2.1 Uncertainty principles for finite combinations of Hermite functions

The main result in this work is the following uncertainty principles for finite combinations of Hermite functions:

Theorem 2.1. Let $\rho : \mathbb {R}^{n} \longrightarrow (0,+\infty )$ be a $\frac {1}{2}$ -Lipschitz positive function with $\mathbb {R}^{n}$ being equipped with the Euclidean norm, such that there exist some positive constants $0< \varepsilon \leq 1, m>0, R \geq m$ such that

$$ \begin{align*} \forall x \in \mathbb{R}^{n}, \quad 0<m \leq \rho(x) \leq R{\left\langle x\right\rangle}^{1-\varepsilon}, \end{align*} $$

with $\langle x \rangle =\left (1+\lVert x\rVert ^{2}\right )^{\frac {1}{2}}$ and $\lVert \cdot \rVert $ the Euclidean norm on $\mathbb {R}^{n}$ . Let $\omega $ be a measurable subset of $\mathbb {R}^{n}$ which is $\gamma $ -thick with respect to the density $\rho $ – that is,

(2.1) $$ \begin{align} \exists 0< \gamma \leq 1, \forall x \in \mathbb{R}^{n}, \quad \lvert\omega \cap B(x,\rho(x))\rvert \geq \gamma \lvert B(x,\rho(x))\rvert, \end{align} $$

where $B(y,r)$ denotes the Euclidean ball centered at $y \in \mathbb {R}^{n}$ with radius $r>0$ , and $\lvert \cdot \rvert $ denotes the Lebesgue measure. Then there exist some positive constants $\kappa _{n}(m, R, \gamma , \varepsilon )>0, \tilde {C}_{n}(\varepsilon , R) >0$ and a positive universal constant $\tilde {\kappa }_{n}>0$ depending only on the dimension such that

$$ \begin{align*} \forall N \geq 1, \forall f \in \mathcal{E}_{N}, \quad \lVert f\rVert_{L^{2}(\mathbb{R}^{n})} \leq \kappa_{n}(m, R, \gamma, \varepsilon) \left( \frac{\tilde{\kappa}_{n}}{\gamma} \right)^{\tilde{C}_{n}(\varepsilon, R) N^{1-\frac{\varepsilon}{2}}} \lVert f\rVert_{L^{2}(\omega)}, \end{align*} $$

with $\mathcal E_{N}$ being the finite-dimensional vector space spanned by the Hermite functions $(\Phi _{\alpha })_{\lvert \alpha \rvert \leq N}$ .

Taking the parameter $\varepsilon =1$ , Theorem 2.1 allows us to recover the quantitative spectral estimate of Logvinenko–Sereda type (1.22), established in [Reference Beauchard, Jaming and Pravda-Starov7, Theorem 2.1], as condition (2.1) is then equivalent to the thickness property (1.3). Contrary to the thick case ( $\varepsilon =1$ ), notice that when $0<\varepsilon <1$ , condition (2.1) allows control subsets to have holes with diameters tending to infinity. Theorem 2.1 applies, for instance, with the family of unbounded densities

$$ \begin{align*} \rho_{\varepsilon}(x)= R_{\varepsilon} \left\langle x \right\rangle^{1-\varepsilon}, \quad x \in \mathbb{R}^{n}, \end{align*} $$

when $0 < \varepsilon <1$ and $0< R_{\varepsilon } \leq \frac {1}{2(1-\varepsilon )}$ , as $\rho _{\varepsilon }$ is then a $\frac {1}{2}$ -Lipschitz positive function (see Section A.4). However, the case $\varepsilon =0$ , corresponding to a possible linear dependence of the radius, is not covered by Theorem 2.1.

The following result shows that the regularity assumptions on the density $\rho $ can be slightly weakened by allowing it to fail to be a Lipschitz function, while strengthening on the $\gamma $ -thickness condition with respect to $\rho $ by imposing some constraints on the lower bound for the parameter $0<\gamma \leq 1$ :

Corollary 2.2. Let $\rho : \mathbb {R}^{n} \longrightarrow (0,+\infty )$ be a continuous positive function verifying

(2.2) $$ \begin{align} \exists 0< \varepsilon \leq 1, \exists 0 < R_{\varepsilon} \leq \frac{1}{2(1-\varepsilon)}, \forall x \in \mathbb{R}^{n}, \quad 0<\rho(x) \leq R_{\varepsilon} {\left\langle x\right\rangle}^{1-\varepsilon}, \end{align} $$

with (by convention) no upper-bound condition on $R_{\varepsilon }>0$ in the case when $\varepsilon =1$ . If $\omega $ is a measurable subset of $\mathbb {R}^{n}$ that is $\gamma $ -thick with respect to the density $\rho $ – that is,

(2.3) $$ \begin{align} \forall x \in \mathbb{R}^{n}, \quad \lvert\omega \cap B(x,\rho(x))\rvert \geq \gamma \lvert B(x,\rho(x))\rvert, \end{align} $$

with $1-\frac {1}{6^{n}}<\gamma \leq 1$ , where $B(y,r)$ denotes the Euclidean ball centered at $y \in \mathbb {R}^{n}$ with radius $r>0$ – then there exist some positive constants $\kappa _{n} (R_{\varepsilon }, \gamma , \varepsilon )>0, \tilde {C}_{n} (\varepsilon , R_{\varepsilon }) >0$ and a positive universal constant $\tilde {\kappa }_{n}>0$ depending only on the dimension such that

$$ \begin{align*} \forall N \geq 1, \forall f \in \mathcal{E}_{N}, \quad \lVert f\rVert_{L^{2}(\mathbb{R}^{n})} \leq \kappa_{n} (R_{\varepsilon}, \gamma, \varepsilon) \left( \frac{\tilde{\kappa}_{n}}{\gamma} \right)^{\tilde{C}_{n} \left(\varepsilon, R_{\varepsilon}\right) N^{1-\frac{\varepsilon}{2}}} \lVert f\rVert_{L^{2}(\omega)}. \end{align*} $$

The lower-bound condition $1-\frac {1}{6^{n}}<\gamma \leq 1$ can be unexpected. We actually do not know if this assumption is really relevant, or if Corollary 2.2 holds true as well without this technical condition. Let us only mention that this lower-bound condition is somehow related to the smallness condition on the positive parameter $0<\varepsilon \leq \varepsilon _{0}$ , with $0<\varepsilon _{0} \ll 1$ sufficiently small, in the result of Kovrijkine [Reference Kovrizhkin27, Theorem 1.1], where it is established that a pair $(S, \Sigma )$ is a strong annihilating one when S and $\Sigma $ are measurable subsets satisfying the following $\varepsilon $ -thinness condition:

(2.4) $$ \begin{align} \forall x \in \mathbb{R}^{n}, \quad &\lvert S \cap B(x,\rho_{1} (\lVert x\rVert))\rvert \leq \varepsilon \lvert B(x,\rho_{1}(\lVert x\rVert))\rvert, \end{align} $$

(2.5) $$ \begin{align} \forall x \in \mathbb{R}^{n}, \quad &\lvert\Sigma \cap B(x,\rho_{2} (\lVert x\rVert))\rvert \leq \varepsilon \lvert B(x,\rho_{2}(\lVert x\rVert))\rvert, \end{align} $$

when $\rho _{1}, \rho _{2} : \mathbb {R}_{+} \longrightarrow (0,+\infty )$ are continous nonincreasing functions satisfying

$$ \begin{align*} \exists C_{1}, C_{2}>0, \forall t \in \mathbb{R}_{+}, \quad \frac{C_{2}}{\rho_{2}\left(\frac{C_{1}}{\rho_{1}(t)} \right)} \geq t, \end{align*} $$

with $0<\varepsilon \leq \varepsilon _{0}$ . Corollary 2.2 is a direct consequence of Theorem 2.1 using the density $\rho _{\varepsilon }(x)=R_{\varepsilon } {\left \langle x\right \rangle }^{1-\varepsilon }$ , with $x \in \mathbb {R}^{n}$ and $ 0 < R_{\varepsilon } \leq \frac {1}{2(1-\varepsilon )}$ , together with Lemma A.6.

2.2 Null-controllability of hypoelliptic non-self-adjoint quadratic equations

This section is devoted to the study of null-controllability for evolution equations associated to certain classes of non-self-adjoint quadratic operators enjoying some global subelliptic properties. The main result in this section is Corollary 2.4. This result is a consequence of the new uncertainty principles established in Theorem 2.1 and the abstract observability result given by Theorem 1.6. It extends to any control subset that is thick with respect to an unbounded Lipschitzian density with almost-linear growth at infinity, the result of null-controllability proved by Beauchard, Jaming and the second author in [Reference Beauchard, Jaming and Pravda-Starov7, Theorem 2.2].

2.2.1 Miscellaneous facts about quadratic differential operators

Quadratic operators are pseudodifferential operators defined in the Weyl quantization

(2.6) $$ \begin{align} q^{w}(x,D_{x}) f(x) =\frac{1}{(2\pi)^{n}}\int_{\mathbb{R}^{2n}}{e^{i(x-y) \cdot \xi}q\left(\frac{x+y}{2},\xi\right)f(y)dyd\xi} \end{align} $$

by symbols $q(x,\xi )$ , with $(x,\xi ) \in \mathbb {R}^{n} \times \mathbb {R}^{n}, n \geq 1$ , which are complex-valued quadratic forms

$$ \begin{align*} q : \mathbb{R}_{x}^{n} \times \mathbb{R}_{\xi}^{n} &\rightarrow \mathbb{C}\\ (x,\xi) & \mapsto q(x,\xi). \end{align*} $$

These operators are actually differential operators with simple and fully explicit expression, since the Weyl quantization of the quadratic symbol $x^{\alpha } \xi ^{\beta }$ , with $(\alpha ,\beta ) \in \mathbb {N}^{2n}, \lvert \alpha +\beta \rvert = 2$ , is given by the differential operator

$$ \begin{align*} \frac{x^{\alpha}D_{x}^{\beta}+D_{x}^{\beta} x^{\alpha}}{2}, \quad D_{x}=i^{-1}\partial_{x}. \end{align*} $$

Notice that these operators are non-self-adjoint as soon as their Weyl symbols have a nonzero imaginary part. The maximal closed realization of the quadratic operator $q^{w}(x,D_{x})$ on $L^{2}(\mathbb {R}^{n})$ – that is, the operator equipped with the domain

(2.7) $$ \begin{align} D(q^{w})=\left\{f \in L^{2}(\mathbb{R}^{n}) : q^{w}(x,D_{x})f \in L^{2}(\mathbb{R}^{n})\right\}, \end{align} $$

where $q^{w}(x,D_{x})f$ is defined in the distribution sense – is known to coincide with the graph closure of its restriction to the Schwartz space [Reference Hörmander24, pp. 425–426],

$$ \begin{align*} q^{w}(x,D_{x}) : \mathscr{S}(\mathbb{R}^{n}) \rightarrow \mathscr{S}(\mathbb{R}^{n}). \end{align*} $$

Classically, to any quadratic form $q : \mathbb {R}_{x}^{n} \times \mathbb {R}_{\xi }^{n} \rightarrow \mathbb {C}$ defined on the phase space is associated a matrix $F \in M_{2n}(\mathbb {C})$ called its Hamilton map, or its fundamental matrix, which is the unique matrix satisfying the identity

(2.8) $$ \begin{align} \forall (x,\xi) \in \mathbb R^{2n},\forall (y,\eta) \in \mathbb R^{2n}, \quad q((x,\xi),(y,\eta))=\sigma((x,\xi),F(y,\eta)), \end{align} $$

where $q(\cdot ,\cdot )$ is the polarized form associated with the quadratic form q and where $\sigma $ stands for the standard symplectic form

(2.9) $$ \begin{align} \sigma((x,\xi),(y,\eta))=\langle \xi, y \rangle -\langle x, \eta\rangle=\sum_{j=1}^{n}\left(\xi_{j} y_{j}-x_{j} \eta_{j}\right), \end{align} $$

with $x=(x_{1},\dotsc ,x_{n}), y=(y_{1},\dotsc ,y_{n}), \xi =(\xi _{1},\dotsc ,\xi _{n}), \eta =(\eta _{1},\dotsc ,\eta _{n}) \in \mathbb {C}^{n}$ . We observe from the definition that

$$ \begin{align*} F=\frac{1}{2}\begin{pmatrix} \nabla_{\xi}\nabla_{x} q & \nabla_{\xi}^{2}q \\ -\nabla_{x}^{2}q & -\nabla_{x}\nabla_{\xi} q \end{pmatrix}, \end{align*} $$

where the matrices $\nabla _{x}^{2}q=\left (a_{i,j}\right )_{1 \leq i,j \leq n}, \nabla _{\xi }^{2}q=\left (b_{i,j}\right )_{1 \leq i,j \leq n}, \nabla _{\xi }\nabla _{x} q =\left (c_{i,j}\right )_{1 \leq i,j \leq n}, \nabla _{x}\nabla _{\xi } q=\left (d_{i,j}\right )_{1 \leq i,j \leq n}$ are defined by the entries

$$ \begin{align*} a_{i,j}=\partial_{x_{i},x_{j}}^{2} q, \qquad b_{i,j}=\partial_{\xi_{i},\xi_{j}}^{2}q, \qquad c_{i,j}=\partial_{\xi_{i},x_{j}}^{2}q, \qquad d_{i,j}=\partial_{x_{i},\xi_{j}}^{2}q. \end{align*} $$

The notion of singular space was introduced in [Reference Hitrik and Pravda-Starov18] by Hitrik and the second author by pointing out the existence of a particular vector subspace in the phase space $S \subset \mathbb {R}^{2n}$ , which is intrinsically associated with a given quadratic symbol q. This vector subspace is defined as the following finite intersection of kernels:

(2.10) $$ \begin{align} S=\left( \bigcap_{j=0}^{2n-1}\mathrm{Ker} \left[\mathrm{Re}F(\mathrm{Im}F)^{j} \right]\right)\cap \mathbb{R}^{2n}, \end{align} $$

where $\mathrm {Re}F$ and $\mathrm {Im}F$ stand respectively for the real and imaginary parts of the Hamilton map F associated with the quadratic symbol q:

$$ \begin{align*} \mathrm{Re}F=\frac{1}{2}\left(F+\overline{F}\right), \qquad \mathrm{Im}F=\frac{1}{2i}\left(F-\overline{F}\right). \end{align*} $$

As pointed out in [Reference Hitrik and Pravda-Starov18, Reference Hitrik, Pravda-Starov and Viola21, Reference Hitrik, Pravda-Starov and Viola22, Reference Ottobre, Pavliotis and Pravda-Starov34, Reference Pravda-Starov35, Reference Pravda-Starov, Rodino and Wahlberg36, Reference Viola44], the notion of singular space plays a basic role in the understanding of the spectral and hypoelliptic properties of the (possibly) nonelliptic quadratic operator $q^{w}(x,D_{x})$ , as well as the spectral and pseudospectral properties of certain classes of degenerate doubly characteristic pseudodifferential operators [Reference Hitrik and Pravda-Starov19, Reference Hitrik and Pravda-Starov20, Reference Viola42, Reference Viola43]. In particular, [Reference Hitrik and Pravda-Starov18, Theorem 1.2.2] provides a complete description for the spectrum of any nonelliptic quadratic operator $q^{w}(x,D_{x})$ whose Weyl symbol q has a nonnegative real part $\mathrm {Re}q \geq 0$ and satisfies a condition of partial ellipticity along its singular space S,

(2.11) $$ \begin{align} (x,\xi) \in S, \qquad q(x,\xi)=0 \Rightarrow (x,\xi)=0. \end{align} $$

Under these assumptions, the spectrum of the quadratic operator $q^{w}(x,D_{x})$ is shown to be composed of a countable number of eigenvalues with finite algebraic multiplicities, and the structure of this spectrum is similar to the one known for elliptic quadratic operators [Reference Sjöstrand39]. This condition of partial ellipticity is generally weaker than the condition of ellipticity, $S \subsetneq \mathbb {R}^{2n}$ , and allows one to deal with more degenerate situations. An important class of quadratic operators satisfying condition (2.11) are those with zero singular spaces $S=\{0\}$ . In this case, the condition of partial ellipticity trivially holds. More specifically, these quadratic operators have been shown in [Reference Pravda-Starov35, Theorem 1.2.1] to be hypoelliptic and to enjoy global subelliptic estimates of the type

(2.12) $$ \begin{align} & \exists C > 0, \forall f \in \mathscr{S}(\mathbb{R}^{n}), \\\nonumber & \qquad\quad \left\lVert\langle(x,D_{x})\rangle^{2(1-\delta)} f\right\rVert_{L^{2}(\mathbb{R}^{n})} \leq C\left(\lVert q^{w}(x,D_{x}) f\rVert_{L^{2}(\mathbb{R}^{n})}+\lVert f\rVert_{L^{2}(\mathbb{R}^{n})}\right), \end{align} $$

where $\langle (x,D_{x})\rangle ^{2}=1+\lVert x\rVert ^{2}+\lVert D_{x}\rVert ^{2}$ , with a sharp loss of derivatives $0 \leq \delta <1$ with respect to the elliptic case (case $\delta =0$ ), which can be explicitly derived from the structure of the singular space.

In this work, we study the class of quadratic operators whose Weyl symbols have nonnegative real parts $\mathrm {Re}q \geq 0$ , and zero singular spaces $S=\{0\}$ . These quadratic operators are also known [Reference Hitrik and Pravda-Starov18, Theorem 1.2.1] to generate strongly continuous contraction semigroups $\left (e^{-tq^{w}}\right )_{t \geq 0}$ on $L^{2}(\mathbb {R}^{n})$ , which are smoothing in the Schwartz space for any positive time:

$$ \begin{align*} \forall t>0, \forall f \in L^{2}(\mathbb{R}^{n}), \quad e^{-t q^{w}}f \in \mathscr{S}(\mathbb{R}^{n}). \end{align*} $$

In [Reference Hitrik, Pravda-Starov and Viola22, Theorem 1.2], these regularizing properties were sharpened and these contraction semigroups were shown to be actually smoothing for any positive time in the Gelfand–Shilov space $S_{1/2}^{1/2}(\mathbb {R}^{n})$ : $\exists C>0, \exists t_{0} > 0, \forall f \in L^{2}(\mathbb {R}^{n}), \forall \alpha , \beta \in \mathbb {N}^{n}, \forall 0<t \leq t_{0}$ ,

(2.13) $$ \begin{align} \left\lVert x^{\alpha}\partial_{x}^{\beta}\left(e^{-tq^{w}}f\right)\right\rVert_{L^{\infty}(\mathbb{R}^{n})} \leq \frac{C^{1+\lvert\alpha\rvert+\lvert\beta\rvert}}{t^{\frac{2k_{0}+1}{2}(\lvert\alpha\rvert+\lvert\beta\rvert+2n+s)}}(\alpha!)^{1/2}(\beta!)^{1/2}\lVert f\rVert_{L^{2}(\mathbb{R}^{n})}, \end{align} $$

where s is a fixed integer verifying $s> n/2$ and where $0 \leq k_{0} \leq 2n-1$ is the smallest integer satisfying

(2.14) $$ \begin{align} \left( \bigcap_{j=0}^{k_{0}}\mathrm{Ker}\left[\mathrm{Re}F(\mathrm{Im}F)^{j} \right]\right)\cap \mathbb{R}^{2n}=\{0\}. \end{align} $$

Thanks to this Gelfand–Shilov smoothing effect (2.13), Beauchard and the second author established in [Reference Beauchard and Pravda-Starov8, Proposition 4.1] that for any quadratic form $q : \mathbb {R}_{x,\xi }^{2n} \rightarrow \mathbb {C}$ with a nonnegative real part $\mathrm {Re}q \geq 0$ and a zero singular space $S=\{0\}$ , the following dissipation estimate holds:

(2.15) $$ \begin{align} & \exists C_{0}>1, \exists t_{0}>0, \forall t \geq 0, \forall k \geq 0, \forall f \in L^{2}(\mathbb{R}^{n}),\\\nonumber & \qquad\qquad\qquad\qquad \left\lVert(1-\pi_{k})\left(e^{-tq^{w}}f\right)\right\rVert_{L^{2}(\mathbb{R}^{n})} \leq C_{0}e^{- \delta(t)k} \lVert f\rVert_{L^{2}(\mathbb{R}^{n})},\qquad\qquad \end{align} $$

with

(2.16) $$ \begin{align} \delta(t)=\frac{\inf(t,t_{0})^{2k_{0}+1}}{C_{0}} \geq 0, \quad t \geq 0, \end{align} $$

where $0 \leq k_{0} \leq 2n-1$ is the smallest integer satisfying equation (2.14) and where $(\pi _{k})_{k \geq 0}$ are the Hermite orthogonal projections defined in equation (1.15). Combining these dissipation estimates with the quantitative spectral estimate of Logvinenko–Sereda type (1.22) established in [Reference Beauchard, Jaming and Pravda-Starov7, Theorem 2.1], Beauchard, Jaming and the second author derived from the abstract observability result [Reference Beauchard and Pravda-Starov8, Theorem 2.1] the following result of null-controllability [Reference Beauchard, Jaming and Pravda-Starov7, Theorem 2.2]:

Theorem 2.3 Beauchard, Jaming, and Pravda-Starov

Let $q : \mathbb {R}_{x}^{n} \times \mathbb {R}_{\xi }^{n} \rightarrow \mathbb {C}$ be a complex-valued quadratic form with a nonnegative real part $\mathrm {Re}q \geq 0$ and a zero singular space $S=\{0\}$ . If $\omega $ is a measurable thick subset of $\mathbb {R}^{n}$ – that is, if condition (1.3) holds for some $L>0$ and $0<\gamma \leq 1$ – then the evolution equation

with $q^{w}(x,D_{x})$ being the quadratic differential operator defined by the Weyl quantization of the symbol q, is null-controllable from the set $\omega $ in any positive time $T>0$ .

Thanks to the new uncertainty principles established in Theorem 2.1 and the abstract observability result given by Theorem 1.6, Theorem 2.3 can be generalized to any control subset that is thick with respect to an unbounded Lipschitzian density with almost-linear growth at infinity.

If $\rho : \mathbb {R}^{n} \longrightarrow (0,+\infty )$ is a $\frac {1}{2}$ -Lipschitz positive function with $\mathbb {R}^{n}$ being equipped with the Euclidean norm such that there exist some positive constants $0 < \varepsilon \leq 1, m>0, R \geq m$ such that

$$ \begin{align*} \forall x \in \mathbb{R}^{n}, \quad 0 < m \leq \rho (x) \leq R \left\langle x \right\rangle^{1-\varepsilon}, \end{align*} $$

and if $\omega \subset \mathbb {R}^{n}$ is a measurable subset that is $\gamma $ -thick with respect to the density $\rho $ for some $0<\gamma \leq 1$ – that is, if condition (2.1) holds – we can apply Theorem 1.6 together with Theorem 2.1 for the following choices of parameters: $\Omega =\mathbb {R}^{n}$ ; $A=q^{w}(x,D_{x})$ ; $0<a=1-\frac {\varepsilon }{2}< b=1$ ; $t_{0}>0$ as in formula (2.15); $m_{1}=2k_{0}+1$ , where $k_{0}$ is defined in equation (2.14); $m_{2}=0$ ; any constant $c_{1}>0$ satisfying

$$ \begin{align*} \forall k \geq 1, \quad \kappa_{n} (m, R, \gamma, \varepsilon) \left( \frac{\tilde{\kappa}_{n}}{\gamma} \right)^{\tilde{C}_{n} (\varepsilon, R)k^{1-\frac{\varepsilon}{2}}} \leq e^{c_{1} k^{1-\frac{\varepsilon}{2}}}, \end{align*} $$

where the positive constants $\kappa _{n} (m, R, \gamma , \varepsilon )>0, \tilde {C}_{n} (\varepsilon , R)$ , $\tilde {\kappa }_{n}>0$ are given by Theorem 2.1; $c_{1}^{\prime }=c_{2}^{\prime }=1$ ; and $c_{2}= \frac {1}{C_{0}}>0$ , where $C_{0}>1$ is defined in formula (2.15). We therefore obtain the following observability estimate in any positive time:

$$ \begin{align*} & \exists C>1, \forall T>0, \forall g \in L^{2}(\mathbb{R}^{n}), \\ & \qquad\qquad\qquad \left\lVert e^{-Tq^{w}} g\right\rVert_{L^{2}(\mathbb{R}^{n})}^{2} \leq C\exp\left(\frac{C}{T^{\left(\frac{2}{\varepsilon}-1\right)\left(2k_{0}+1\right)}}\right) \int_{0}^{T} \left\lVert e^{-tq^{w}} g\right\rVert_{L^{2}(\omega)}^{2} dt. \end{align*} $$

After we note on one hand that the $L^{2}(\mathbb {R}^{n})$ -adjoint of the quadratic operator $(q^{w},D(q^{w}))$ is the quadratic operator $\left (\overline {q}^{w},D\left (\overline {q}^{w}\right )\right )$ , whose Weyl symbol is the complex conjugate of q, and that on the other hand the symbol $\overline {q}$ is also a complex-valued quadratic form with a nonnegative real part and a zero singular space, the Hilbert uniqueness method allows us to obtain the following result of null-controllability:

Corollary 2.4. Let $q : \mathbb {R}_{x}^{n} \times \mathbb {R}_{\xi }^{n} \rightarrow \mathbb {C}$ be a complex-valued quadratic form with a nonnegative real part $\mathrm {Re}q \geq 0$ and a zero singular space $S=\{0\}$ . Let $\rho : \mathbb {R}^{n} \longrightarrow (0,+\infty )$ be a $\frac {1}{2}$ -Lipschitz positive function, with $\mathbb {R}^{n}$ being equipped with the Euclidean norm, such that there exist some positive constants $0 < \varepsilon \leq 1, m>0, R \geq m$ such that

$$ \begin{align*} \forall x \in \mathbb{R}^{n}, \quad 0 < m \leq \rho (x) \leq R \left\langle x \right\rangle^{1-\varepsilon} \end{align*} $$

and $\omega $ is a measurable subset of $\mathbb {R}^{n}$ . If $\omega $ is $\gamma $ -thick with respect to the density $\rho $ – that is,

$$ \begin{align*} \exists 0< \gamma \leq 1, \forall x \in \mathbb{R}^{n}, \quad \lvert\omega \cap B(x,\rho(x))\rvert \geq \gamma \lvert B(x,\rho(x))\rvert, \end{align*} $$

where $B(y,r)$ denotes the Euclidean ball centered at $y \in \mathbb {R}^{n}$ with radius $r>0$ – then the evolution equation

with $q^{w}(x,D_{x})$ being the quadratic differential operator defined by the Weyl quantization of the symbol q – is null-controllable from the control subset $\omega $ in any positive time $T>0$ .

2.3 Null-controllability of evolution equations enjoying Gelfand–Shilov smoothing effects

Given an abstract evolution equation enjoying some Gelfand–Shilov regularizing effects, we aim now to figure out sufficient geometric conditions on control subsets to ensure null-controllability in any positive time.

Let us consider the evolution equation

(2.17)

associated to A a closed operator on $L^{2}(\mathbb {R}^{n})$ that is the infinitesimal generator of a strongly continuous contraction semigroup $\left (e^{-tA}\right )_{t \geq 0}$ on $L^{2}(\mathbb {R}^{n})$ enjoying some Gelfand–Shilov smoothing effects for any positive time – that is, verifying

(2.18) $$ \begin{align} \forall t>0, \forall u \in L^{2}(\mathbb{R}^{n}), \quad e^{-tA}u \in S^{1/(2s)}_{1/(2s)} (\mathbb{R}^{n}), \end{align} $$

with $\frac {1}{2}<s \leq 1$ . We assume more specifically that the contraction semigroup $\left (e^{-tA}\right )_{t \geq 0}$ enjoys the following quantitative regularizing estimates: There exist some constants $\frac {1}{2}<s \leq 1, C_{s}>1, 0<t_{0} \leq 1, m_{1},m_{2} \in \mathbb {R}$ with $m_{1}>0, m_{2} \geq 0$ such that

(2.19) $$ \begin{align} & \forall 0< t \leq t_{0}, \forall \alpha, \beta \in \mathbb{N}^{n}, \forall g \in L^{2}(\mathbb{R}^{n}),\\\nonumber & \qquad\qquad\qquad \left\lVert x^{\alpha} \partial_{x}^{\beta}\left( e^{-t A}g\right)\right\rVert \leq \frac{C_{s}^{1+\lvert\alpha\rvert+\lvert\beta\rvert}}{t^{m_{1} \left(\lvert\alpha\rvert+\lvert\beta\rvert\right)+m_{2}}} (\alpha !)^{\frac{1}{2s}}(\beta!)^{\frac{1}{2s}} \lVert g\rVert_{L^{2}(\mathbb{R}^{n})}, \quad \end{align} $$

where here (and only here) the norm $\lVert \cdot \rVert $ denotes either the $L^{\infty }(\mathbb {R}^{n})$ -norm or the $L^{2}(\mathbb {R}^{n})$ -norm. Lemma A.8 shows that if the estimates (2.19) hold with the $L^{\infty }(\mathbb {R}^{n})$ -norm, then they also hold with the $L^{2}(\mathbb {R}^{n})$ -norm with the same constants $\frac {1}{2}<s \leq 1, 0<t_{0} \leq 1$ , but with different values for the constants $C_{s}>1, m_{1}>0, m_{2} \geq 0$ . The following result provides sufficient geometric conditions on control subsets related to the index of symmetric Gelfand–Shilov regularity $\frac {1}{2s}$ to ensure the null-controllability of the adjoint system:

Theorem 2.5. Let A be a closed operator on $L^{2}(\mathbb {R}^{n})$ which is the infinitesimal generator of a strongly continuous contraction semigroup $\left (e^{-tA}\right )_{t \geq 0}$ on $L^{2}(\mathbb {R}^{n})$ that satisfies the quantitative smoothing estimates (2.19) for some $\frac {1}{2}< s \leq 1$ . Let $\rho : \mathbb {R}^{n} \longrightarrow (0,+\infty )$ be a $\frac {1}{2}$ -Lipschitz positive function with $\mathbb {R}^{n}$ being equipped with the Euclidean norm, such that there exist some constants $0 \leq \delta < 2s-1, m>0, R \geq m$ such that

$$ \begin{align*} \forall x \in \mathbb{R}^{n}, \quad 0<m \leq \rho(x) \leq R{\left\langle x\right\rangle}^{\delta}. \end{align*} $$

If $\omega $ is a measurable subset of $\mathbb {R}^{n}$ which is $\gamma $ -thick with respect to the density $\rho $ – that is,

$$ \begin{align*} \exists 0< \gamma \leq 1, \forall x \in \mathbb{R}^{n}, \quad \lvert\omega \cap B(x,\rho(x))\rvert \geq \gamma \lvert B(x,\rho(x))\rvert, \end{align*} $$

where $B(y,r)$ denotes the Euclidean ball centered at $y \in \mathbb {R}^{n}$ with radius $r>0$ – then the evolution equation associated to the $L^{2}(\mathbb {R}^{n})$ -adjoint operator $A^{*}$ ,

(2.20)

is null-controllable from the control subset $\omega $ in any positive time $T>0$ .

As recalled in the previous section, strongly continuous contraction semigroups generated by accretive non-self-adjoint quadratic operators with zero singular spaces enjoy smoothing effects in the Gelfand–Shilov space $S^{1/2}_{1/2}(\mathbb {R}^{n})$ . More specifically, Alphonse and Bernier established in [Reference Alphonse and Bernier2, Theorem 1.6] that such contraction semigroups $\left (e^{-tq^{w}}\right )_{t \geq 0}$ on $L^{2}(\mathbb {R}^{n})$ satisfy the following quantitative regularizing estimates: There exist some constants $C> 1, 0<t_{0} \leq 1$ such that

(2.21) $$ \begin{align} & \forall 0< t \leq t_{0}, \forall k \geq 1, \forall X_{1},\dotsc,X_{k} \in \mathbb{R}^{2n}, \forall g \in L^{2}(\mathbb{R}^{n}),\\\nonumber & \qquad\qquad\quad \left\lVert L_{X_{1}}\dotsm L_{X_{k}}\left( e^{-t q^{w}}g\right)\right\rVert_{L^{2}(\mathbb{R}^{n})} \leq \frac{C^{1+k}}{t^{\frac{2k_{0}+1}{2}k}} \left(\prod_{j=1}^{k} \left\lVert X_{j}\right\rVert \right)(k!)^{\frac{1}{2}} \lVert g\rVert_{L^{2}(\mathbb{R}^{n})}, \end{align} $$

with $0 \leq k_{0} \leq 2n-1$ the smallest integer satisfying equation (2.14), where $\lVert X_{0}\rVert $ is the Euclidean norm of $X_{0} \in \mathbb {R}^{2n}$ and where $L_{X_{j}}$ is the first-order differential operator

$$ \begin{align*} L_{X_{j}}=\left\langle x_{j}, x \right\rangle + \left\langle \xi_{j}, \partial_{x} \right\rangle, \quad X_{j}=\left(x_{j},\xi_{j}\right) \in \mathbb{R}^{2n}, \end{align*} $$

with $\langle \cdot ,\cdot \rangle $ the Euclidean dot product. The estimates (2.21) imply in particular that for all $0< t \leq t_{0}, \alpha , \beta \in \mathbb {N}^{n}, g \in L^{2}(\mathbb {R}^{n})$ ,

(2.22) $$ \begin{align} \left\lVert x^{\alpha} \partial_{x}^{\beta}\left(e^{-t q^{w}}g\right)\right\rVert_{L^{2}(\mathbb{R}^{n})} \leq \frac{C\left((2n)^{\frac{1}{2}}C\right)^{\lvert\alpha\rvert+\lvert\beta\rvert}}{t^{\frac{2k_{0}+1}{2}(\lvert\alpha\rvert+\lvert\beta\rvert)}}(\alpha !)^{\frac{1}{2}}(\beta!)^{\frac{1}{2}} \lVert g\rVert_{L^{2}(\mathbb{R}^{n})}. \end{align} $$

Indeed, we observe that

$$ \begin{align*} x^{\alpha}\partial_{x}^{\beta}= \left(\prod_{j=1}^{n} L_{e_{j}}^{\alpha_{j}}\right)\left(\prod_{k=1}^{n} L_{\varepsilon_{k}}^{\beta_{k}}\right),\quad \alpha=(\alpha_{1},\dotsc,\alpha_{n}), \beta=(\beta_{1},\dotsc,\beta_{n}) \in \mathbb{N}^{n}, \end{align*} $$

where $(e_{1},\dotsc ,e_{n},\varepsilon _{1},\dotsc ,\varepsilon _{n})$ denotes the canonical basis of $\mathbb {R}_{x}^{n}\times \mathbb {R}_{\xi }^{n}$ , and that the basic estimate (3.44) implies that

$$ \begin{align*} \forall \alpha, \beta \in \mathbb{N}^{n}, \quad (\lvert\alpha\rvert+\lvert\beta\rvert)! \leq 2^{\lvert\alpha\rvert+\lvert\beta\rvert}(\lvert\alpha\rvert)!(\lvert\beta\rvert)! \leq (2n)^{\lvert\alpha\rvert+\lvert\beta\rvert}\alpha!\beta!, \end{align*} $$

since

(2.23) $$ \begin{align} \frac{(\lvert\alpha\rvert+\lvert\beta\rvert)!}{(\lvert\alpha\rvert)!(\lvert\beta\rvert)!}=\binom{\lvert\alpha\rvert+\lvert\beta\rvert}{\lvert\alpha\rvert} \leq \sum_{k=0}^{\lvert\alpha\rvert+\lvert\beta\rvert}\binom{\lvert\alpha\rvert+\lvert\beta\rvert}{k}=2^{\lvert\alpha\rvert+\lvert\beta\rvert}. \end{align} $$

The strongly continuous contraction semigroup generated by the $L^{2}(\mathbb {R}^{n})$ -adjoint operator $(q^{w})^{*}=\left (\overline {q}\right )^{w}$ satisfies the very same quantitative regularizing estimates (2.21), since the quadratic symbol $\overline {q}$ also has a nonnegative real part with a zero singular space. Thanks to these smoothing estimates, the result of Corollary 2.4 can therefore be recovered in applying Theorem 2.5.

As noted at the end of the proof of Theorem 2.5 (§4), the conclusions of Theorem 2.5 hold true as well when the quantitative regularizing estimates (2.19) holding for some $\frac {1}{2}< s \leq 1$ are replaced by the following assumption:

(2.24) $$ \begin{align} & \exists m_{1}, m_{2}>0, \exists C_{1}, C_{2} >0, \exists 0<t_{0} \leq 1, \forall 0 < t \leq t_{0}, \forall g \in L^{2}(\mathbb{R}^{n}),\\\nonumber & \qquad\qquad\qquad\qquad \sum_{\alpha \in \mathbb{N}^{n}}e^{\frac{2 t^{m_{1}}}{C_{1}} \left(2 \lvert\alpha\rvert+n\right)^{s}} \left\lvert\left\langle e^{-t A}g, \Phi_{\alpha}\right\rangle_{L^{2}(\mathbb{R}^{n})}\right\rvert^{2} \leq \frac{C_{2}^{2}}{t^{2m_{2}}} \lVert g\rVert_{L^{2}(\mathbb{R}^{n})}^{2},\quad \end{align} $$

with $(\Phi _{\alpha })_{\alpha \in \mathbb {N}^{n}}$ the $L^{2}(\mathbb {R}^{n})$ -Hermite basis. As an application of this remark, we consider the fractional harmonic operator

(2.25) $$ \begin{align} \forall u \in D(\mathcal{H}^{s} ), \quad \mathcal{H}^{s} u=\left(-\Delta_{x}+\lVert x\rVert^{2}\right)^{s}u = \sum_{\alpha \in \mathbb{N}^{n}} (2 \lvert\alpha\rvert+n)^{s} \langle u,\Phi_{\alpha} \rangle_{L^{2}(\mathbb{R}^{n})} \Phi_{\alpha}, \end{align} $$

with $\frac {1}{2}< s \leq 1$ , equipped with the domain

(2.26) $$ \begin{align} D(\mathcal{H}^{s} )= \left\{ u \in L^{2}(\mathbb{R}^{n}): \sum_{\alpha \in \mathbb{N}^{n}} (2 \lvert\alpha\rvert+n)^{2s} \left\lvert\langle u,\Phi_{\alpha}\rangle_{L^{2}(\mathbb{R}^{n})}\right\rvert^{2} < + \infty \right\}. \end{align} $$

The fractional harmonic oscillator $\mathcal {H}^{s}$ is a self-adjoint operator generating a strongly continuous contraction semigroup $\left (e^{-t \mathcal {H}^{s}}\right )_{t \geq 0}$ on $L^{2}(\mathbb {R}^{n})$ explicitly given by

(2.27) $$ \begin{align} \forall t \geq 0, \forall u \in L^{2}(\mathbb{R}^{n}), \quad e^{-t \mathcal{H}^{s}} u = \sum_{\alpha \in \mathbb{N}^{n}} e^{-t\left(2 \lvert\alpha\rvert+n\right)^{s}} \langle u, \Phi_{\alpha} \rangle_{L^{2}(\mathbb{R}^{n})} \Phi_{\alpha} \end{align} $$

(see, e.g. [Reference Tucsnak and Weiss41, Propositions 2.6.2 and 2.6.5]). As assumption (2.24) trivially holds for the fractional harmonic oscillator, Theorem 2.5 allows us to derive the following result of null-controllability:

Corollary 2.6. Set $\frac {1}{2}< s \leq 1$ and let $\rho : \mathbb {R}^{n} \longrightarrow (0,+\infty )$ be a $\frac {1}{2}$ -Lipschitz positive function with $\mathbb {R}^{n}$ being equipped with the Euclidean norm, such that there exist some constants $0\leq \delta < 2s-1, m>0, R \geq m$ such that

$$ \begin{align*} \forall x \in \mathbb{R}^{n}, \quad 0<m \leq \rho(x) \leq R{\left\langle x\right\rangle}^{\delta}. \end{align*} $$

If $\omega $ is a measurable subset of $\mathbb {R}^{n}$ which is $\gamma $ -thick with respect to the density $\rho $ – that is,

$$ \begin{align*} \exists 0< \gamma \leq 1, \forall x \in \mathbb{R}^{n}, \quad \lvert\omega \cap B(x,\rho(x))\rvert \geq \gamma \lvert B(x,\rho(x))\rvert, \end{align*} $$

where $B(y,r)$ denotes the Euclidean ball centered at $y \in \mathbb {R}^{n}$ with radius $r>0$ – then the evolution equation associated to the fractional harmonic oscillator $\mathcal {H}^{s}=\left (-\Delta _{x}+\lVert x\rVert ^{2}\right )^{s}$ ,

is null-controllable from the control subset $\omega $ in any positive time $T>0$ .

2.4 Outline of the work

Section 3 is devoted to the proof of Theorem 2.1. It is the core of the present work. Theorem 2.5 is then proved in §4, and Appendix A gathers miscellaneous facts about the gamma function, Hermite functions, slowly varying metrics, and Gelfand–Shilov regularity. Some proofs of technical results as Bernstein-type estimates are also given in the appendix.

3.1 Step 1: Bad and good balls

Using Theorem A.5, we can find a sequence $(x_{k})_{k \geq 0}$ in $\mathbb {R}^{n}$ such that

(3.4) $$ \begin{align} \exists N_{0} \in \mathbb{N}, \forall \left(i_{1}, \dotsc, i_{N_{0}+1}\right) \in \mathbb{N}^{N_{0}+1} \text{ with } i_{k} \neq i_{l} \text{ if }1 \leq k \neq l \leq N_{0}+1, \quad \bigcap \limits_{k=1}^{N_{0}+1} {B_{i_{k}}}=\emptyset \end{align} $$

and

(3.5) $$ \begin{align} \mathbb{R}^{n}=\bigcup_{k=0}^{+\infty} {B_{k}}, \end{align} $$

where

(3.6) $$ \begin{align} B_{k}=\left\{y \in \mathbb{R}^{n}: \lVert y-x_{k}\rVert_{x_{k}} <1\right\}=\{y \in \mathbb{R}^{n}: \lVert y-x_{k}\rVert <\rho(x_{k})\}=B(x_{k},\rho(x_{k})). \end{align} $$

Let us notice from Theorem A.5 that the nonnegative integer $N_{0}$ depends only on the dimension n and the constant $C \geq 1$ appearing in the slowness condition (A.36), which can be taken equal to $C=2$ here, as $\rho $ is a $\frac {1}{2}$ -Lipschitz function. The integer $N_{0}=N_{0}(n)$ is therefore independent on the function $\rho $ and depends only on the dimension $n \geq 1$ . It follows from equations (3.4) and (3.5) that

(3.7)

where

denotes the characteristic function of $B_{k}$ . We deduce from formula (3.7) that for all $g \in L^{2}(\mathbb {R}^{n})$ ,

(3.8) $$ \begin{align} \lVert g\rVert_{L^{2}(\mathbb{R}^{n})}^{2} = \int_{\mathbb{R}^{n}}\lvert g(x)\rvert^{2}dx \leq \sum_{k=0}^{+\infty}\int_{B_{k}}\lvert g(x)\rvert^{2}dx \leq N_{0} \lVert g\rVert_{L^{2}(\mathbb{R}^{n})}^{2}. \end{align} $$

Let $N \in \mathbb {N}$ be a nonnegative integer and set $f \in \mathcal E_{N} \setminus \{0\}$ , with $\mathcal E_{N}$ being the finite-dimensional vector space spanned by the Hermite functions $(\Phi _{\alpha })_{\lvert \alpha \rvert \leq N}$ defined in equation (1.14). Let $0<\delta \leq 1$ be a positive constant to be chosen later. We divide the family of balls $(B_{k})_{k \geq 0}$ into families of good and bad balls. A ball $B_{k}$ , with $k \in \mathbb {N}$ , is said to be good if it satisfies

(3.9) $$ \begin{align} & \forall \left(\beta,\tilde{\beta} \right) \in \mathbb{N}^{n} \times \mathbb{N}^{n}, \left\lvert\tilde{\beta}\right\rvert \leq n, \nonumber\\ & \quad \int_{B_{k}}\left\lvert\left\langle x\right\rangle^{(1-\varepsilon)(\left\lvert\beta\right\rvert+n)}\partial_{x}^{\beta+\tilde{\beta}}f(x)\right\rvert^{2}dx \leq 4^{n} (2(2^{n} N_{0}+1))^{\left\lvert \beta \right\rvert+1} M_{\beta,\tilde{\beta},N}(\delta)^{2} \int_{B_{k}}\lvert f(x)\rvert^{2}dx, \quad \end{align} $$

where the positive constants $M_{\beta ,\tilde {\beta },N}(\delta )>0$ also depend on the fixed positive parameter $0<\varepsilon \leq 1$ and the dimension $n \geq 1$ , and are defined by

(3.10) $$ \begin{align} & M_{\beta,\tilde{\beta},N}(\delta) \nonumber\\ &\ \ =\tilde{K}_{\varepsilon,\delta} K_{\varepsilon}^{(2-\varepsilon)\lvert\beta\rvert+(1-\varepsilon)n+\left\lvert\tilde{\beta}\right\rvert}\delta^{\lvert\beta\rvert+\left\lvert\tilde{\beta}\right\rvert} ( n+1 )^{\frac{(1-\varepsilon)\left(n+\lvert\beta\rvert\right)}{2}} \Gamma\left(\lvert\beta\rvert+\frac{(1-\varepsilon)n+\left\lvert\tilde{\beta}\right\rvert}{2-\varepsilon}+3\right) e^{\frac{N^{1-\frac{\varepsilon}{2}}}{\delta^{2-\varepsilon}}}, \end{align} $$

with the constants $\tilde {K}_{\varepsilon ,\delta }>1$ and $K_{\varepsilon }>1$ defined in Proposition A.3. On the other hand, a ball $B_{k}$ , with $k \in \mathbb {N}$ , which is not good is said to be bad – that is, when

(3.11) $$ \begin{align} & \exists \left(\beta,\tilde{\beta} \right) \in \mathbb{N}^{n} \times \mathbb{N}^{n}, \left\lvert\tilde{\beta}\right\rvert \leq n, \nonumber\\ &\quad \int_{B_{k}}\left\lvert\left\langle x\right\rangle^{(1-\varepsilon)\left(\left\lvert\beta\right\rvert+n\right)}\partial_{x}^{\beta+\tilde{\beta}}f(x)\right\rvert^{2}dx>4^{n} (2(2^{n} N_{0}+1))^{\left\lvert \beta \right\rvert+1} M_{\beta,\tilde{\beta},N}(\delta)^{2} \int_{B_{k}}\lvert f(x)\rvert^{2}dx. \end{align} $$

If $B_{k}$ is a bad ball, it follows from formula (3.11) that there exists $\left (\beta _{0},\tilde {\beta }_{0} \right ) \in \mathbb {N}^{n} \times \mathbb {N}^{n}, \left \lvert \tilde {\beta }_{0}\right \rvert \leq n$ such that

(3.12) $$ \begin{align} &\int_{B_{k}}\lvert f(x)\rvert^{2}dx \leq \frac{1}{4^{n}(2(2^{n} N_{0}+1))^{\left\lvert\beta_{0}\right\rvert+1}M_{\beta_{0},\tilde{\beta}_{0},N}(\delta)^{2}}\int_{B_{k}}\left\lvert\left\langle x\right\rangle^{(1-\varepsilon)\left(\left\lvert\beta_{0}\right\rvert+n\right)}\partial_{x}^{\beta_{0}+\tilde{\beta}_{0}}f(x)\right\rvert^{2}dx \notag\\ &\quad \leq \sum_{\substack{\beta \in \mathbb{N}^{n}, \\ \tilde{\beta} \in \mathbb{N}^{n}, \left\lvert\tilde{\beta}\right\rvert \leq n}} \frac{1}{4^{n}(2(2^{n} N_{0}+1))^{\left\lvert\beta\right\rvert+1}M_{\beta, \tilde{\beta},N}(\delta)^{2}}\int_{B_{k}}\left\lvert\left\langle x\right\rangle^{(1-\varepsilon)(\lvert\beta\rvert+n)}\partial_{x}^{\beta+\tilde{\beta}}f(x)\right\rvert^{2}dx. \end{align} $$

By summing over all the bad balls and using from equation (3.4) that

(3.13)

we deduce from formula (3.12) and the Fubini–Tonelli theorem that

(3.14) $$ \begin{align} &\int_{\bigcup_{\text{bad balls}} B_{k}}\lvert f(x)\rvert^{2}dx \leq \sum_{\text{bad balls}}\int_{B_{k}}\lvert f(x)\rvert^{2}dx \notag\\ &\ \ \leq \sum_{\substack{\beta \in \mathbb{N}^{n} \\ \tilde{\beta} \in \mathbb{N}^{n}, \left\lvert\tilde{\beta}\right\rvert \leq n}} \frac{N_{0}}{4^{n} (2(2^{n} N_{0}+1))^{\left\lvert\beta\right\rvert+1}M_{\beta,\tilde{\beta},N}(\delta)^{2}} \int_{\bigcup_{\text{bad balls}} B_{k}} \left\lvert{\left\langle x\right\rangle}^{(1-\varepsilon)\left(\left\lvert\beta\right\rvert+n\right)} \partial_{x}^{\beta+\tilde{\beta}} f(x)\right\rvert^{2}dx \notag\\ & \ \ \leq \sum_{\substack{\beta \in \mathbb{N}^{n} \\ \tilde{\beta} \in \mathbb{N}^{n}, \left\lvert\tilde{\beta}\right\rvert \leq n}} \frac{N_{0}}{4^{n} (2(2^{n} N_{0}+1))^{\left\lvert\beta\right\rvert+1}M_{\beta,\tilde{\beta},N}(\delta)^{2}} \int_{\mathbb{R}^{n}}\left\lvert\left\langle x\right\rangle^{(1-\varepsilon)\left(\left\lvert\beta\right\rvert+n\right)}\partial_{x}^{\beta+\tilde{\beta}} f(x)\right\rvert^{2}dx. \end{align} $$

Using the fact that the number of solutions to the equation $\beta _{1}+\dotsb +\beta _{n}=k$ , with ${k \geq 0}$ , $n \geq 1$ , and unknown $\beta =(\beta _{1},\dotsc ,\beta _{n}) \in \mathbb {N}^{n}$ , is given by $\binom {k+n-1}{k}$ , we obtain from the Bernstein-type estimates in Proposition A.3 and formulas (3.10) and (3.14) that

(3.15) $$ \begin{align} &\int_{\bigcup_{\text{bad balls}} B_{k}}\lvert f(x)\rvert^{2}dx \leq \left(\sum_{\substack{\beta \in \mathbb{N}^{n}, \\ \tilde{\beta} \in \mathbb{N}^{n}, \left\lvert\tilde{\beta}\right\rvert \leq n}} \frac{N_{0}}{4^{n} (2(2^{n}N_{0}+1))^{\lvert\beta\rvert+1}}\right) \lVert f\rVert_{L^{2}(\mathbb{R}^{n})}^{2}\notag \\ \notag &\quad = \left(\sum_{\beta \in \mathbb{N}^{n}} \frac{N_{0}}{(2(2^{n}N_{0}+1))^{\lvert\beta\rvert+1}}\right) \left(\sum_{\tilde{\beta} \in \mathbb{N}^{n}, \left\lvert\tilde{\beta}\right\rvert \leq n} \frac{1}{4^{n}} \right) \lVert f\rVert_{L^{2}(\mathbb{R}^{n})}^{2}\\ \notag &\quad = \left(\sum_{k=0}^{+\infty}\binom{k+n-1}{k} \frac{N_{0}}{2^{k+1}(2^{n}N_{0}+1)^{k+1}}\right) \left(\sum_{j=0}^{n}\frac{1}{4^{n}}\binom{j+n-1}{j}\right) \lVert f\rVert_{L^{2}(\mathbb{R}^{n})}^{2}\\ &\quad \leq 2^{n-2}\left(\sum_{k=1}^{+\infty}\frac{N_{0}}{(2^{n}N_{0}+1)^{k}}\right) \left(\sum_{j=0}^{n}\frac{2^{j+n-1}}{4^{n}}\right) \lVert f\rVert_{L^{2}(\mathbb{R}^{n})}^{2} \leq \frac{1}{4}\lVert f\rVert_{L^{2}(\mathbb{R}^{n})}^{2}, \end{align} $$

since

(3.16) $$ \begin{align} \binom{k+n-1}{k} \leq \sum_{j=0}^{k+n-1}\binom{k+n-1}{j}=2^{k+n-1}. \end{align} $$

Recalling from equation (3.5) that

we notice that

(3.17) $$ \begin{align} \lVert f\rVert_{L^{2}(\mathbb{R}^{n})}^{2} \leq \int_{\bigcup_{\text{good balls}} B_{k}}\lvert f(x)\rvert^{2}dx+ \int_{\bigcup_{\text{bad balls}} B_{k}}\lvert f(x)\rvert^{2}dx. \end{align} $$

It follows from formulas (3.15) and (3.17) that

(3.18) $$ \begin{align} \lVert f\rVert_{L^{2}(\mathbb{R}^{n})}^{2} \leq \frac{4}{3} \int_{\bigcup_{\text{good balls}} B_{k}}\lvert f(x)\rvert^{2}dx. \end{align} $$

3.2 Step 2: Properties on good balls

As the ball $B(0,1)$ is an Euclidean ball, the Sobolev embedding

(see, e.g., [Reference Adams and Fournier1, Theorem 4.12]) implies that there exists a positive constant $C_{n}>0$ depending only on the dimension $n \geq 1$ such that

(3.19) $$ \begin{align} \forall u \in W^{n,2}(B(0,1)), \quad \lVert u\rVert_{L^{\infty}(B(0,1))} \leq C_{n} \lVert u\rVert_{W^{n,2}(B(0,1))}. \end{align} $$

By translation invariance and the homogeneity of the Lebesgue measure, it follows from formulas (3.1), (3.6), and (3.19) that for all $u \in {W^{n,2}(B_{k})}$ ,

$$ \begin{align*} \lVert u\rVert^{2}_{L^{\infty}\left(B_{k}\right)} &=\lVert x \mapsto u(x_{k}+x \rho(x_{k}))\rVert^{2}_{L^{\infty}(B(0,1))} \leq C_{n}^{2} \lVert x \mapsto u(x_{k}+x \rho(x_{k}))\rVert^{2}_{W^{n,2}(B(0,1))} \\ &=C_{n}^{2} \sum_{\substack{\alpha \in \mathbb{N}^{n}, \\ \lvert\alpha\rvert \leq n}} \int_{B_{k}} \rho(x_{k})^{2\lvert\alpha\rvert-n} \left\lvert\partial^{\alpha}_{x} u(x)\right\rvert^{2} dx \\ & =C_{n}^{2} \sum_{\substack{\alpha \in \mathbb{N}^{n}, \\ \lvert\alpha\rvert \leq n}} \int_{B_{k}}m^{2\lvert\alpha\rvert-n} \left( \frac{\rho(x_{k})}{m} \right)^{2\lvert\alpha\rvert-n} \left\lvert\partial^{\alpha}_{x} u(x)\right\rvert^{2} dx\end{align*} $$

and

(3.20) $$ \begin{align} \lVert u\rVert^{2}_{L^{\infty}\left(B_{k}\right)} &\leq C_{n}^{2} \max\left(m,m^{-1}\right)^{n} \sum_{\substack{\alpha \in \mathbb{N}^{n}, \\ \lvert\alpha\rvert \leq n}} \int_{B_{k}} \left( \frac{\rho(x_{k})}{m} \right)^{n} \left\lvert\partial^{\alpha}_{x} u(x)\right\rvert^{2} dx \nonumber\\ &= C_{n}^{2} \max\left(1,m^{-1}\right)^{2n} \rho(x_{k})^{n} \sum_{\substack{\alpha \in \mathbb{N}^{n}, \\ \lvert\alpha\rvert \leq n}} \int_{B_{k}} \left\lvert\partial^{\alpha}_{x} u(x)\right\rvert^{2} dx. \end{align} $$

We deduce from formula (3.20) that for all $u \in {W^{n,2}(B_{k})}$ ,

(3.21) $$ \begin{align} \lVert u\rVert_{L^{\infty}\left(B_{k}\right)} \leq C_{n} \max\left(1,m^{-1}\right)^{n} \rho(x_{k})^{\frac{n}{2}} \lVert u\rVert_{W^{n,2}\left(B_{k}\right)}. \end{align} $$

Let $B_{k}$ be a good ball. Using the fact that the mapping $\rho $ is a $\frac {1}{2}$ -Lipschitz positive function, we notice that

(3.22) $$ \begin{align} \forall x \in B_{k}=B(x_{k},\rho(x_{k})), \quad 0 < \rho(x_{k}) \leq 2 \rho(x). \end{align} $$

We deduce from formulas (3.21) and (3.22) that for all $\beta \in \mathbb {N}^{n}$ and $k \in \mathbb {N}$ such that $B_{k}$ is a good ball,

(3.23) $$ \begin{align} &\rho(x_{k})^{\lvert\beta\rvert+ \frac{n}{2}}\left\lVert\partial_{x}^{\beta}f\right\rVert_{L^{\infty}\left(B_{k}\right)} \notag \\ \notag &\quad \leq C_{n} \max\left(1,m^{-1}\right)^{n} \rho(x_{k})^{\lvert\beta\rvert+ n}\left(\sum_{\substack{\tilde{\beta} \in \mathbb{N}^{n}, \left\lvert\tilde{\beta}\right\rvert \leq n}}\left\lVert\partial_{x}^{\beta+\tilde{\beta}}f\right\rVert^{2}_{L^{2}\left(B_{k}\right)}\right)^{\frac{1}{2}}\\ \notag &\quad = C_{n} \max\left(1,m^{-1}\right)^{n} \left(\sum_{\substack{\tilde{\beta} \in \mathbb{N}^{n}, \left\lvert\tilde{\beta}\right\rvert \leq n}}\left\lVert \rho(x_{k})^{\lvert\beta\rvert+ n}\partial_{x}^{\beta+\tilde{\beta}}f\right\rVert^{2}_{L^{2}\left(B_{k}\right)}\right)^{\frac{1}{2}} \\ &\quad \leq C_{n} \max\left(1,m^{-1}\right)^{n} 2^{\lvert\beta\rvert+n} \left(\sum_{\substack{\tilde{\beta} \in \mathbb{N}^{n}, \left\lvert\tilde{\beta}\right\rvert \leq n}}\left\lVert \rho(x)^{\lvert\beta\rvert+ n}\partial_{x}^{\beta+\tilde{\beta}}f\right\rVert^{2}_{L^{2}\left(B_{k}\right)}\right)^{\frac{1}{2}}. \end{align} $$

Using formula (3.1) and the definition of good balls (3.9), it follows from formula (3.23) that for all $\beta \in \mathbb {N}^{n}$ and $k \in \mathbb {N}$ such that $B_{k}$ is a good ball,

(3.24) $$ \begin{align} &\rho(x_{k})^{\lvert\beta\rvert+ \frac{n}{2}}\left\lVert\partial_{x}^{\beta}f\right\rVert_{L^{\infty}\left(B_{k}\right)} \notag\\ \notag &\quad \leq C_{n} \max\left(1,m^{-1}\right)^{n} (2R)^{\lvert\beta\rvert+n} \left(\sum_{\substack{\tilde{\beta} \in \mathbb{N}^{n}, \\ \left\lvert\tilde{\beta}\right\rvert \leq n}}\left\lVert {\left\langle x \right\rangle}^{(1-\varepsilon)(\lvert\beta\rvert+ n)}\partial_{x}^{\beta+\tilde{\beta}}f\right\rVert^{2}_{L^{2}\left(B_{k}\right)}\right)^{\frac{1}{2}} \\ &\quad \leq C_{n} \max\left(1,m^{-1}\right)^{n} (2R)^{\lvert\beta\rvert+n} 2^{n} \sqrt{2(2^{n}N_{0}+1)}^{\lvert\beta\rvert+1} \left(\sum_{\substack{\tilde{\beta} \in \mathbb{N}^{n}, \\ \left\lvert\tilde{\beta}\right\rvert \leq n}} M_{\beta,\tilde{\beta},N}(\delta)^{2} \right)^{\frac{1}{2}}\lVert f\rVert_{L^{2}\left(B_{k}\right)}. \end{align} $$

Using the fact that the gamma function is increasing on $[2,+\infty )$ (see Section A.1), we obtain from equation (3.10) that for all $\beta \in \mathbb {N}^{n}, \tilde {\beta } \in \mathbb {N}^{n}, \left \lvert \tilde {\beta }\right \rvert \leq n, 0<\delta \leq 1$ ,

(3.25) $$ \begin{align} \ M_{\beta,\tilde{\beta},N}(\delta) \leq \tilde{K}_{\varepsilon,\delta} K_{\varepsilon}^{(2-\varepsilon)\lvert\beta\rvert+n(1-\varepsilon)+\left\lvert\tilde{\beta}\right\rvert}\delta^{\lvert\beta\rvert+\left\lvert\tilde{\beta}\right\rvert} (n+1)^{\frac{(1-\varepsilon)\left(\lvert\beta\rvert+n\right)}{2}} \Gamma (\lvert\beta\rvert+n+3 ) e^{\frac{N^{1-\frac{\varepsilon}{2}}}{\delta^{2-\varepsilon}}}. \end{align} $$

Recalling that $K_{\varepsilon }>1$ and $0<\delta \leq 1$ , it follows from formulas (3.24) and (3.25) that for all $\beta \in \mathbb {N}^{n}$ and $k \in \mathbb {N}$ such that $B_{k}$ is a good ball,

(3.26) $$ \begin{align} & \rho(x_{k})^{\lvert\beta\rvert+ \frac{n}{2}}\left\lVert\partial_{x}^{\beta}f\right\rVert_{L^{\infty}\left(B_{k}\right)} \nonumber\\ & \qquad\qquad\qquad \leq C_{n}(\delta, \varepsilon, m, R) \left(\delta \tilde{C}_{n}(\varepsilon, R) \right)^{\lvert\beta\rvert} \Gamma ( \lvert\beta\rvert+n+3 ) e^{\frac{N^{1-\frac{\varepsilon}{2}}}{\delta^{2-\varepsilon}}} \lVert f\rVert_{L^{2}\left(B_{k}\right)}, \end{align} $$

with

(3.27) $$ \begin{align} C_{n}(\delta, \varepsilon, m, R)=\tilde{K}_{\varepsilon,\delta} C_{n} \max\left(1,m^{-1}\right)^{n} (4R)^{n} K_{\varepsilon}^{(2-\varepsilon)n} (n+1)^{(1-\varepsilon)\frac{n}{2}} \sqrt{2(2^{n} N_{0}+1)}>0 \end{align} $$

and

(3.28) $$ \begin{align} \tilde{C}_{n}(\varepsilon, R)=2 R\sqrt{2(2^{n}N_{0}+1)} K_{\varepsilon}^{2-\varepsilon} (n+1)^{\frac{1-\varepsilon}{2}}>0. \end{align} $$

Let $B_{k}$ be a good ball. Recalling that f is a finite combination of Hermite functions, we deduce from the continuity of the function f and the compactness of $\overline {B_{k}}$ that there exists $y_{k} \in \overline {B_{k}}$ such that

(3.29) $$ \begin{align} \lVert f\rVert_{L^{\infty}\left(B_{k}\right)}=\lvert f(y_{k})\rvert. \end{align} $$

Using spherical coordinates centered at $y_{k} \in \overline {B_{k}}$ and the fact that the Euclidean diameter of the ball $B_{k}=B(x_{k},\rho (x_{k}))$ is $2\rho (x_{k})$ , we observe that

(3.30)

where denotes the characteristic function of the measurable set $\omega \cap B_{k}$ . Using the Fubini–Tonelli theorem, we deduce from equation (3.30) that

(3.31)

where

(3.32) $$ \begin{align} I_{\sigma}=\{r \in [0,1] : y_{k}+2\rho(x_{k})r \sigma \in \omega \cap B_{k}\}. \end{align} $$

The estimate (3.31) implies that there exists $\sigma _{0}(k) \in \mathbb {S}^{n-1}$ such that

(3.33) $$ \begin{align} \lvert\omega \cap B_{k}\rvert \leq (2\rho(x_{k}))^{n}\left\lvert\mathbb{S}^{n-1}\right\rvert \left\lvert I_{\sigma_{0}(k)}\right\rvert. \end{align} $$

Recalling that $B_{k}=B(x_{k}, \rho (x_{k}) )$ and using the property (3.2), it follows from formula (3.33) that

(3.34) $$ \begin{align} 0< \frac{\gamma \lvert B(0,1)\rvert}{2^{n}\left\lvert\mathbb{S}^{n-1}\right\rvert} \leq \frac{\lvert\omega \cap B(x_{k},\rho(x_{k}))\rvert}{(2\rho(x_{k}))^{n}\left\lvert\mathbb{S}^{n-1}\right\rvert} \leq \left\lvert I_{\sigma_{0}(k)}\right\rvert \leq 1. \end{align} $$

3.3 Step 3: Recovery of the $L^{2}(\mathbb {R}^{n})$ -norm

Let $B_{k}$ be a good ball. We first notice that $\lVert f\rVert _{L^{2}\left (B_{k}\right )} \neq 0$ , since f is a nonzero entire function. We consider the entire function

(3.35) $$ \begin{align} \forall z \in \mathbb{C}, \quad \phi(z)=\lvert B_{k}\rvert^{\frac{1}{2}}\frac{f(y_{k}+2\rho(x_{k}) z \sigma_{0}(k))}{\lVert f\rVert_{L^{2}\left(B_{k}\right)}}, \end{align} $$

where $y_{k}$ and $\sigma _{0}(k)$ are defined in formulas (3.29) and (3.33). We observe from equation (3.29) that

$$ \begin{align*} \lvert\phi(0)\rvert=\lvert B_{k}\rvert^{\frac{1}{2}}\frac{\lvert f(y_{k})\rvert}{\lVert f\rVert_{L^{2}\left(B_{k}\right)}}=\lvert B_{k}\rvert^{\frac{1}{2}} \frac{\lVert f\rVert_{L^{\infty}\left(B_{k}\right)}}{{\lVert f\rVert_{L^{2}\left(B_{k}\right)}}} \geq 1. \end{align*} $$

Instrumental in the proof of Theorem 2.1 is the following lemma, proved by Kovrijkine [Reference Kovrijkine26, Lemma 1]:

Lemma 3.1. Let $I \subset \mathbb {R}$ be an interval of length $1$ such that $0 \in I$ , and let $E \subset I$ be a subset of positive measure $\lvert E\rvert>0$ . There exists a positive constant $C>1$ such that for all analytic functions $\Phi $ on the open ball $B_{\mathbb {C}}(0,5)$ centered in zero with radius $5$ such that $\lvert \Phi (0)\rvert \geq 1$ ,

$$ \begin{align*} \sup_{x \in I}\lvert\Phi(x)\rvert \leq \left(\frac{C}{\lvert E\rvert}\right)^{\frac{\ln M}{\ln 2}}\sup_{x \in E}\lvert\Phi(x)\rvert, \end{align*} $$

with $M=\sup _{\lvert z\rvert \leq 4}\lvert \Phi (z)\rvert \geq 1$ .

Applying Lemma 3.1 with $I=[0,1]$ , $E=I_{\sigma _{0}(k)} \subset [0,1]$ verifying $\lvert E\rvert =\left \lvert I_{\sigma _{0}(k)}\right \rvert>0$ according to formula (3.34), and the analytic function $\Phi =\phi $ defined in equation (3.35) satisfying $\lvert \phi (0)\rvert \geq 1$ , we obtain

(3.36) $$ \begin{align} &\lvert B_{k}\rvert^{\frac{1}{2}}\frac{\sup_{x \in [0,1]}\lvert f(y_{k}+2\rho(x_{k}) x \sigma_{0}(k))\rvert}{\lVert f\rVert_{L^{2}\left(B_{k}\right)}} \nonumber\\ & \qquad\qquad\quad \leq \left(\frac{C}{\left\lvert I_{\sigma_{0}(k)}\right\rvert}\right)^{\frac{\ln M}{\ln 2}}\lvert B_{k}\rvert^{\frac{1}{2}}\frac{\sup_{x \in I_{\sigma_{0}(k)}}\lvert f(y_{k}+2\rho(x_{k}) x \sigma_{0}(k))\rvert}{\lVert f\rVert_{L^{2}\left(B_{k}\right)}}, \end{align} $$

with

(3.37) $$ \begin{align} 1 \leq M=\lvert B_{k}\rvert^{\frac{1}{2}}\frac{\sup_{\lvert z\rvert \leq 4}\lvert f(y_{k}+2\rho(x_{k}) z \sigma_{0}(k))\rvert}{\lVert f\rVert_{L^{2}\left(B_{k}\right)}}. \end{align} $$

It follows from formulas (3.34) and (3.36) that

(3.38) $$ \begin{align} \sup_{x \in [0,1]}\lvert f(y_{k}+2\rho(x_{k}) x \sigma_{0}(k))\rvert & \leq \left(\frac{2^{n} C \left\lvert\mathbb{S}^{n-1}\right\rvert}{\gamma \lvert B(0,1)\rvert}\right)^{\frac{\ln M}{\ln 2}}\sup_{x \in I_{\sigma_{0}(k)}}\lvert f(y_{k}+2\rho(x_{k}) x \sigma_{0}(k))\rvert \notag\\ & \leq M^{\frac{1}{\ln 2}\ln\left(\frac{2^{n} C \left\lvert\mathbb{S}^{n-1}\right\rvert}{\gamma \lvert B(0,1)\rvert}\right)} \sup_{x \in I_{\sigma_{0}(k)}}\lvert f(y_{k}+2\rho(x_{k}) x \sigma_{0}(k))\rvert. \end{align} $$

According to equation (3.32), we notice that

(3.39) $$ \begin{align} \sup_{x \in I_{\sigma_{0}(k)}}\lvert f(y_{k}+2\rho(x_{k}) x \sigma_{0}(k))\rvert \leq \lVert f\rVert_{L^{\infty}\left(\omega \cap B_{k}\right)}. \end{align} $$

On the other hand, we deduce from equation (3.29) that

(3.40) $$ \begin{align} \lVert f\rVert_{L^{\infty}\left(B_{k}\right)}= \lvert f(y_{k})\rvert \leq \sup_{x \in [0,1]}\lvert f(y_{k}+2\rho(x_{k}) x \sigma_{0}(k))\rvert. \end{align} $$

It follows from formulas (3.38), (3.39), and (3.40) that

(3.41) $$ \begin{align} \lVert f\rVert_{L^{\infty}\left(B_{k}\right)}\leq M^{\frac{1}{\ln 2}\ln\left(\frac{2^{n} C \left\lvert\mathbb{S}^{n-1}\right\rvert}{\gamma \lvert B(0,1)\rvert}\right)}\lVert f\rVert_{L^{\infty}\left(\omega \cap B_{k}\right)}. \end{align} $$

Using the analyticity of the entire function f, we observe that

(3.42) $$ \begin{align} \forall z \in \mathbb{C}, \quad f(y_{k}+2\rho(x_{k})z \sigma_{0}(k)) =\sum_{\beta \in \mathbb{N}^{n}}\frac{\left(\partial_{x}^{\beta}f\right)(y_{k})}{\beta!}\sigma_{0}(k)^{\beta} (2\rho(x_{k}) )^{\lvert\beta\rvert}z^{\lvert\beta\rvert}. \end{align} $$

Using the facts that $B_{k}=B(x_{k}, \rho (x_{k}) )$ is a good ball and $y_{k} \in \overline {B_{k}}$ , and the continuity of the functions $\partial _{x}^{\beta }f$ , we deduce from formulas (3.26) and (3.42) that for all $\lvert z\rvert \leq 4$ ,

(3.43) $$ \begin{align} & \lvert B_{k}\rvert^{\frac{1}{2}}\lvert f(y_{k}+2\rho(x_{k})z \sigma_{0}(k))\rvert \nonumber \\ &\quad = \rho(x_{k})^{\frac{n}{2}} \lvert B(0,1)\rvert^{\frac{1}{2}} \lvert f(y_{k}+2\rho(x_{k})z \sigma_{0}(k))\rvert \nonumber\\ &\quad \leq \lvert B(0,1)\rvert^{\frac{1}{2}} \sum_{\beta \in \mathbb{N}^{n}} \rho(x_{k})^{\lvert\beta\rvert+\frac{n}{2}}\frac{\left\lvert\left(\partial_{x}^{\beta}f\right)(y_{k})\right\rvert}{\beta!} 8^{\lvert\beta\rvert} \nonumber\\ &\quad\leq \lvert B(0,1)\rvert^{\frac{1}{2}}\sum_{\beta \in \mathbb{N}^{n}} \rho(x_{k})^{\lvert\beta\rvert+\frac{n}{2}}\frac{\left\lVert\partial_{x}^{\beta}f\right\rVert_{L^{\infty}\left(B_{k}\right)}}{\beta!} 8^{\lvert\beta\rvert} \nonumber\\ &\quad\leq \lvert B(0,1)\rvert^{\frac{1}{2}}C_{n}(\delta,\varepsilon, m, R) e^{\frac{N^{1-\frac{\varepsilon}{2}}}{\delta^{2-\varepsilon}}}\left(\sum_{\beta \in \mathbb{N}^{n}}\frac{\Gamma (\lvert\beta\rvert+n+3 )}{\beta!}\left(8\delta \tilde{C}_{n}(\varepsilon, R) \right)^{\lvert\beta\rvert} \right)\lVert f\rVert_{L^{2}\left(B_{k}\right)}. \end{align} $$

We recall the estimate

(3.44) $$ \begin{align} \forall \beta \in \mathbb{N}^{n}, \quad \lvert\beta\rvert! \leq n^{\lvert\beta\rvert} \beta !, \end{align} $$

which is obtained using the Newton formula [Reference Nicola and Rodino33, formula (0.3.3)]. Using anew the facts that the number of solutions to the equation $\beta _{1}+\dotsb +\beta _{n}=k$ , with $k \geq 0$ , $n \geq 1$ , and unknown $\beta =(\beta _{1},\dotsc ,\beta _{n}) \in \mathbb {N}^{n}$ , is given by $\binom {k+n-1}{k}$ , and that

$$ \begin{align*} \Gamma ( \lvert\beta\rvert +n+3 )= (\lvert\beta\rvert+n+2)!, \qquad (\lvert\beta\rvert+n+2)^{n+2} \leq (n+2)! e^{\lvert\beta\rvert+n+2} \end{align*} $$

according to formulas (A.3) and (A.21), we notice from formula (3.16) that

(3.45) $$ \begin{align} \sum_{\beta \in \mathbb{N}^{n}}\frac{\Gamma (\lvert\beta\rvert+n+3 )}{\beta!}\left(8 \delta \tilde{C}_{n}(\varepsilon, R) \right)^{\lvert\beta\rvert} & = \sum_{\beta \in \mathbb{N}^{n}}\frac{(\lvert\beta\rvert+n+2)!}{\beta!}\left(8 \delta \tilde{C}_{n}(\varepsilon, R) \right)^{\lvert\beta\rvert} \nonumber\\ &\leq \sum_{\beta \in \mathbb{N}^{n}} (\lvert\beta\rvert+n+2)^{n+2}\frac{\lvert\beta\rvert!}{\beta!}\left(8 \delta \tilde{C}_{n}(\varepsilon, R) \right)^{\lvert\beta\rvert} \nonumber \\ & \leq e^{n+2} (n+2)! \sum_{\beta \in \mathbb{N}^{n}}\left(8 \delta n e \tilde{C}_{n}(\varepsilon, R) \right)^{\lvert\beta\rvert}\nonumber\\ &=e^{n+2} (n+2)! \sum_{k=0}^{+\infty}\binom{k+n-1}{k}\left(8\delta n e \tilde{C}_{n}(\varepsilon, R) \right)^{k} \nonumber \\ &\leq e^{n+2}(n+2)! 2^{n-1}\sum_{k=0}^{+\infty}\left(16 \delta n e \tilde{C}_{n}(\varepsilon, R) \right)^{k}. \end{align} $$

We can now make a choice for the positive parameter $0<\delta \leq 1$ , which is fixed from now on and taken to be equal to

(3.46) $$ \begin{align} 0<\delta=\delta_{n,\varepsilon,R}= \min\left(1,\frac{1}{32 n e \tilde{C}_{n}(\varepsilon, R)}\right) \leq 1. \end{align} $$

Setting $D_{n}(\varepsilon , m, R)=C_{n}\left (\delta _{n,\varepsilon ,R},\varepsilon , m, R\right )>0$ , it follows from formulas (3.37), (3.43), (3.45), and (3.46) that

(3.47) $$ \begin{align} 1 \leq M \leq \lvert B(0,1)\rvert^{\frac{1}{2}}(n+2)! D_{n}(\varepsilon, m, R) e^{n+2} 2^{n} e^{\delta_{n,\varepsilon,R}^{\varepsilon-2} N^{1-\frac{\varepsilon}{2}}}. \end{align} $$

We notice from formula (3.34) that

(3.48) $$ \begin{align} \frac{2^{n} C\left\lvert\mathbb{S}^{n-1}\right\rvert}{\gamma \lvert B(0,1)\rvert}> 1, \end{align} $$

since the positive constant given by Lemma 3.1 satisfies $C>1$ . With this choice, we deduce from formulas (3.41) and (3.47) that

(3.49) $$ \begin{align} \lVert f\rVert_{L^{\infty}\left(B_{k}\right)}\leq \left(\frac{2^{n} C\left\lvert\mathbb{S}^{n-1}\right\rvert}{\gamma \lvert B(0,1)\rvert}\right)^{\frac{\ln\left(D_{n}\left(\varepsilon, m, R\right) e^{n+2} 2^{n} \lvert B(0,1)\rvert^{\frac{1}{2}}(n+2)!\right)}{\ln 2}+\frac{\delta_{n,\varepsilon,R}^{\varepsilon-2}}{\ln 2} N^{1-\frac{\varepsilon}{2}}} \lVert f\rVert_{L^{\infty}\left(\omega \cap B_{k}\right)}. \end{align} $$

Recalling from the property (3.2) that

(3.50) $$ \begin{align} \lvert\omega \cap B_{k}\rvert \geq \gamma \lvert B_{k}\rvert>0 \end{align} $$

as $B_{k}=B(x_{k}, \rho (x_{k}))$ , and setting

(3.51) $$ \begin{align} \tilde{\omega}_{k}=\left\{x \in \omega \cap B_{k} : \lvert f(x)\rvert \leq \frac{2}{\lvert\omega \cap B_{k}\rvert}\int_{\omega \cap B_{k}}\lvert f(y)\rvert dy\right\}, \end{align} $$

we observe that

(3.52) $$ \begin{align} \int_{\omega \cap B_{k}}\lvert f(x)\rvert dx \geq \int_{\left(\omega \cap B_{k}\right)\setminus \tilde{\omega}_{k}}\lvert f(x)\rvert dx \geq \frac{2\left\lvert(\omega \cap B_{k})\setminus \tilde{\omega}_{k}\right\rvert}{\lvert\omega \cap B_{k}\rvert}\int_{\omega \cap B_{k}}\lvert f(x)\rvert dx. \end{align} $$

Using the fact that the integral

$$ \begin{align*} \int_{\omega \cap B_{k}}\lvert f(x)\rvert dx> 0 \end{align*} $$

is positive,Footnote ¹ since f is a nonzero entire function and $\lvert \omega \cap B_{k}\rvert>0$ , we obtain

$$ \begin{align*}\left\lvert(\omega \cap B_{k})\setminus \tilde{\omega}_{k}\right\rvert \leq \frac{1}{2}\lvert\omega \cap B_{k}\rvert, \end{align*} $$

which implies that

(3.53) $$ \begin{align} \left\lvert\tilde{\omega}_{k}\right\rvert=\lvert\omega \cap B_{k}\rvert- \left\lvert(\omega \cap B_{k})\setminus \tilde{\omega}_{k}\right\rvert \geq \frac{1}{2}\lvert\omega \cap B_{k}\rvert\geq \frac{1}{2} \gamma\lvert B_{k}\rvert= \frac{1}{2}\gamma \rho(x_{k})^{n} \lvert B(0,1)\rvert>0, \end{align} $$

thanks to formula (3.50). Using again spherical coordinates as in formulas (3.30) and (3.31), we observe that

(3.54)

where

(3.55) $$ \begin{align} \tilde{I}_{\sigma}=\left\{r \in [0,1] : y_{k}+ 2\rho(x_{k})r \sigma \in \tilde{\omega}_{k} \cap B_{k}\right\}. \end{align} $$

As in formula (3.33), the estimate (3.54) implies that there exists $\tilde {\sigma }_{0}(k) \in \mathbb {S}^{n-1}$ such that

(3.56) $$ \begin{align} \left\lvert\tilde{\omega}_{k}\right\rvert \leq ( 2\rho(x_{k}) )^{n}\left\lvert\mathbb{S}^{n-1}\right\rvert \left\lvert\tilde{I}_{\tilde{\sigma}_{0}(k)}\right\rvert. \end{align} $$

We deduce from formulas (3.53) and (3.56) that

(3.57) $$ \begin{align} 1 \geq\left\lvert\tilde{I}_{\tilde{\sigma}_{0}(k)}\right\rvert \geq \frac{\gamma \lvert B(0,1)\rvert}{2^{n+1}\left\lvert\mathbb{S}^{n-1}\right\rvert}>0. \end{align} $$

Applying anew Lemma 3.1 with $I=[0,1]$ , $E=\tilde {I}_{\tilde {\sigma }_{0}(k)} \subset [0,1]$ verifying $\lvert E\rvert =\left \lvert \tilde {I}_{\tilde {\sigma }_{0}(k)}\right \rvert>0$ , and the analytic function $\Phi =\phi $ defined in equation (3.35) with $\sigma _{0}(k)$ replaced by $\tilde {\sigma }_{0}(k)$ satisfying $\lvert \phi (0)\rvert \geq 1$ , we obtain

(3.58) $$ \begin{align} & \lvert B_{k}\rvert^{\frac{1}{2}}\frac{\sup_{x \in [0,1]}\left\lvert f(y_{k}+2\rho(x_{k}) x \tilde{\sigma}_{0}(k))\right\rvert}{\lVert f\rVert_{L^{2}\left(B_{k}\right)}} \nonumber\\ & \qquad\qquad\quad \leq \left(\frac{C}{\left\lvert\tilde{I}_{\tilde{\sigma}_{0}(k)}\right\rvert}\right)^{\frac{\ln M}{\ln 2}}\lvert B_{k}\rvert^{\frac{1}{2}}\frac{\sup_{x \in \tilde{I}_{\tilde{\sigma}_{0}(k)}}\left\lvert f(y_{k}+2\rho(x_{k}) x \tilde{\sigma}_{0}(k))\right\rvert}{\lVert f\rVert_{L^{2}\left(B_{k}\right)}}, \end{align} $$

where $M \geq 1$ denotes the constant defined in formula (3.37). It follows from formulas (3.57) and (3.58) that

(3.59) $$ \begin{align} \sup_{x \in [0,1]}\left\lvert f(y_{k}+2\rho(x_{k}) x \tilde{\sigma}_{0}(k))\right\rvert & \leq \left(\frac{2^{n+1} C \left\lvert\mathbb{S}^{n-1}\right\rvert}{\gamma \lvert B(0,1)\rvert}\right)^{\frac{\ln M}{\ln 2}}\sup_{x \in \tilde{I}_{\tilde{\sigma}_{0}(k)}}\left\lvert f(y_{k}+2\rho(x_{k}) x \tilde{\sigma}_{0}(k))\right\rvert \notag\\ & \leq M^{\frac{1}{\ln 2}\ln\left(\frac{2^{n+1} C \left\lvert\mathbb{S}^{n-1}\right\rvert}{\gamma \lvert B(0,1)\rvert}\right)} \sup_{x \in \tilde{I}_{\tilde{\sigma}_{0}(k)}}\left\lvert f(y_{k}+2\rho(x_{k}) x \tilde{\sigma}_{0}(k))\right\rvert. \end{align} $$

According to equation (3.55), we notice that

(3.60) $$ \begin{align} \sup_{x \in \tilde{I}_{\tilde{\sigma}_{0}(k)}}\left\lvert f(y_{k}+2\rho(x_{k}) x \tilde{\sigma}_{0}(k))\right\rvert \leq \lVert f\rVert_{L^{\infty}\left(\tilde{\omega}_{k} \cap B_{k}\right)}. \end{align} $$

It follows from formulas (3.29), (3.59), and (3.60) that

(3.61) $$ \begin{align} \lVert f\rVert_{L^{\infty}\left(B_{k}\right)}=\lvert f(y_{k})\rvert & \leq \sup_{x \in [0,1]}\left\lvert f(y_{k}+2\rho(x_{k}) x \tilde{\sigma}_{0}(k))\right\rvert\nonumber \\ &\leq M^{\frac{1}{\ln 2}\ln\left(\frac{2^{n+1} C \left\lvert\mathbb{S}^{n-1}\right\rvert}{\gamma \lvert B(0,1)\rvert}\right)}\lVert f\rVert_{L^{\infty}\left(\tilde{\omega}_{k} \cap B_{k}\right)}. \end{align} $$

On the other hand, it follows from equation (3.51) that

(3.62) $$ \begin{align} \lVert f\rVert_{L^{\infty}\left(\tilde{\omega}_{k} \cap B_{k}\right)} \leq \frac{2}{\lvert\omega \cap B_{k}\rvert}\int_{\omega \cap B_{k}}\lvert f(x)\rvert dx. \end{align} $$

We deduce from formulas (3.61) and (3.62) and the Cauchy–Schwarz inequality that

(3.63) $$ \begin{align} \lVert f\rVert_{L^{2}\left(B_{k}\right)} & \leq \lvert B_{k}\rvert^{\frac{1}{2}} \lVert f\rVert_{L^{\infty}\left(B_{k}\right)} \notag\\ \notag & \leq \frac{2\lvert B_{k}\rvert^{\frac{1}{2}}}{\lvert\omega \cap B_{k}\rvert} M^{\frac{1}{\ln 2}\ln\left(\frac{2^{n+1} C \left\lvert\mathbb{S}^{n-1}\right\rvert}{\gamma \lvert B(0,1)\rvert}\right)} \int_{\omega \cap B_{k}}\lvert f(x)\rvert dx\\ & \leq \frac{2\lvert B_{k}\rvert^{\frac{1}{2}}}{\lvert\omega \cap B_{k}\rvert^{\frac{1}{2}}} M^{\frac{1}{\ln 2}\ln\left(\frac{2^{n+1} C \left\lvert\mathbb{S}^{n-1}\right\rvert}{\gamma \lvert B(0,1)\rvert}\right)} \lVert f\rVert_{L^{2}\left(\omega \cap B_{k}\right)}. \end{align} $$

By the property (3.50), it follows from formulas (3.47), (3.48), and (3.63) that

(3.64) $$ \begin{align} & \lVert f\rVert_{L^{2}\left(B_{k}\right)}^{2} \leq \frac{4}{\gamma} M^{\frac{2}{\ln 2}\ln\left(\frac{2^{n+1} C \left\lvert\mathbb{S}^{n-1}\right\rvert}{\gamma \lvert B(0,1)\rvert}\right)} \lVert f\rVert_{L^{2}\left(\omega \cap B_{k}\right)}^{2}\nonumber\\ &\quad \leq \frac{4}{\gamma} \left(\left\lvert B(0,1)\right\rvert^{\frac{1}{2}} (n+2)! D_{n}(\varepsilon, m, R) e^{n+2} 2^{n} e^{\delta_{n,\varepsilon,R}^{\varepsilon-2} N^{1-\frac{\varepsilon}{2}}}\right)^{\frac{2}{\ln 2}\ln\left(\frac{2^{n+1} C \left\lvert\mathbb{S}^{n-1}\right\rvert}{\gamma \lvert B(0,1)\rvert}\right)} \lVert f\rVert_{L^{2}\left(\omega \cap B_{k}\right)}^{2}. \end{align} $$

Setting

(3.65) $$ \begin{align} \kappa_{n}(m, R,\gamma,\varepsilon)=\frac{4}{\sqrt{3}\gamma^{\frac{1}{2}}} \left(\frac{2^{n+1} C \left\lvert\mathbb{S}^{n-1}\right\rvert}{\gamma \lvert B(0,1)\rvert}\right)^{\frac{\ln\left(\left\lvert B(0,1)\right\rvert^{\frac{1}{2}} (n+2)! D_{n}\left(\varepsilon, m, R\right) e^{n+2} 2^{n}\right)}{\ln 2}}>0, \end{align} $$

we deduce from formula (3.64) that there exists a positive universal constant $\tilde {\kappa }_{n}>1$ such that for any good ball $B_{k}$ ,

(3.66) $$ \begin{align} \lVert f\rVert_{L^{2}\left(B_{k}\right)}^{2}\leq \frac{3}{4}\kappa_{n}(m, R,\gamma,\varepsilon)^{2}\left(\frac{\tilde{\kappa}_{n}}{\gamma}\right)^{\frac{2}{\ln 2}\delta_{n,\varepsilon,R}^{\varepsilon-2} N^{1-\frac{\varepsilon}{2}}} \lVert f\rVert_{L^{2}\left(\omega \cap B_{k}\right)}^{2}. \end{align} $$

Using anew from equation (3.4) that

(3.67)

it follows from formulas (3.18) and (3.66) that

(3.68) $$ \begin{align} \lVert f\rVert_{L^{2}(\mathbb{R}^{n})}^{2} & \leq \frac{4}{3} \int_{\bigcup_{\text{good balls}} B_{k}}\lvert f(x)\rvert^{2}dx \leq \frac{4}{3} \sum_{\text{good balls}}\lVert f\rVert_{L^{2}\left(B_{k}\right)}^{2}\notag\\ \notag & \leq \kappa_{n}(m, R,\gamma, \varepsilon)^{2}\left(\frac{\tilde{\kappa}_{n}}{\gamma}\right)^{\frac{2}{\ln 2}\delta_{n,\varepsilon,R}^{\varepsilon-2} N^{1-\frac{\varepsilon}{2}}} \sum_{\text{good balls}} \lVert f\rVert_{L^{2}\left(\omega \cap B_{k}\right)}^{2}\\ \notag & \leq N_{0} \kappa_{n}(m, R,\gamma, \varepsilon)^{2}\left(\frac{\tilde{\kappa}_{n}}{\gamma}\right)^{\frac{2}{\ln 2}\delta_{n,\varepsilon,R}^{\varepsilon-2} N^{1-\frac{\varepsilon}{2}}} \int_{\omega \cap \left(\bigcup_{\text{good balls}} B_{k}\right)}\lvert f(x)\rvert^{2}dx\\ & \leq N_{0} \kappa_{n}(m, R,\gamma,\varepsilon)^{2}\left(\frac{\tilde{\kappa}_{n}}{\gamma}\right)^{\frac{2}{\ln 2}\delta_{n,\varepsilon,R}^{\varepsilon-2}N^{1-\frac{\varepsilon}{2}}}\lVert f\rVert_{L^{2}(\omega)}^{2}. \end{align} $$

This ends the proof of Theorem 2.1.