node$\bullet$;
(-1.05,1.2) node $M_{2}=t_{\rho_{1}}(M_{1})$;
(g) node$\bullet$;
(2.6,.47) node $M_{1}=t_{\rho_{1}}(M_{2})$;

(h) node$\bullet$;
(i) node$\bullet$;
(1.5,-.4) node $s_{0}(M)$;

[fill=none](h) circle (.4) ;
(.79,1.4) node $O$;
(-.35,.5) node $\Gamma$;

(j) at (-.56,.35);
(k) at (2.15,-.45);
(l) at (0.36,0.08);
(l) node$\bullet$;
(.18,-0.05) node $t_{\rho_{1}}(M)$;

Figure 1 : The convex curve $\Gamma$, the circle $\T$, the points $M, M_{1}, M_{2}$ on $\Gamma$,

and some of their transforms, as defined in the text.

Moreover, let

$$\tilde t_{\rho}:=p\circ t_{\rho}\circ p^{-1}:\T\to\T,\qquad\tilde g=g\circ p^{-1}\in L^{1}(\T),$$

where we notice that $\tilde t_{\rho}$ is a $C^{2}$ diffeomorphism of $\T$. Indeed, $\Gamma$ being a curve of class $C^{2}$, all of the bijective maps $s_{0}, s_{\rho}, t_{\rho}, p$ are diffeomorphisms of class $C^{2}$.

First, it is clear that, when $\rho=0$, $t_{0}$ is the identity map of $\Gamma$, and $\tilde t_{0}$ the identity map of $\T$, with rotation number 0. Second, consider the unique points $M_{1},M_{2}$ alluded to above, e.g. $M_{1}$ with the largest $x_{1}$-coordinate on $\Gamma$, and let choose $\rho=\rho_{1}$ such that the segment $(M_{1},M_{2})$ makes an angle $\rho_{1}+\pi/2$ with the oriented horizontal axis. Then,

t_{\rho_{1}}(M_{1})=s_{\rho_{1}}(M_{1})=M_{2}\quad\text{and}\quad t_{\rho_{1}}(M_{2})=s_{\rho_{1}}(M_{2})=M_{1}.

Hence, $M_{1}$ is a 2-periodic point of $t_{\rho_{1}}$, and, similarly, $p(M_{1})$ is a 2-periodic point of $\tilde t_{\rho_{1}}$, which implies that the rotation number of $\tilde t_{\rho_{1}}$ is $1/2$. From the continuity of the map $\rho\mapsto \tau(\tilde t_{\rho})$, recall Lemma <ref>, we derive that there exists some angle $0<\rho_{2}<\rho_{1}$ such that $\tau_{2}:=\tau(\tilde t_{\rho_{2}})\not\in\Q$. From Denjoy's theorem <ref>, there exists a homeomorphism $h:\T\to\T$ such that $h\circ\tilde t_{\rho_{2}}\circ h^{-1}$ is the rotation $R_{\tau_{2}}$ of angle
$2\pi\tau_{2}$. It then follows, with (<ref>), that

(\tilde g\circ h^{-1})(x)=(\tilde g\circ h^{-1})(R_{\tau_{2}}(x)),\qquad x\in\T.

By the fact that $\tau_{2}$ is irrational, and the uniqueness of the Fourier coefficients of $\tilde g\circ h^{-1}$, we obtain as above that the function $\tilde g\circ h^{-1}$ is constant on $\T$, hence also the function $g$ on $\Gamma$. From the first equality in (<ref>), we finally conclude that $g=0$ almost everywhere on $\Gamma$. Applying property (<ref>) with a rotation finishes the proof of Theorem <ref>.

§ APPENDIX

We give a proof of Theorem <ref> about the continuity of single layer potential with weakly singular kernels.

First, when $x_{0}\not\in\Gamma$, $\Phi g$ is continuous at $x_{0}$ since

– the map $x\mapsto \vphi(x-y)g(y)$ is continuous at $x_{0}$, for all $y\in\Gamma$,

– $|\vphi(x-y)g(y)|\leq Mg(y)\in L^{p}(\Gamma)\subset L^{1}(\Gamma)$, where $M$ is the sup of $|\vphi(x-y)|$ when $x$ is in a neighborhood of $x_{0}$ that does not intersect $\Gamma$, and $y\in\Gamma$.

Second, assume $x_{0}\in\Gamma$.

Let $\rm T\Gamma_{x_{0}}$ and $n_{x_{0}}$ be the $(n-1)$-dimensional tangeant space to $\Gamma$ at $x_{0}$, and the normal unit vector to $\Gamma$ at $x_{0}$. We write $y=x_{0}+\tau_{y}+\eta_{y}n_{x_{0}}\in\Gamma$ with $\tau\in \rm T\Gamma_{x_{0}}$, and we let $\eta=\gamma(\tau)$ be the equation of $\Gamma$ in a neighborhood of $x_{0}$.
For some small enough $\delta>0$, let

$$\Gamma(x_{0},\delta)=\{y\in\Gamma,~\tau_{y}\in B(x_{0},\delta)\},$$

where $B(x_{0},\delta)$ is the ball of radius $\delta$ in $\rm T\Gamma_{x_{0}}$. Finally, for $x$ sufficiently close to $x_{0}$, we write

Then, we have

\begin{multline}\label{diff-Phi}
\Phi g(x)-\Phi g(x_{0})=\int_{y\in\Gamma(x_{0},\delta)}\vphi(x-y)g(y)d\sigma(y)-\int_{y\in\Gamma(x_{0},\delta)}\vphi(x_{0}-y)g(y)d\sigma(y)\\[5pt]
\end{multline}

For the first two integrals, for $x$ possibly equal to $x_{0}$, we have, by Hölder inequality, with $q$ the exponent conjugate to $p$,

\begin{multline*}
\left|\int_{y\in\Gamma(x_{0},\delta)}\vphi(x-y)g(y)d\sigma(y)\right|\\
\leq\left(\int_{y\in\Gamma(x_{0},\delta)}|\vphi(x-y)|^{q}d\sigma(y)\right)^{1/q}
\left(\int_{y\in\Gamma(x_{0},\delta)}|g(y)|^{p}d\sigma(y)\right)^{1/p}.
\end{multline*}

The last integral is finite since $g\in L^{p}_{d\sigma}(\Gamma)$. For the second one, note that

for $\delta$ small (recall that $\gamma'$ is continuous and $\gamma'(0)=0$). Hence

\left|\int_{y\in\Gamma(x_{0},\delta)}|\vphi(x-y)|^{q}d\sigma(y)\right|\leq 2C\int_{B(x_{0},\delta)}
\frac{d\tau}{|\tau_{x}-\tau|^{\nu q}},

(with $\tau_{x}=0$ if $x=x_{0}$) where the last integral over the $(n-1)$-dimensional ball $B(x_{0},\delta)$ is convergent since

$$ 1+\nu/(n-1-\nu)<p\quad\iff\quad \nu q<n-1.

Hence, by taking $\delta$ small, it can be made as small as we wish (uniformly in $\tau_{x}$).

It remains to show that the third integral in (<ref>) can be made small. For $x$ close to $x_{0}$ so that $x\notin\Gamma\setminus\Gamma(x_{0},\delta)$, the function

is continuous at $x_{0}$ (same argument as in the first case), and thus, the third integral is small when $x$ is sufficiently close to $x_{0}$.

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