A generalizaton of Hardy’s uncertainty principle on compact extensions of $\mathbb R ^n$

Abstract

Let $K$ be a compact subgroup of automorphisms of $\mathbb R ^n$. We prove in this paper a generalization of Hardy’s uncertainty principle on the semi-direct product $K\ltimes \mathbb R ^n$, and we solve the sharpness problem. As a consequence, a complete analogue of classical Hardy’s theorem is obtained. The representation theory and the Plancherel formula play an important role in the proofs.

1 Introduction

A classical theorem of Hardy proved in 1933 in the case of the real line says that an integrable function $f$ and its Fourier transform $\hat{f}$ cannot both have arbitrary Gaussian decay unless $f$ is identically zero. More precisely, if both $f(x)e^{\alpha \Vert x\Vert ^2}$ and $\hat{f}(\xi )e^{\beta \Vert \xi \Vert ^2}$ are in $L^\infty (\mathbb{R }^n)$ for some $\alpha , \beta > 0$, then the following conclusions hold:

• 1. $f = 0$ whenever $\alpha \beta > 1/4.$

• 2. The function $f$ is a constant multiple of $e^{-\alpha \Vert x\Vert ^2}$ when $\alpha \beta = 1/4$.

• 3. When $\alpha \beta < 1/4$, there are infinitely many linearly independent functions satisfying both conditions.

Here, the Fourier transform $\widehat{f}$ is defined by

$$\begin{aligned} \widehat{f}(y) = (2\pi )^{-n/2}\int \limits _{R^n} f(x)\exp (- i \langle x,y \rangle )\text{ d}x \end{aligned}$$

and $\big \Vert x\big \Vert = \langle x,x \rangle ^{1/2}$ is the Euclidean norm. Actually, the bound $\frac{1}{4}$ is sharp and the strong Gaussian decay of $f$ and $\widehat{f}$ is only required with respect to a one-dimensional subgroup of $\mathbb R ^n$. For a detailed proof of Hardy’s theorem, see [8]. During last decade, there has been much effort to prove Hardy-like theorems for various classes of non-abelian connected Lie groups ([2, 3, 7, 15–20]). Specifically, analogues of Hardy’s theorem have been shown for motion groups ([7, 17]). The Euclidean norms on $\mathbb R ^n$ and on $\widehat{\mathbb{R ^n}}$ have to be replaced by certain norm-functions on $G$ and the Hilbert–Schmidt norm of the operator-valued Fourier transform on the unitary dual $\widehat{G}$ of $G$, respectively, depending upon the parameters involved in the problem in question.

Note that in the references above, only results concerning the first assertion recorded above are considered, which concerns the case $\alpha \beta > 1/4.$ In this paper, we prove a generalization of Hardy’s uncertainty principle for a general compact extension $K\ltimes \mathbb R ^n$, where $K$ stands for a compact subgroup of the automorphisms group $\text{ Aut}(\mathbb R ^n)$ of $\mathbb R ^n$, providing evidence to the three assertions cited above.

The idea is to endow $\mathbb{R }^n$ with an Euclidean scalar product which embeds the compact group $K$ as a subgroup of orthogonal transformations (for details see [13]). We then make use of a reduction technique to transfer the study to a Hardy’s uncertainty problem on $\mathbb{R }^n$. We hope that our results can be generalized to encompass compact extensions of nilpotent Lie groups.

2 The main result

2.1 Case of the Euclidean motion groups

Let $M_n=\text{ SO}(n)\ltimes \mathbb{R }^n$ be the Euclidean motion group. It is well known that every infinite dimensional irreducible unitary representation of $M_n$ is equivalent to the induced representation $\pi _{r,\sigma }:=\mathrm{ind}_{\mathrm{SO}(n-1)\ltimes \mathbb{R }^n}^{\mathrm{SO}(n)\ltimes \mathbb{R }^n} (\sigma \otimes \chi _r)$ for some $r>0$ and $\sigma \in \widehat{\text{ SO}(n-1)}$, where $\chi _r$ is the character on $\mathbb{R }^n$ defined by $\chi _r(x)=e^{-irx_n}$, for all $x=(x_1,\cdots ,x_n)\in \mathbb{R }^n$. Furthermore, the set of generic representations in $\widehat{M_n}$ which appear in the Plancherel measure of $M_n$ is identified to the set of all induced representations $\pi _{r,\sigma }$ when the couple $(r,\sigma )$ runs the set $\mathbb{R }^*_+\times \widehat{\text{ SO}(n-1)}$. Sundari [17] proved in that if a measurable function $f$ on $M_n$ satisfies the assertions:

• 1. $|f(k,x)|\le \ C e^{-\alpha \big \Vert x\big \Vert ^2},\ (k,x)\in M_n$

• 2. $\big \Vert \pi _{r,\sigma }(f)\big \Vert _{HS}\le \ C_\sigma e^{-\beta r^2},\ (r,\sigma )\in \mathbb{R }^*_+\times \widehat{\text{ SO}(n-1)}$

for some positive constants $C_\sigma , \alpha , \beta$ and $C$, where $C_\sigma$ depends only on $\sigma$ and $\alpha \beta >\frac{1}{4}$, then $f=0$ almost everywhere.

As a generalization of this result, an analogue of Hardy’s theorem for the Cartan motion group $G=K\ltimes \mathfrak p$ has been proved by Eguchi et al. [7], where $\mathfrak p =\{X\in \mathfrak g : \theta (X)=-X\}$ with $\theta$ is the Cartan involution on the Lie algebra $\mathfrak g$ of a connected semi-simple Lie group $G_0$. It was proved in this case that given a measurable function $f$ on $G$ such that the mappings $(k,x)\mapsto |f(k,x)|e^{\alpha \big \Vert x\big \Vert ^2}$ and $\ell \mapsto \big \Vert \pi _\ell (f)\big \Vert _\mathrm{HS}e^{\beta \big \Vert \ell \big \Vert ^2}$ are both bounded on $G$ and on the dual $\mathfrak p ^*$ of $\mathfrak p$, respectively, for some positive real numbers $\alpha ,\beta$ such that $\alpha \beta >\frac{1}{4}$, then $f=0$ almost everywhere. Here, for $\ell \in \mathfrak p ^*$, $\pi _\ell$ denotes the (reducible) unitary representation of $G$ defined on $L^2(K)$ by $\pi _\ell (k,x)\varphi (s)=e^{-i \ell (\mathrm{Ad}(s)^{-1}(x))}\varphi (k^{-1}s)$, for all $(k,s,x)\in K\times K\times \mathfrak p$.

Recently, the second author of this paper and Kaniuth [3] treated in the case of a second countable and unimodular locally compact group $G$ containing a closed normal vector subgroup $N$, $N=\mathbb{R }^n$ say, and for which we identify the group of automorphisms $x\mapsto y^{-1}x y$, $y\in G$, of $N$ with the orthogonal group $\text{ SO}(n)$. Using the notation above, if $S$ denotes the Borel cross-section of the cosets of $N$, then there is no non-zero measurable function $f$ on $G$ satisfying the following estimates:

• 1. $|f(xs)|\le \varphi (s)e^{-\alpha \big \Vert x\big \Vert ^2}$ for all $x\in \mathbb{R }^n$ and $s\in S$, where $\varphi \in L^1(S)\cap L^2(S)$,

• 2. $\big \Vert \pi _{r,\sigma }(f)\big \Vert _\mathrm{HS}\le \psi _r(\sigma ) e^{-\beta r^2}$

for all $r\in \mathbb{R }^*_+$ and $\sigma \in \widehat{\text{ SO}(n-1)}$ and for some positive constants $\alpha ,\beta$ so that $\alpha \beta >\frac{1}{4}$, and where $\psi _r \in l^2(\widehat{\text{ SO}(n-1)})$ for which there exist $0<\gamma <\beta$ and a constant $c>0$ such that $\alpha \gamma >\frac{1}{4}$ and $\big \Vert \psi _r\big \Vert _{l^2(\widehat{\text{ SO}(n-1)})}\le c e^{(\beta -\gamma )r^2}$ for all $r\in \mathbb{R }^*_+$.

2.2 On compact extensions of $\mathbb R ^n$

We assume in this paper that $G:=K\ltimes \mathbb R ^n$, where $K$ stands for a compact subgroup of the automorphisms group $\text{ Aut}(\mathbb R ^n)$. In this case and in the whole paper, $\mathbb{R }^n$ is equipped with a Euclidean scalar product $\langle .,.\rangle$ which embeds the compact group $K$ as a subgroup of orthogonal transformations (cf. [13]). We fix once for all a Haar measure $\text{ d}g:=\text{ d}\nu (k)\otimes \text{ d}x$ of $G$, where $\text{ d}\nu (k)$ denotes the normalized Haar measure of $K$ and $\text{ d}x=\text{ d}x_1\ldots \text{ d}x_n$ the Lebesgue measure on $\mathbb{R }^n$. We recall that since $K$ is compact, $dx$ is invariant under the action of $K$ on $\mathbb{R }^n$ given by $x\rightarrow k^{-1} x k$, for $k\in K$.

2.2.1 The unitary dual of $G$

The unitary dual space of such a group has been described by Mackey [12] using the little group method. We give here a brief view, and for full details, the reader may consult the reference.

For $\ell$ a non-zero real linear form on $\mathbb R ^n$, we define the unit character $\chi _{\ell }$ of $\mathbb{R }^n$ by $\chi _{\ell }(x)=e^{-i\langle \ell , x\rangle }$. We denote by $g.\ell$ the action of $G$ on $\mathbb{R }^n$ given by $\langle g.\ell ,x\rangle =\langle \ell ,g^{-1}x g\rangle$ for $g\in G$ and $x\in \mathbb{R }^n$. Therefore, we get $g.\chi _\ell =\chi _{g.\ell }$, where $g$ acts on $\widehat{\mathbb{R }^n}$ by $g.\chi _\ell (x):=\chi _\ell (g^{-1}x g)$. We set thus by $K_\ell \ltimes \mathbb{R }^n$ the stabilizer of $\chi _\ell$ where $K_\ell :=\{k\in K;\ k.\chi _\ell =\chi _\ell \}$. We take the normalized Haar measure $\text{ d}\nu _\ell$ on $K_\ell$ and $K$-invariant measure $\text{ d}\dot{\nu }_\ell$ on $K/K_\ell$ such that

$$\begin{aligned} \int \limits _K\xi (k) \text{ d}\nu (k)=\int \limits _{K/K_\ell }\int \limits _{K_\ell }\xi (kk^{\prime }) \text{ d}\nu _\ell (k^{\prime })\text{ d}\dot{\nu }_\ell (kK_\ell ). \end{aligned}$$

(2.1)

We note that the measure $\text{ d}\dot{\nu }_\ell$ is also normalized as $\int _{K/K_\ell }\text{ d}\dot{\nu }_\ell =1$.

Concerning the action of $K$ on $\widehat{\mathbb{R }^n}\simeq \mathbb{R }^n$, let $d\bar{\ell }$ be the image of the Lebesgue measure on $\mathbb{R }^n/K$ by the canonical projection $\mathbb{R }^n\ni \ell \mapsto \bar{\ell }:=K.\ell \in \mathbb{R }^n/K$ so that

$$\begin{aligned} \int \limits _{\mathbb{R }^n}\varphi (\ell ) \text{ d}\ell =\int \limits _{\mathbb{R }^n/K}\int \limits _{K}\varphi (k.\ell ) \text{ d}\nu (k)\text{ d}\bar{\ell }. \end{aligned}$$

(2.2)

Let $(\sigma ,{\fancyscript{H}}_\sigma )$ be a unitary and irreducible representation of $K_\ell$ and ${\fancyscript{H}}_{\ell ,\sigma }$ be the completion of the vector space of all continuous mapping $\xi : K\rightarrow {\fancyscript{H}}_\sigma$ for which $\xi (ks)=\sigma (s)^*(\xi (k))$ for $k\in K$ and $s\in K_\ell$ with respect to the norm

$$\begin{aligned} \Vert \xi \Vert _2=\Big (\int \limits _{K}\Vert \xi (k)\Vert ^2_{{\fancyscript{H}}_\sigma }\text{ d}\nu (k)\Big )^{\frac{1}{2}}. \end{aligned}$$

The induced representation $\pi _{\ell ,\sigma }:=\text{ ind}_{K_\ell \ltimes \mathbb{R }^n}^{G}(\sigma \otimes \chi _{\ell })$, realized on the Hilbert space ${\fancyscript{H}}_{\ell ,\sigma }$ by

$$\begin{aligned} \pi _{\ell ,\sigma }(k,x)\xi (s)=e^{-i\langle \ell ,s^{-1}xs\rangle }\xi (k^{-1}s)=e^{-i\langle s.\ell ,x\rangle }\xi (k^{-1}s), \end{aligned}$$

for $\xi \in {\fancyscript{H}}_{\ell ,\sigma }$, $(k,x)\in G$ and $s\in K$, is an irreducible representation of $G$. Furthermore, every infinite dimensional irreducible unitary representation of $G$ is equivalent to some $\pi _{\ell ,\sigma }$. We remark that the representations $\pi _{\ell ,\sigma }$ and $\pi _{\ell ^{\prime },\sigma ^{\prime }}$ are equivalent if and only if $\ell$ and $\ell ^{\prime }$ belong to the same $K$-orbit and the representations $\sigma \in \widehat{K_{\ell }}$ and $\sigma ^{\prime }\in \widehat{K_{\ell ^{\prime }}}$ are equivalent under the obvious identification of $K_\ell$ with $K_{\ell ^{\prime }}$ since there exists $k\in K$ such that $\ell ^{\prime }=k.\ell$ and hence $K_{\ell ^{\prime }}=k K_\ell k^{-1}$, and $\sigma ^{\prime }$ is determined by $\sigma ^{\prime }(k^{\prime })=\sigma (k^{-1}k^{\prime } k)$, for every $k^{\prime }\in K_{\ell ^{\prime }}$ (cf. [12] paragraph 3.9).

Furthermore, every irreducible unitary representation $\tau$ of $K$ can be trivially extended to an irreducible representation (also denoted by $\tau$) of the group $G$ as follows

$$\begin{aligned} \tau (k,x):=\tau (k),\ k\in K\, \text{ and} \, x\in \mathbb{R }^n. \end{aligned}$$

(2.3)

In this way, we obtain that the unitary dual of the semi-direct product $K\ltimes \mathbb R ^n$ is in bijection with the set of parameters

$$\begin{aligned} \Sigma :=\underset{\ell \in \mathbb{R }^n}{\coprod } \Sigma _\ell \coprod \widehat{K}, \end{aligned}$$

where $\Sigma _\ell$ is the set of all pairs $(\ell ,\sigma )$ as $\sigma$ runs the set $\widehat{K_\ell }$.

2.2.2 The Plancherel measure

We recall that the group $G$ is a separable locally compact unimodular group of type I, and we endow the unitary dual of $G$ with the Mackey Borel structure. We denote by $L^p(G)=L^p(G, \text{ d}g)$ the space of $L^p$-functions on $G$ for $1\le p$, and we define

$$\begin{aligned} \pi (f)=\int \limits _G\pi (g)f(g)\> \text{ d}g, \quad \pi \in \widehat{G}, \quad f\in L^1(G). \end{aligned}$$

Then by the abstract Plancherel theorem, there exists a unique Borel measure $\mu$ on $\widehat{G}$ such that for any function $f \in L^1(G)\cap L^2(G)$,

$$\begin{aligned} \int \limits _G|f(g)|^2\> \text{ d}g=\int \limits _{\widehat{G}}\text{ Tr}(\pi (f)^*\pi (f))\> \text{ d}\mu (\pi ). \end{aligned}$$

(2.4)

The representations of $G$ defined as in formula (2.3) do not contribute to the support of the Plancherel measure of $G$ (see [10, 11] for more details). For any function $f \in L^1(G)$, the operator $\pi _{\ell ,\sigma }(f)$ is Hilbert–Schmidt. It follows from Theorem 2.3 of [11] (see also [10] Theorem 4.4) that the Plancherel formula 2.4 for $G$ precisely reads

$$\begin{aligned} \int \limits _K\int \limits _{\mathbb{R }^n}|f(k,x)|^2\text{ d}x\text{ d}\nu (k)= \int \limits _{\mathbb{R }^n/K}\Big (\underset{\sigma \in \widehat{K_\ell }}{\sum }\big \Vert \pi _{\ell ,\sigma }(f)\big \Vert ^2_\mathrm{HS}\Big )\text{ d}\bar{\ell }, \end{aligned}$$

(2.5)

for all $f\in L^1(G)\cap L^2(G)$.

2.3 The main result

Our objective is to generalize the results recorded in Sect. 2.1, writing down a generalized analogue (allowing polynomial growth) of Hardy’s uncertainty principle on any compact extension $K\ltimes \mathbb{R }^n$ of $\mathbb R ^n$. Our result is as follows:

Theorem 2.1

Let $q\in \mathbb N$, $\alpha$ and $\beta$ be positive real numbers and let $f$ be a measurable function on $G$ satisfying the following decay conditions:

• (i) $|f(k,x)|\le \varphi (k)(1+\big \Vert x\big \Vert ^2)^q e^{-\alpha \big \Vert x\big \Vert ^2}$ for all $x\in \mathbb{R }^n$ and $k\in K$, where $\varphi \in L^2(K)$,

• (ii) $\big \Vert \pi _{\ell ,\sigma }(f)\big \Vert _\mathrm{HS}\le \psi _\ell (\sigma ) (1+\big \Vert \ell \big \Vert ^2)^q e^{-\beta \big \Vert \ell \big \Vert ^2}$, for all $\ell \in \mathbb{R }^n$ and all $\sigma \in \widehat{K_\ell }$, with $\big \Vert \psi _\ell \big \Vert _{l^2(\widehat{K_\ell })}\le C$ for some positive constant $C$ independent of $\ell$. Then, the following conclusions hold:

• 1. $f = 0$ almost everywhere whenever $\alpha \beta > 1/4.$

• 2. If $\alpha \beta < 1/4$, then there are infinitely many linearly independent functions satisfying both conditions (i) and (ii).

• 3. When $\alpha \beta = 1/4$, the function $f$ is of the form $f(k,x)=\zeta (k,x)e^{-\alpha \big \Vert x\big \Vert ^2}$ where $x\mapsto \zeta (k,x)$ is polynomial on $x$ of degree $\le 2q$ and $k\mapsto \zeta (k,x)$ is an $L^2$-function on $K$.

In this context, we mention that Sarkar and Thangavelu [14] gave an analogue of Hardy’s theorem for the group $M_n$ by means of the heat kernel. Indeed, denoting by $\Delta _n$ the standard Laplacian on $\mathbb R ^n$ and by

$$\begin{aligned} p_t^n(x) = (4 \pi t)^{-\frac{n}{2}} e^{-\frac{1}{4t} \big \Vert x \big \Vert ^2} \end{aligned}$$

the associated heat kernel, they proved the following result:

Theorem 2.2

Let $f \in L^1(M_n)$ satisfy the following conditions:

• 1. $|f(k,x)| \le c (1+ \big \Vert x \big \Vert )^N p_s^n(x), \; (k,x) \in M_n$.

• 2. $\big \Vert \pi _{r,\sigma }(f)\big \Vert _\mathrm{HS} \le c (1+r)^N e^{-\mathrm{tr}^2} \; (r,\sigma )\in \mathbb{R }_+^* \times \widehat{\text{ SO}(n-1)}$.

Then $f=0$ whenever $s < t$. When $s=t$, $f$ can be expressed as a finite linear combination of functions of the form

$$\begin{aligned} P_{m,j}(x)(-\Delta _{n+2 m} )^{\frac{j-m}{2}} p_t^{n+2 m} (x) g_{m,j} (k) \end{aligned}$$

where $P_{m,j}$ are solid harmonics of degree $m$ and $g_{m,j}$ are certain bounded functions in $L^2(\mathrm{SO}(n))$.

Let us now comment the discrepancy between the two upshots. Clearly, the condition $s < t$ (and $s=t$ respectively) corresponds to the setting $\alpha \beta > \frac{1}{4}$ (and $\alpha \beta = \frac{1}{4}$ respectively) in Theorem 2.1. Regarding the optimal setting in Theorem 2.2 where $s = t$, they obtained limiting functions of the form

$$\begin{aligned} e^{-\frac{1}{4t}\big \Vert x \big \Vert ^2} \zeta (k,x)=e^{-\frac{1}{4t}\big \Vert x \big \Vert ^2} \underset{\eta }{\sum }Q_{\eta }(x) g_{\eta }(k) \end{aligned}$$

for some polynomials $Q_{\eta }$ on $\mathbb{R }^n$ and some bounded functions $g_{\eta }$ on $\text{ SO}(n)$. So, clearly for fixed $k \in K$, $x \mapsto \zeta (k,x)$ is a polynomial on $\mathbb{R }^n$, and for $x \in \mathbb{R }^n$, $k \mapsto \zeta (k,x)$ belongs to $L^2(\text{ SO}(n))$.

Let us precise in this context that the second decay condition considered in Theorem 2.2 is simpler than that of Theorem 2.1, which allows them to use some tools from complex analysis.

3 Proof of the main result

We start by determining the kernel of the operator $\pi _{\ell ,\sigma }(f)$, for $f \in L^1(G)$, $\ell \in \mathbb R ^n$ and $\sigma \in \widehat{K_\ell }$. We first prove the following:

Lemma 3.1

For $f \in L^1(G)$, $\ell \in \mathbb R ^n$ and $\sigma \in \widehat{K_\ell }$, the operator $\pi _{\ell ,\sigma }(f)$ is a kernel operator. Its kernel is defined for $(s,u)\in K/K_{\ell }\times K/K_{\ell }$ as

$$\begin{aligned} K_{\ell ,\sigma }^f(s,u)= (2\pi )^\frac{n}{2}\int \limits _{K_\ell }\widehat{f}^2(sv u^{-1},s.\ell )\sigma (v)\mathrm{d}\nu _\ell (v). \end{aligned}$$

(3.1)

where $\widehat{f}^2$ denotes the partial Fourier transform of the function $f$ with respect to the Euclidean variable.

Proof

Let $\xi \in {\fancyscript{H}}_{\ell ,\sigma }$ and $s \in K$. A direct computation shows the following:

$$\begin{aligned} \pi _{\ell ,\sigma }(f)\xi (s)&= \int \limits _K\int \limits _{\mathbb{R }^n}f(k,x)\pi _{\ell ,\sigma }(k,x)\xi (s) \text{ d}x\text{ d}\nu (k)\\&= \int \limits _{K}\int \limits _{\mathbb{R }^n}f(k,x)e^{-i\langle s.\ell ,x\rangle }\xi (k^{-1}s)\text{ d}x\text{ d}\nu (k)\\&= (2\pi )^\frac{n}{2}\int \limits _{K}\widehat{f}^2(sk^{-1},s.\ell )\xi (k)\text{ d}\nu (k)\\&(\text{ using} \text{ equation} (2.1))\\&= (2\pi )^\frac{n}{2}\int \limits _{K/K_\ell }\int \limits _{K_\ell }\widehat{f}^2(sv^{-1}u^{-1},s.\ell ) \sigma (v)^*(\xi (u))\text{ d}\nu _\ell (v)\text{ d}\dot{\nu }_\ell (u K_\ell )\\&= (2\pi )^\frac{n}{2}\int \limits _{K/K_\ell }\int \limits _{K_\ell }\widehat{f}^2(sv u^{-1},s.\ell ) \sigma (v)(\xi (u))\text{ d}\nu _\ell (v)\text{ d}\dot{\nu }_\ell (u K_\ell )\ \\&= \int \limits _{K/K_\ell } K_{\ell ,\sigma }^f(s,u)(\xi (u))\text{ d}\dot{\nu }_\ell (u K_\ell ), \end{aligned}$$

where $K_{\ell ,\sigma }^f$ is as in formula (3.1) as needed. $\square$

For $k\in K$, let $f_k\in L^1(\mathbb{R }^n)$ denote the function defined by $f_k(x)=f(k,x)$, and define the measurable functions $g$ and $\tilde{g}$ on $\mathbb{R }^n$ by

$$\begin{aligned} g(x)=\int \limits _K(f_k*f_k^{\star })(x)\text{ d}\nu (k), \end{aligned}$$

where $*$ stands for the usual convolution of functions on $\mathbb{R }^n$, and $\star$ denotes the involution of functions defined as follows: $\mathrm for \; g \in L^1(\mathbb{R }^n) \; \; g^{\star }(x)=\overline{g(-x)}$.

Let

$$\begin{aligned} \tilde{g}(x)= \int \limits _{\mathrm{SO}(n)}g(ax)\text{ d}a,\\ \nonumber \end{aligned}$$

(3.2)

where $\text{ SO}(n)$ denotes the rotation group of degree $n$ equipped with normalized Haar measure $\text{ d}a$.

Obviously, $\widetilde{g} \in L^1(\mathbb{R }^n)$ and

$$\begin{aligned} \widehat{\tilde{g}}(x)&= \int \limits _{\mathbb{R }^n} \int \limits _{\mathrm{SO}(n)}g(ay)\; \text{ d}a \; e^{-i\langle x,y \rangle } dy\nonumber \\ \nonumber&= \int \limits _{\mathrm{SO}(n)} \int \limits _{\mathbb{R }^n} g(ay)\; e^{-i\langle x,y \rangle } \text{ d}y \; \text{ d}a\\ \nonumber&= \int \limits _{\mathrm{SO}(n)} \int \limits _{\mathbb{R }^n} g(y)\; e^{-i\langle a x,y \rangle } \text{ d}y\; \text{ d}a\\&= \widetilde{\widehat{g}}(x) \end{aligned}$$

(3.3)

Lemma 3.2

Suppose that $f$ satisfies the decay condition (i) in Theorem 2.1 and let $\tilde{g}$ be defined as in formula (3.2). Then

$$\begin{aligned} |\tilde{g}(x)|\le c (1+ \big \Vert x\big \Vert ^2)^{2q} e^{-\frac{\alpha }{2}\big \Vert x\big \Vert ^2}, \end{aligned}$$

for some constant $C>0$ and all $x\in \mathbb{R }^n$.

Proof For all $x\in \mathbb{R }^n$, we have

$$\begin{aligned} g(x)&= \int \limits _K\int \limits _{\mathbb{R }^n}f_k(y)f^{\star }_k(x-y)\text{ d}y\text{ d}\nu (k)\nonumber \\&= \int \limits _{K}\int \limits _{\mathbb{R }^n}f(k,y)\overline{f(k,y-x)}\text{ d}y\text{ d}\nu (k).\nonumber \end{aligned}$$

Hence,

$$\begin{aligned} |g(x)|&\le \int \limits _{K}\int \limits _{\mathbb{R }^n}|f(k,y)||f(k,y-x)|\text{ d}y\text{ d}\nu (k)\nonumber \\&\le \int \limits _{K}\int \limits _{\mathbb{R }^n}|\varphi (k)|^2 \left(1+ \big \Vert y\big \Vert ^2\right)^q \left(1+ \big \Vert y-x\big \Vert ^2\right)^qe^{-\alpha \left(\big \Vert y\big \Vert ^2+\big \Vert y-x\big \Vert ^2\right)}\text{ d}y\text{ d}\nu (k)\nonumber \\&= \big \Vert \varphi \big \Vert _2^2 e^{-\frac{\alpha }{2}\big \Vert x\big \Vert ^2} \int \limits _{\mathbb{R }^n} \left(1+ \big \Vert y\big \Vert ^2\right)^q \left(1+ \big \Vert y-x\big \Vert ^2\right)^q e^{-2\alpha \big \Vert y-\frac{x}{2}\big \Vert ^2}\text{ d}y\nonumber \\&= \big \Vert \varphi \big \Vert _2^2 e^{-\frac{\alpha }{2}\big \Vert x\big \Vert ^2} \int \limits _{\mathbb{R }^n} \left(1+ \big \Vert y+\frac{x}{2}\big \Vert ^2\right)^q \left(1+ \big \Vert y-\frac{x}{2}\big \Vert ^2\right)^q e^{-2\alpha \big \Vert y\big \Vert ^2}\text{ d}y\nonumber \\ \end{aligned}$$

Using that for all $x,y\in \mathbb{R }^n$

$$\begin{aligned} |\langle x,y\rangle |\le \frac{1}{2}\left(\big \Vert x\big \Vert ^2+\big \Vert y\big \Vert ^2\right), \end{aligned}$$

we get

$$\begin{aligned} 1+\big \Vert y\pm \frac{x}{2}\big \Vert ^2&= 1+\frac{1}{4}\big \Vert x\big \Vert ^2+\big \Vert y\big \Vert ^2\pm \langle x,y\rangle \nonumber \\&\le 1+\frac{3}{4}\big \Vert x\big \Vert ^2+\frac{3}{2}\big \Vert y\big \Vert ^2\nonumber \\&\le \left(1+\big \Vert x\big \Vert ^2\right)\left(1+2\big \Vert y\big \Vert ^2\right).\nonumber \end{aligned}$$

It follows that

$$\begin{aligned} |g(x)|\le&\big \Vert \varphi \big \Vert _2^2 \left(1+ \big \Vert x\big \Vert ^2\right)^{2q}e^{-\frac{\alpha }{2}\big \Vert x\big \Vert ^2} \int \limits _{\mathbb{R }^n} \left(1+ 2\big \Vert y\big \Vert ^2\right)^{2q}e^{-2\alpha \big \Vert y\big \Vert ^2}\text{ d}y \end{aligned}$$

and thus for $x\in \mathbb{R }^n$,

$$\begin{aligned} |\tilde{g}(x)|\le C \left(1+ \big \Vert x\big \Vert ^2\right)^{2q}e^{-\frac{\alpha }{2}\big \Vert x\big \Vert ^2}. \end{aligned}$$

$\square$

We now investigate the effect that the second decay condition of $f$ has on the function $\tilde{g}$. Remark first that for any function $f \in L^1(G)\cap L^2(G)$, the operator $\pi _{\ell ,\sigma }(f)$ is Hilbert–Schmidt, and its Hilbert–Schmidt norm is given according to Lemma 3.1 as:

$$\begin{aligned} \big \Vert \pi _{\ell ,\sigma }(f)\big \Vert ^2_\mathrm{HS}&= \int \limits _{K/K_\ell } \int \limits _{K/K_\ell }\big \Vert K_{\ell ,\sigma }^f(s,u)\big \Vert ^2_\mathrm{HS}\text{ d}\dot{\nu }_\ell (s K_\ell )\text{ d}\dot{\nu }_\ell (u K_\ell ). \end{aligned}$$

(3.4)

Lemma 3.3

Suppose that $f$ satisfies the decay condition (ii) in Theorem 2.1 and let $\tilde{g}$ be defined as above. Then

$$\begin{aligned} \widehat{\tilde{g}}(y)\le C (1+\big \Vert y\big \Vert ^2)^{2q}e^{-2\beta \big \Vert y\big \Vert ^2}, \end{aligned}$$

for some constant $C>0$ and all $y\in \mathbb{R }^n$.

Proof Let $y_0\in \mathbb{R }^n$. For $p\in \mathbb N$, let

$$\begin{aligned} \fancyscript{C}_p(y_0)=\left\{ y\in \mathbb{R }^n:\ \big \Vert y_0\big \Vert -\frac{1}{2p}\le \big \Vert y\big \Vert \le \big \Vert y_0\big \Vert +\frac{1}{2p}\right\} \end{aligned}$$

denote the annulus in $\mathbb{R }^n$ and $v_p$ its volume. For every $p\in \mathbb N$, there exists a sequence $(h_{p,m})_m$ of $L^1$-functions on $\mathbb{R }^n$ with the following properties:

• 1. $0\le \widehat{h_{p,m}}\le 1$ and $h_{p,m}=h^{\star }_{p,m}$,

• 2. $(\widehat{h_{p,m}})_m$ converges pointwise to the characteristic function $\chi _{\fancyscript{C}_p(y_0)}$ of $\fancyscript{C}_p(y_0)$.For each $p,m\in \mathbb N$, we consider the function $g_{p,m}$ defined by $g_{p,m}=h_{p,m}*h^{\star }_{p,m}*g$. Using property (1), we get

$$\begin{aligned} g_{p,m}=h_{p,m}*h^{\star }_{p,m}* \int \limits _K(f_k*f_k^{\star })\text{ d}k=\int \limits _K(h_{p,m}*f_k)* (h_{p,m}*f_k)^{\star }\text{ d}\nu (k) \end{aligned}$$

and thus,

$$\begin{aligned} g_{p,m}(0)=\int \limits _K\int \limits _{\mathbb{R }^n}|h_{p,m}*f_k(x)|^2\text{ d}x\text{ d}\nu (k). \end{aligned}$$

It follows from the inversion formula for $\mathbb{R }^n$ and the Plancherel formula for $G$ that the Fourier transform of $\tilde{g}$

$$\begin{aligned} \widehat{\tilde{g}}(y_0)&= \underset{p}{\lim }\ v_p^{-1}\int \limits _{\fancyscript{C}_p(y_0)}\widehat{\tilde{g}}(y)\text{ d}y\\&= \underset{p}{\lim }\ v_p^{-1}\int \limits _{\fancyscript{C}_p(y_0)}\int \limits _{\mathrm{SO}(n)}\widehat{g}(ay)\text{ d}a\text{ d}y\\&= \underset{p}{\lim }\ v_p^{-1}\int \limits _{\fancyscript{C}_p(y_0)}\widehat{g}(y)\text{ d}y\\&= \underset{p}{\lim }\ v_p^{-1}\int \limits _{\mathbb{R }^n}\underset{m}{\lim }\ \big (\widehat{h_{p,m}}(y)\big )^2\widehat{g}(y)\text{ d}y\\&= \underset{p}{\lim }\ v_p^{-1}\int \limits _{\mathbb{R }^n}\underset{m}{\lim }\ \widehat{g_{p,m}}(y)\text{ d}y\\&= \underset{p}{\lim }\ v_p^{-1}\underset{m}{\lim }\ g_{p,m}(0)\\&= \underset{p}{\lim }\ \underset{m}{\lim } \ v_p^{-1}\int \limits _K\int \limits _{\mathbb{R }^n}|h_{p,m}*f(k,x)|^2\text{ d}x\text{ d}\nu (k)\\&= \underset{p}{\lim }\ \underset{m}{\lim } \ v_p^{-1}\big \Vert h_{p,m}*f\big \Vert ^2_{L^2(G)}\\&= \underset{p}{\lim }\ \underset{m}{\lim } \ v_p^{-1}\int \limits _{\mathbb{R }^n/K}\Big (\underset{\sigma \in \widehat{K_\ell }}{\sum }\big \Vert \pi _{\ell ,\sigma }(h_{p,m}*f)\big \Vert ^2_\mathrm{HS}\Big )\text{ d}\bar{\ell }. \end{aligned}$$

Furthermore, we have for $\xi \in L^2(K/K_\ell , \sigma )$

$$\begin{aligned} \pi _{\ell ,\sigma }(h_{p,m}*f)\xi (s)&\!=\!\int \limits _{K}\int \limits _{\mathbb{R }^n}h_{p,m}*f(k,x)e^{-i\langle s.\ell ,x\rangle }\xi (k^{-1}s)\text{ d}x\text{ d}\nu (k)\\&= \int \limits _{K}\int \limits _{\mathbb{R }^n}\left(\,\,\int \limits _{\mathbb{R }^n}h_{p,m}(y)f(k,x-y)\text{ d}y\right)e^{-i\langle s.\ell ,x\rangle }\xi (k^{-1}s)\text{ d}x\text{ d}\nu (k)\nonumber \\&= \ \widehat{h_{p,m}}(s.\ell )\int \limits _{K}\int \limits _{\mathbb{R }^n}f(k,x)e^{-i\langle s.\ell ,x\rangle }\xi (k^{-1}s)\text{ d}x\text{ d}\nu (k)\nonumber \\&\!=\!(2\pi )^\frac{n}{2}\!\int \limits _{K/K_\ell }\!\Big (\widehat{h_{p,m}}(s.\ell )\!\int \limits _{K_\ell }\!\widehat{f}^2(sv u^{-1},s.\ell ) \sigma (v)d\nu _\ell (v)\Big )(\xi (u))\text{ d}\dot{\nu }_\ell (u K_\ell ) \end{aligned}$$

Let $B( s , u ,\ell ) = \big \Vert \int \limits _{K_\ell }\widehat{f}^2(sv u^{-1},s.\ell )\sigma (v)\text{ d}\nu _\ell (v)\big \Vert _\mathrm{HS}^2$. We obtain from above

$$\begin{aligned} \big \Vert \pi _{\ell ,\sigma }(h_{p,m}*f)\big \Vert _\mathrm{HS}^2&= \;(2\pi )^n \int \limits _{K/K_\ell }\int \limits _{K/K_\ell }\widehat{h_{p,m}}(s.\ell )^2 B(s , u,\ell ) \text{ d}\dot{\nu }_\ell (u K_\ell ) \text{ d}\dot{\nu }_\ell (s K_\ell )\\&\le \;(2\pi )^n \widehat{h_{p,m}}(s_\ell .\ell )^2\int \limits _{K/K_\ell }\int \limits _{K/K_\ell }B( s , u ,\ell ) d\dot{\nu }_\ell (u K_\ell ) \text{ d}\dot{\nu }_\ell (s K_\ell )\\&= \; \widehat{h_{p,m}}(s_\ell .\ell )^2\big \Vert \pi _{\ell ,\sigma }(f)\big \Vert _{HS}^2 \end{aligned}$$

for some $s_\ell \in K$ since the mapping $s\mapsto \widehat{h_{p,m}}(s.\ell )$ is continuous. We deduce that

$$\begin{aligned} \widehat{\tilde{g}}(y_0)&= \underset{p}{\lim }\ \underset{m}{\lim } \mathfrak v _p^{-1}\int \limits _{\mathbb{R }^n/K}\left(\underset{\sigma \in \widehat{K_\ell }}{\sum }\big \Vert \pi _{\ell ,\sigma }(h_{p,m}*f)\big \Vert ^2_\mathrm{HS}\right)d\bar{\ell }\nonumber \\&\le \underset{p}{\lim }\ \underset{m}{\lim } \ v_p^{-1}\int \limits _{\mathbb{R }^n/K}\widehat{h_{p,m}}(s_\ell .\ell )^2\left(\underset{\sigma \in \widehat{K_\ell }}{\sum }\big \Vert \pi _{\ell ,\sigma }(f)\big \Vert ^2_\mathrm{HS}\right)d\bar{\ell }\nonumber \\&\le \underset{p}{\lim }\ \underset{m}{\lim } \ v_p^{-1}\int \limits _{\mathbb{R }^n/K}\widehat{h_{p,m}}(s_\ell .\ell )^2\big \Vert \psi _{\ell }\big \Vert _{l^2(\widehat{K_\ell })}^2\ (1+\big \Vert \ell \big \Vert ^2)^{2q}e^{-2\beta \big \Vert \ell \big \Vert ^2} \text{ d}\bar{\ell }\nonumber \\&\le C\ \underset{p}{\lim }\ v_p^{-1}\int \limits _{\mathbb{R }^n/K}\chi _{\fancyscript{C}_p(y_0)}(\ell )^2 (1+\big \Vert \ell \big \Vert ^2)^{2q}e^{-2\beta \big \Vert \ell \big \Vert ^2} \text{ d}\bar{\ell }\nonumber \\&= C\ \underset{p}{\lim }\ v_p^{-1}\int \limits _{\mathbb{R }^n/K}\int \limits _{K}\chi _{\fancyscript{C}_p(y_0)}(k.\ell ))^2 (1+\big \Vert k.\ell \big \Vert ^2)^{2q}e^{-2\beta \big \Vert k.\ell \big \Vert ^2} \text{ d}\nu (k)\text{ d}\bar{\ell }\nonumber \\&= C\ \underset{p}{\lim }\ v_p^{-1}\int \limits _{\mathbb{R }^n}\chi _{\fancyscript{C}_p(y_0)}(\ell ))^2 (1+\big \Vert \ell \big \Vert ^2)^{2q}e^{-2\beta \big \Vert \ell \big \Vert ^2}\text{ d}\bar{\ell }\ (\text{ using} \text{ equation} \,(2.2)\nonumber \\&= C \ \underset{p}{\lim }\ v_p^{-1}\int \limits _{\fancyscript{C}_p(y_0)} (1+\big \Vert \ell \big \Vert ^2)^{2q} e^{-2\beta \big \Vert \ell \big \Vert ^2} \text{ d}\ell \nonumber \\&= C (1+\big \Vert y_0\big \Vert ^2)^{2q}e^{-2\beta \big \Vert y_0\big \Vert ^2}. \end{aligned}$$

(3.5)

This completes the proof since $y_0$ was taken arbitrary in $\mathbb{R }^n$. $\square$

The proof of the first part of Theorem 2.1 now follows immediately. Indeed, if $\alpha \beta >\frac{1}{4}$, Lemmas 3.2 and 3.3 give that the function $\tilde{g}$ satisfies the hypotheses of Hardy’s theorem for $\mathbb{R }^n$, which implies that $\tilde{g}$ is zero almost everywhere. Therefore, we get

$$\begin{aligned} 0&= \int \limits _{\mathbb{R }^n}\widehat{\tilde{g}}(x)\text{ d}x\\&= \int \limits _{\mathbb{R }^n}\widetilde{\widehat{g}}(x)\text{ d}x \; (\text{ using} \text{ equation} \, (3.3))\\&= \int \limits _{\mathbb{R }^n} \int \limits _{\mathrm{SO}(n)}\widehat{g}(ax) \; \text{ d}a\; \text{ d}x\\&= \int \limits _{\mathrm{SO}(n)} \int \limits _{\mathbb{R }^n} \widehat{g}(x) \; \text{ d}x \; \text{ d}a\\&= \int \limits _{\mathbb{R }^n} \widehat{g}(x) \; \text{ d}x\\&= \int \limits _K\int \limits _{\mathbb{R }^n}\widehat{f_k*f_k^{\star }}(x)\text{ d}x\text{ d}\nu (k)\\&= \int \limits _K\Big (\int \limits _{\mathbb{R }^n}|\widehat{f_k}(x)|^2\text{ d}x\Big )\text{ d}\nu (k). \end{aligned}$$

Then, for almost all $k\in K$, $\widehat{f_k}(x)=0$ for all $x\in \mathbb{R }^n$, whence $f=0$ almost everywhere.

Let now $\alpha \beta <\frac{1}{4}$. We show in this case that the function $f_{\varphi ,r}$ defined on $G$ by

$$\begin{aligned} f_{\varphi ,r}(k,x)=\varphi (k)(1+\big \Vert x\big \Vert ^2)^q e^{-r \big \Vert x\big \Vert ^2}\ \text{ for} \text{ all} (k,x)\in G, \end{aligned}$$

satisfies conditions (i) and (ii) of Theorem 2.1 for any $r\in [\alpha , \frac{1}{4\beta }]$ and $\varphi \in L^2(K)$. In fact, for any $\ell \in \mathbb{R }^n$ and any $\sigma \in \widehat{K_\ell }$, we can see by Lemma 3.1 that for any element $\xi \in L^2(K/K_\ell , \sigma )$ and any $s\in K$, we have by the Hecke–Bochner identity (Theorem 1.3.4 in [20])

$$\begin{aligned} \pi _{\ell ,\sigma }(f_{\varphi ,r})\xi (s)&= (2\pi )^\frac{n}{2}\int \limits _{K/K_\ell }\left(\int \limits _{K_\ell } \widehat{f_{\varphi ,r}}^2(svu^{-1}, s.\ell )\sigma (v)\text{ d}\nu _\ell (v)\right)\xi (u)\text{ d}\dot{\nu }_\ell (u K_\ell )\\&\!=\!C_q (1\!+\!\big \Vert s.\ell \big \Vert ^2)^q e^{-\frac{1}{4r}\big \Vert s.\ell \big \Vert ^2}\int \limits _{K/K_\ell }\left(\int \limits _{K_\ell } \varphi (svu^{-1})\sigma (v)\text{ d}\nu _\ell (v)\right)\\ \xi (u)\text{ d}\dot{\nu }_\ell (u K_\ell )&= C_q (1+\big \Vert \ell \big \Vert ^2)^q e^{-\frac{1}{4r}\big \Vert \ell \big \Vert ^2}\int \limits _{K/K_\ell }\sigma ( _s\varphi _u)(\xi (u))\text{ d}\dot{\nu }_\ell (u K_\ell ), \end{aligned}$$

where $_s\varphi _u\in L^2(K_\ell )$ defined by $_s\varphi _u(v)=\varphi (svu^{-1})$ for all $v\in K_\ell$. Hence,

$$\begin{aligned} \big \Vert \pi _{\ell ,\sigma }(f_r)\big \Vert _\mathrm{HS}=&C_q \psi _\ell (\sigma )(1+\big \Vert \ell \big \Vert ^2)^q e^{-\frac{1}{4r}\big \Vert \ell \big \Vert ^2}, \end{aligned}$$

where

$$\begin{aligned} \psi _\ell (\sigma )=\left(\int \limits _{K/K_\ell }\int \limits _{K/K_\ell } \big \Vert \sigma ( _s\varphi _u)\big \Vert _\mathrm{HS}^2\ \text{ d}\dot{\nu }_\ell (s K_\ell )\text{ d}\dot{\nu }_\ell (u K_\ell )\right)^\frac{1}{2}, \end{aligned}$$

which satisfies that

$$\begin{aligned} \underset{\sigma \in \widehat{K_\ell }}{\sum }\psi _\ell (\sigma )^2&= \int \limits _{K/K_\ell } \int \limits _{K/K_\ell }\underset{\sigma \in \widehat{K_\ell }}{\sum } \big \Vert \sigma ( _s\varphi _u)\big \Vert _{HS}^2\ \text{ d}\dot{\nu }_\ell (s K_\ell )\text{ d}\dot{\nu }_\ell (u K_\ell )\\&= \int \limits _{K/K_\ell }\int \limits _{K/K_\ell }\big \Vert _s\varphi _u\big \Vert _2^2\ \text{ d}\dot{\nu }_\ell (s K_\ell )\text{ d}\dot{\nu }_\ell (u K_\ell )\\&= \int \limits _{K/K_\ell }\int \limits _{K/K_\ell }\int \limits _{K_\ell }| \varphi (svu^{-1})|^2 \text{ d}\nu _\ell (v) \text{ d}\dot{\nu }_\ell (s K_\ell )\text{ d}\dot{\nu }_\ell (u K_\ell )\\&\le \int \limits _{K}| \varphi (s)|^2 \text{ d}\nu (s)=\big \Vert \varphi \big \Vert _2^2. \end{aligned}$$

We now prove the last assertion of Theorem 2.1. Here, $\beta = \frac{1}{4 \alpha }$. Let $f$ be a measurable function on $G$ satisfying conditions (i) and (ii). For $(\delta ,\gamma )\in L^2(K)\times L^2(\text{ SO}(n))$, let

$$\begin{aligned} F_{\delta ,\gamma }(x)=\int \limits _{\mathrm{SO}(n)}\int \limits _K f(k,ax)\overline{\delta (k)\gamma (a)}\text{ d}\nu (k)\text{ d}a, \ \ x\in \mathbb{R }^n. \end{aligned}$$

Hence,

$$\begin{aligned} |F_{\delta ,\gamma }(x)|&\le \int \limits _K \int \limits _{\mathrm{SO}(n)}|f(k,ax)||\gamma (a)\delta (k)|\text{ d}a \text{ d}\nu (k)\\&\le \int \limits _K |\varphi (k)\delta (k)|d\nu (k)\int \limits _{\mathrm{SO}(n)}|\gamma (a)|(1+\big \Vert x\big \Vert ^2)^q e^{-\alpha \big \Vert x\big \Vert ^2}\text{ d}a\\&\le \big \Vert \delta \big \Vert _2\big \Vert \varphi \big \Vert _2\big \Vert \gamma \big \Vert _2 (1+\big \Vert x\big \Vert ^2)^q e^{-\alpha \big \Vert x\big \Vert ^2}, \end{aligned}$$

for all $x\in \mathbb{R }^n$. In addition, it follows from (3.2) and (3.5) that for all $y\in \mathbb{R }^n$

$$\begin{aligned} |\widehat{F_{\delta ,\gamma }}(y)|&\le \int \limits _{\mathrm{SO}(n)} \left(\int \limits _K|\widehat{f}^2(k,ay)\delta (k)|\text{ d}\nu (k)\right)|\gamma (a)|\text{ d}a\\&\le \big \Vert \delta \big \Vert _2\int \limits _{\mathrm{SO}(n)}\left(\int \limits _K |\widehat{f_k}(ay)|^2 \text{ d}\nu (k)\right)^{\frac{1}{2}}|\gamma (a)|\text{ d}a\\&= \ \big \Vert \delta \big \Vert _2\big \Vert \gamma \big \Vert _2\left(\int \limits _{\mathrm{SO}(n)}\int \limits _K \widehat{f_k*f_k^{\star }}(ay)\text{ d}\nu (k) \text{ d}a\right)^{\frac{1}{2}}\\&\le C\ (1+\big \Vert y\big \Vert ^2)^q e^{-\frac{1}{4\alpha }\big \Vert y\big \Vert ^2}. \end{aligned}$$

Applying now Theorem 1.4.4 in [20], which says that the only measurable functions $g$ on $\mathbb{R }^n$ satisfying $|g(x)|\le C_1\ (1+\big \Vert x\big \Vert ^2)^q e^{-\alpha \big \Vert x\big \Vert ^2}$ and $|\widehat{g}(y)|\le C_2\ (1+\big \Vert y\big \Vert ^2)^q e^{-\frac{1}{4\alpha }\big \Vert y\big \Vert ^2}$, for some positive constants $C_1$ and $C_2$, are of the form $g(x)=P(x) e^{-\alpha \big \Vert x\big \Vert ^2}$, where $P$ is a polynomial of degree $\le 2q$, we can deduce that there exists some polynomial $P_{\delta ,\gamma }$ of degree at most $2q$ such that $F_{\delta ,\gamma }(x)=P_{\delta ,\gamma }(x) e^{-\alpha \big \Vert x\big \Vert ^2}$, for all $x\in \mathbb{R }^n$.

For $\tau \in \widehat{K}$, let $u_{ij}^{\tau }$ be the matrix coefficients of $\tau$ in an orthonormal basis $\{e_j^\tau :\ 1\le j \le d_\tau \}$ of its associated Hilbert space ${\fancyscript{H}}_\tau$ of dimension $d_\tau$. In other words, $u^{\tau }_{ij}(k) = \langle \tau (k)e_i^\tau , e_j^\tau \rangle$, for each $k\in K$.The Peter–Weyl theorem now asserts that the set of functions $\left\{ \sqrt{\text{ d}_{\tau }}u^{\tau }_{ij}\mid \, \tau \in \widehat{K},\ 1\le i,j\le \text{ d}_{\tau }\right\}$ is an orthonormal basis of $L^2(K)$. Similarly, let $\left\{ \sqrt{d_{\rho }}v^{\rho }_{ij}\mid \, \rho \in \widehat{\text{ SO}(n)},\ 1\le i,j\le d_{\rho }\right\}$ be an orthonormal basis of $L^2(\text{ SO}(n))$. Allowing now $\delta$ and $\gamma$ to vary over these bases, respectively, we obtain

$$\begin{aligned} \int \limits _{\mathrm{SO}(n)}\int \limits _K f(k,ax)\overline{u_{ij}^\tau (k)v^{\rho }_{i^{\prime }j^{\prime }}(a)}\text{ d}\nu (k)\text{ d}a=P_{i,i^{\prime },j,j^{\prime }}^{\tau ,\rho }(x) e^{-\alpha \big \Vert x\big \Vert ^2} \end{aligned}$$

for some polynomials $P_{i,i^{\prime },j,j^{\prime }}^{\tau ,\rho }$ of degree $\le 2q$.

As a final step, we write for all $(a,k,x) \in \text{ SO}(n) \times K \times \mathbb{R }^n$

$$\begin{aligned} f(k,ax)=&\underset{\tau \in \widehat{K}}{\sum }\underset{1\le i,j \le \text{ d}_\tau }{\sum }\underset{\rho \in \widehat{\text{ SO}(n)}}{\sum }\underset{1\le i^{\prime },j^{\prime } \le \text{ d}_\rho }{\sum } P_{i,i^{\prime },j,j^{\prime }}^{\tau ,\rho }(x) u_{ij}^\tau (k)v^{\rho }_{i^{\prime }j^{\prime }}(a) e^{-\alpha \big \Vert x\big \Vert ^2} .\\ \end{aligned}$$

Then, setting $a=Id$, this entails that

$$\begin{aligned} f(k,x)=\zeta (k,x)e^{-\alpha \big \Vert x\big \Vert ^2} \end{aligned}$$

with

$$\begin{aligned} \zeta (k,x)=&\underset{\tau \in \widehat{K}}{\sum }\underset{1\le i,j \le \text{ d}_\tau }{\sum }\underset{\rho \in \widehat{\text{ SO}(n)}}{\sum }\underset{1\le i^{\prime },j^{\prime } \le \text{ d}_\rho }{\sum } \underset{|\theta | \le 2q}{\sum } c_{i,i^{\prime },j,j^{\prime }}^{\tau ,\rho }(\theta ) x^{\theta } u_{ij}^\tau (k)\\ =&\underset{|\theta | \le 2q}{\sum } \Big ( \underset{\tau \in \widehat{K}}{\sum }\underset{1\le i,j \le \text{ d}_\tau }{\sum }\underset{\rho \in \widehat{\text{ SO}(n)}}{\sum }\underset{1\le i^{\prime },j^{\prime } \le \text{ d}_\rho }{\sum } c_{i,i^{\prime },j,j^{\prime }}^{\tau ,\rho }(\theta ) u_{ij}^\tau (k)\Big ) x^{\theta } \end{aligned}$$

where $\theta =(\theta _1, \ldots , \theta _{n}) \in \mathbb N ^{n}$ and $|\theta |= \theta _1+ \cdots + \theta _{n}$ its length. Here, $x^{\theta }= x_1^{\theta _1} \cdots x_n^{\theta _n}$ for $x=(x_1,\ldots ,x_n)\in \mathbb R ^{n}$.

So, for fixed $x \in \mathbb{R }^n$, $k \mapsto \zeta (k,x)$ is an element of $L^2(K)$, and for fixed $k\in K$, the function $x \mapsto \zeta (k,x)$ is a polynomial of degree at most $2q$. This completes the proof of the theorem.

4 Concluding remarks

As a first consequence of Theorem 2.1, a complete analogue of classical Hardy’s theorem can be written as follows:

Theorem 4.1

Let $\alpha$ and $\beta$ be positive real numbers and let $f$ be a measurable function on $G$ satisfying the following decay conditions:

• (i) $|f(k,x)|\le \varphi (k)e^{-\alpha \big \Vert x\big \Vert ^2}$ for all $x\in \mathbb{R }^n$ and $k\in K$, where $\varphi \in L^2(K)$,

• (ii) $\big \Vert \pi _{\ell ,\sigma }(f)\big \Vert _\mathrm{HS}\le \psi _\ell (\sigma ) e^{-\beta \big \Vert \ell \big \Vert ^2}$, for all $\ell \in \mathbb{R }^n$ and all $\sigma \in \widehat{K_\ell }$, with $\big \Vert \psi _\ell \big \Vert _{l^2(\widehat{K_\ell })}\le C$ for some positive constant $C$.

Then, the following conclusions hold:

• 1. If $\alpha \beta > 1/4$, then $f = 0$ almost everywhere.

• 2. When $\alpha \beta < 1/4$, there are infinitely many linearly independent functions satisfying both conditions (i) and (ii).

• 3. The function $f$ is of the form $f(k,x)=\zeta (k) e^{-\alpha \big \Vert x\big \Vert ^2}$ for some $\zeta \in L^2(K)$, whenever $\alpha \beta = 1/4$.

We finally describe an analogue of Hardy’s theorem for the universal covering of the Euclidean motion group. None of the groups $\text{ SO}(n)$ or $M(n)$ is simply connected. The universal covering groups of the orthogonal groups are called Spin groups. The Lie group ${Spin} (n) \ltimes \mathbb R ^n$ is, therefore, the universal covering group of $M(n)$, where the action of ${Spin}(n)$ on $\mathbb R ^n$ is simply the pullback of the action of $\text{ SO}(n)$ on $\mathbb R ^n$. We have the following:

Theorem 4.2

An analogue of Hardy’s uncertainty principle holds on the group $G= \text{ Spin} (n) \ltimes \mathbb R ^n$ for any $n\ge 2$.

Proof When $n=2$, ${Spin}(n)= \mathbb R$ and $G$ is a three-dimensional non-exponential solvable Lie group with a trivial centre. In this case, the result is the subject of Theorem 3.1 in [3]. Otherwise ($n\ge 3$), ${Spin}(n)$ is compact and an application of Theorem 4.1 completes the proof. $\square$