# Generalized analogs of the Heisenberg uncertainty inequality

## Abstract

We investigate locally compact topological groups for which a generalized analog of the Heisenberg uncertainty inequality hold. In particular, it is shown that this inequality holds for $$\mathbb{R}^{n} \times K$$ (where K is a separable unimodular locally compact group of type I), Euclidean motion group and several general classes of nilpotent Lie groups which include thread-like nilpotent Lie groups, 2-NPC nilpotent Lie groups and several low-dimensional nilpotent Lie groups.

## Introduction

In 1927, Heisenberg presented a principle related to the uncertainties in the measurements of position and momentum of microscopic particles. This principle is known as Heisenberg uncertainty principle and can be stated as follows:

It is impossible to know simultaneously the exact position and momentum of a particle. That is, the more exactly the position is determined, the less known the momentum, and vice versa.

In 1933, Wiener gave the following mathematical formulation of the Heisenberg uncertainty principle:

A nonzero function and its Fourier transform cannot both be sharply localized.

Heisenberg’s uncertainty inequality is a precise quantitative formulation of the above principle.

The Fourier transform of $$f \in L^{1}(\mathbb{R}^{n})$$ is given by

$$\hat{f}(\xi) =\int_{\mathbb{R}^{n}}{f(x) e^{-2\pi i \langle {x,\xi}\rangle}}\, dx,$$

where $$\langle{\cdot,\cdot}\rangle$$ denotes the usual inner product on $$\mathbb{R}^{n}$$. This definition of Fourier transform holds for functions in $$L^{1}(\mathbb{R}^{n}) \cap L^{2}(\mathbb{R}^{n})$$. Since $$L^{1}(\mathbb{R}^{n}) \cap L^{2}(\mathbb{R}^{n})$$ is dense in $$L^{2}(\mathbb{R}^{n})$$, the definition of Fourier transform can be extended to the functions in $$L^{2}(\mathbb{R}^{n})$$.

The following theorem gives the Heisenberg uncertainty inequality for the Fourier transform on $$\mathbb{R}^{n}$$. For a proof of the theorem, see .

### Theorem 1.1

For any $$f \in L^{2}(\mathbb{R}^{n})$$, we have

$$\frac{n\|f\|_{2}^{2}}{4\pi} \leq \biggl(\int_{\mathbb {R}^{n}}{\|x\| ^{2} \bigl\vert f(x)\bigr\vert ^{2}}\, dx \biggr)^{1/2} \biggl( \int_{\mathbb{R}^{n}}{\| y\|^{2} \bigl\vert \hat{f}(y)\bigr\vert ^{2}}\, dy \biggr)^{1/2},$$
(1.1)

where $$\|\cdot\|_{2}$$ denotes the $$L^{2}$$-norm and $$\|\cdot\|$$ denotes the Euclidean norm.

The Heisenberg uncertainty inequality has been established for the Fourier transform on the Heisenberg group by Thangavelu . Further generalizations of the inequality on the Heisenberg group have been established by Sitaram et al.  and Xiao and He . For some more details, see .

The inequality given below can be proved using Hölder’s inequality and the inequality (1.1).

### Theorem 1.2

For any $$f \in L^{2}(\mathbb{R}^{n})$$ and $$a,b \geq1$$, we have

$$\frac{n\|f\|_{2}^{ (\frac{1}{a}+\frac{1}{b} )}}{4\pi} \leq \biggl(\int_{\mathbb{R}^{n}}{\|x\|^{2a} \bigl\vert f(x)\bigr\vert ^{2}} \, dx \biggr)^{\frac {1}{2a}} \biggl(\int_{\mathbb{R}^{n}}{\|y\|^{2b} \bigl\vert \hat{f}(y) \bigr\vert ^{2}}\, dy \biggr)^{\frac{1}{2b}},$$

where $$\|\cdot\|_{2}$$ denotes the $$L^{2}$$-norm and $$\|\cdot\|$$ denotes the Euclidean norm.

In Section 2, we shall prove a generalized analog of the Heisenberg uncertainty inequality for $$\mathbb{R}^{n} \times K$$, where K is a separable unimodular locally compact group of type I. In the next section, a generalized analog of the Heisenberg uncertainty inequality for the Euclidean motion group $$M(n)$$ is proved. The last section deals with a generalized analog of the Heisenberg uncertainty inequality for several general classes of nilpotent Lie groups for which the Hilbert-Schmidt norm of the group Fourier transform $$\pi_{\xi}(f)$$ of f attains a particular form. These classes include thread-like nilpotent Lie groups, 2-NPC nilpotent Lie groups and several low-dimensional nilpotent Lie groups.

## $$\mathbb{R}^{n} \times K$$, K a locally compact group

Consider $$G=\mathbb{R}^{n} \times K$$, where K is a separable unimodular locally compact group of type I. The Haar measure of G is $$dg=dx \,dk$$, where dx is the Lebesgue measure on $$\mathbb{R}^{n}$$ and dk is the left Haar measure on K. The dual $$\widehat{G}$$ of G is $$\mathbb{R}^{n} \times \widehat{K}$$, where $$\widehat{K}$$ is the dual space of K.

The Fourier transform of $$f \in L^{2}(G)$$ is given by

$$\hat{f}(y,\sigma) =\int_{\mathbb{R}^{n}}\int_{K}{f(x,k) e^{-2\pi i \langle{x,y}\rangle} \sigma\bigl(k^{-1}\bigr)} \,dk \,dx,$$

for $$(y,\sigma) \in\mathbb{R}^{n} \times\widehat{K}$$.

### Theorem 2.1

For any $$f \in L^{2}(\mathbb{R}^{n} \times K)$$ (where K is a separable unimodular locally compact group of type I) and $$a,b \geq1$$, we have

\begin{aligned} \frac{n\|f\|_{2}^{ (\frac{1}{a}+\frac{1}{b} )}}{4\pi} \leq& \biggl(\int_{\mathbb{R}^{n}}\int _{K}{\|x\|^{2a} \bigl\vert f(x,k)\bigr\vert ^{2}} \,dk \,dx \biggr)^{\frac{1}{2a}} \\ &{}\times \biggl(\int_{\mathbb {R}^{n}} \int_{\widehat{K}}{\|y\|^{2b} \bigl\Vert \hat{f}(y,\sigma) \bigr\Vert _{\mathrm{HS}}^{2}} \,dy \,d\sigma \biggr)^{\frac{1}{2b}}. \end{aligned}
(2.1)

### Proof

Without loss of generality, we may assume that both integrals on the right-hand side of (2.1) are finite.

Given that $$f \in L^{2}(\mathbb{R}^{n} \times K)$$, there exists $$A \subseteq K$$ of measure zero such that for $$k \in K\setminus A=A'$$ (say), we have

$$\int_{\mathbb{R}^{n}}{\bigl\vert f(x,k)\bigr\vert ^{2}} \,dx < \infty.$$

For all $$k \in A'$$, we define $$f_{k}(x)=f(x,k)$$, for every $$x\in\mathbb{R}^{n}$$.

Clearly, for all $$k\in A'$$, $$f_{k} \in L^{2}(\mathbb{R}^{n})$$, and for all $$y \in\mathbb{R}^{n}$$,

$$\hat{f}_{k}(y) =\int_{\mathbb{R}^{n}}{f(x,k) e^{-2\pi i \langle {x,y}\rangle}} \,dy=\mathscr{F}_{1} f(y,k).$$

By Theorem 1.1, we have

$$\frac{n}{4\pi}\int_{\mathbb{R}^{n}}{\bigl\vert f(x,k)\bigr\vert ^{2}} \,dx \leq \biggl(\int_{\mathbb{R}^{n}}{\|x \|^{2} \bigl\vert f_{k}(x)\bigr\vert ^{2}} \,dx \biggr)^{1/2} \biggl(\int_{\mathbb{R}^{n}}{\| y\|^{2} \bigl\vert \hat{f}_{k}(y)\bigr\vert ^{2}} \,dy \biggr)^{1/2}.$$

Integrating both sides with respect to dk, we obtain

$$\frac{n}{4\pi}\int_{A'}\int_{\mathbb{R} ^{n}}{ \bigl\vert f(x,k)\bigr\vert ^{2}} \,dx \,dk \leq\int _{A'} \biggl(\int_{\mathbb{R}^{n}}{\| x \|^{2} \bigl\vert f_{k}(x)\bigr\vert ^{2}} \,dx \biggr)^{1/2} \biggl(\int_{\mathbb{R}^{n}}{\|y\|^{2} \bigl\vert \hat {f}_{k}(y)\bigr\vert ^{2}} \,dy \biggr)^{1/2} \,dk.$$

The integral on the L.H.S. is equal to $$\|f\|_{2}^{2}$$, so using the Cauchy-Schwarz inequality and Fubini’s theorem, we have

$$\frac{n\|f\|_{2}^{2}}{4\pi} \leq \biggl(\int_{K}\int _{\mathbb {R}^{n}}{\|x\| ^{2} \bigl\vert f(x,k)\bigr\vert ^{2}} \,dx \,dk \biggr)^{1/2} \biggl(\int_{\mathbb{R}^{n}} \|y\| ^{2}\int_{A'}{\bigl\vert \hat{f}_{k}(y)\bigr\vert ^{2}} \,dk \,dy \biggr)^{1/2}.$$
(2.2)

Now, using Hölder’s inequality, we have

\begin{aligned}& \biggl(\int_{\mathbb{R}^{n}}\int_{K}{\|x \|^{2a} \bigl\vert f(x,k)\bigr\vert ^{2}} \,dk \,dx \biggr)^{\frac{1}{a}} \biggl(\int_{\mathbb{R}^{n}}\int _{K}{\bigl\vert f(x,k)\bigr\vert ^{2}} \,dk \,dx \biggr)^{1-\frac{1}{a}} \\& \quad \geq\int_{\mathbb{R}^{n}}\int_{K}{\|x \|^{2} \bigl\vert f(x,k)\bigr\vert ^{\frac {2}{a}}\bigl\vert f(x,k)\bigr\vert ^{2 (1-\frac{1}{a} )}} \,dk \,dx \\& \quad = \int_{\mathbb{R}^{n}}\int_{K}{\|x \|^{2} \bigl\vert f(x,k)\bigr\vert ^{2}} \,dk \,dx, \end{aligned}

which implies

$$\int_{\mathbb{R}^{n}}\int_{K}{\|x\|^{2} \bigl\vert f(x,k)\bigr\vert ^{2}} \,dk \,dx \leq \biggl(\int _{\mathbb{R}^{n}}\int_{K}{\|x\|^{2a} \bigl\vert f(x,k)\bigr\vert ^{2}} \,dk \,dx \biggr)^{\frac{1}{a}} \bigl( \|f\|_{2}^{2} \bigr)^{1-\frac{1}{a}}.$$
(2.3)

Combining (2.2) and (2.3), we obtain

\begin{aligned} \frac{n\|f\|_{2}^{2}}{4\pi} \leq& \biggl(\int_{\mathbb{R}^{n}}\int _{K}{\|x\| ^{2a} \bigl\vert f(x,k)\bigr\vert ^{2}} \,dk \,dx \biggr)^{\frac{1}{2a}} \bigl(\|f\| _{2}^{2} \bigr)^{\frac{1}{2}-\frac{1}{2a}} \\ &{}\times \biggl(\int_{\mathbb {R}^{n}}\|y\| ^{2} \int_{A'}{\bigl\vert \hat{f}_{k}(y)\bigr\vert ^{2}} \,dk \,dy \biggr)^{1/2}. \end{aligned}
(2.4)

Since

$$\int_{\mathbb{R}^{n}}\int_{A'}{\bigl\vert \mathscr {F}_{1}f(y,k)\bigr\vert ^{2}} \,dy \,dk =\int _{\mathbb{R}^{n}}\int_{A'}{\bigl\vert f(x,k)\bigr\vert ^{2}} \,dx \,dk =\|f\|_{2}^{2} < \infty,$$

therefore, $$\mathscr{F}_{1}f \in L^{2}(\mathbb{R}^{n}\times A')$$. Therefore, $$\mathscr{F}_{2}\mathscr{F}_{1} f$$ is well defined a.e. By approximating $$f\in L^{2}(\mathbb{R}^{n} \times A')$$ by functions in $$L^{1}\cap L^{2}(\mathbb{R}^{n}\times A')$$, we have

$$\mathscr{F}_{2}\mathscr{F}_{1} f =\hat{f},$$

for all $$f \in L^{2}(\mathbb{R}^{n} \times A')$$. Applying the Plancherel formula on the locally compact group K, we have

$$\int_{A'}{\bigl\vert \hat{f}_{k}(y)\bigr\vert ^{2}} \,dk =\int_{\widehat{K}}{\bigl\Vert \hat{f}(y, \sigma)\bigr\Vert _{\mathrm{HS}}^{2}} \,d\sigma.$$

Thus, (2.4) can be written as

\begin{aligned} \frac{n\|f\|_{2}^{2}}{4\pi} \leq& \biggl(\int_{\mathbb{R}^{n}}\int _{K}{\|x\| ^{2a} \bigl\vert f(x,k)\bigr\vert ^{2}} \,dk \,dx \biggr)^{\frac{1}{2a}} \bigl(\|f\| _{2}^{2} \bigr)^{\frac{1}{2}-\frac{1}{2a}} \\ &{}\times \biggl(\int_{\mathbb {R}^{n}}\int _{\widehat{K}}{\|y\|^{2}\bigl\Vert \hat{f}(y,\sigma)\bigr\Vert _{\mathrm{HS}}^{2}} \,dy \,d\sigma \biggr)^{1/2}. \end{aligned}
(2.5)

Now, again using Hölder’s inequality, we have

\begin{aligned}& \biggl(\int_{\mathbb{R}^{n}}\int_{\widehat{K}}{\|y \|^{2b} \bigl\Vert \hat {f}(y,\sigma)\bigr\Vert _{\mathrm{HS}}^{2}} \,dy \,d\sigma \biggr)^{\frac {1}{b}} \biggl(\int_{\mathbb{R}^{n}}\int _{\widehat{K}}{\bigl\Vert \hat {f}(y,\sigma)\bigr\Vert _{\mathrm{HS}}^{2}} \,dy \,d\sigma \biggr)^{1-\frac {1}{b}} \\& \quad \geq\int_{\mathbb{R}^{n}}\int_{\widehat{K}}{\|y \|^{2} \bigl\Vert \hat {f}(y,\sigma)\bigr\Vert _{\mathrm{HS}}^{\frac{2}{b}} \bigl\Vert \hat{f}(y,\sigma )\bigr\Vert _{\mathrm{HS}}^{2 (1-\frac{1}{b} )}} \,dy \,d\sigma \\& \quad = \int_{\mathbb{R}^{n}}\int_{\widehat{K}}{\|y \|^{2} \bigl\Vert \hat {f}(y,\sigma )\bigr\Vert _{\mathrm{HS}}^{2}} \,dy \,d\sigma, \end{aligned}

which implies

$$\int_{\mathbb{R}^{n}}\int_{\widehat{K}}{\|y\|^{2} \bigl\Vert \hat {f}(y,\sigma )\bigr\Vert _{\mathrm{HS}}^{2}} \,dy \,d\sigma\leq \biggl(\int_{\mathbb {R}^{n}}\int_{\widehat{K}}{ \|y\|^{2b} \bigl\Vert \hat{f}(y,\sigma)\bigr\Vert _{\mathrm{HS}}^{2}} \,dy \,d\sigma \biggr)^{\frac{1}{b}} \bigl(\|f \|_{2}^{2} \bigr)^{1-\frac{1}{b}}.$$
(2.6)

Combining (2.5) and (2.6), we obtain

\begin{aligned} \frac{n\|f\|_{2}^{2}}{4\pi} \leq& \biggl(\int_{\mathbb{R}^{n}}\int _{K}{\|x\| ^{2a} \bigl\vert f(x,k)\bigr\vert ^{2}} \,dk \,dx \biggr)^{\frac{1}{2a}} \bigl(\|f\| _{2}^{2} \bigr)^{\frac{1}{2}-\frac{1}{2a}} \\ &{}\times \biggl(\int_{\mathbb{R}^{n}}\int_{\widehat{K}}{\|y \| ^{2b} \bigl\Vert \hat{f}(y,\sigma)\bigr\Vert _{\mathrm{HS}}^{2}} \,dy \,d\sigma \biggr)^{\frac {1}{2b}} \bigl(\|f\|_{2}^{2} \bigr)^{\frac{1}{2}-\frac{1}{2b}}, \end{aligned}

which implies

$$\frac{n\|f\|_{2}^{ (\frac{1}{a}+\frac{1}{b} )}}{4\pi} \leq \biggl(\int_{\mathbb{R}^{n}}\int _{K}{\|x\|^{2a} \bigl\vert f(x,k)\bigr\vert ^{2}} \,dk \,dx \biggr)^{\frac{1}{2a}} \biggl(\int_{\mathbb{R}^{n}} \int_{\widehat {K}}{\|y\|^{2b} \bigl\Vert \hat{f}(y,\sigma) \bigr\Vert _{\mathrm{HS}}^{2}} \,dy \,d\sigma \biggr)^{\frac{1}{2b}}.$$

□

## Euclidean motion group $$M(n)$$

Consider $$M(n)$$ to be the semi-direct product of $$\mathbb{R}^{n}$$ with $$K=\operatorname{SO}(n)$$. The group law is given by

$$(z,k) \bigl(w,k'\bigr) =\bigl(z+k\cdot w, kk'\bigr),$$

for $$z,w \in\mathbb{R}^{n}$$ and $$k,k' \in K$$. The group $$M(n)$$ is called the motion group of the Euclidean plane $$\mathbb{R}^{n}$$.

As in , $$M=\operatorname{SO}(n-1)$$ can be considered as a subgroup of K leaving the point $$e_{1}=(1,0,0,\ldots,0)$$ fixed. All the irreducible unitary representations of $$M(n)$$ relevant for the Plancherel formula are parametrized (up to unitary equivalence) by pairs $$(\lambda,\sigma)$$, where $$\lambda>0$$ and $$\sigma\in\widehat {M}$$, the unitary dual of M.

Given $$\sigma\in\widehat{M}$$ realized on a Hilbert space $$H_{\sigma }$$ of dimension $$d_{\sigma}$$, consider the space,

\begin{aligned} L^{2}(K,\sigma) =& \biggl\{ \varphi \Bigm| \varphi: K \rightarrow M_{d_{\sigma}\times d_{\sigma}}, \int{\bigl\Vert \varphi(k)\bigr\Vert ^{2}} \,dk < \infty, \\ & \varphi(uk)=\sigma(u)\varphi(k), \text{for } u \in M \text{ and } k\in K \biggr\} . \end{aligned}

Note that $$L^{2}(K,\sigma)$$ is a Hilbert space under the inner product

$$\langle{\varphi,\psi}\rangle =\int_{K}{\operatorname{tr} \bigl(\varphi(k)\psi (k)^{\ast}\bigr)} \,dk.$$

For each $$\lambda>0$$ and $$\sigma\in\widehat{M}$$, we can define a representation $$\pi_{\lambda,\sigma}$$ of $$M(n)$$ on $$L^{2}(K,\sigma)$$ as follows.

For $$\varphi\in L^{2}(K,\sigma)$$, $$(z,k)\in M(n)$$,

$$\pi_{\lambda,\sigma}(z,k)\varphi(u) =e^{i\lambda\langle {u^{-1}\cdot e_{1},z\rangle}} \varphi(uk),$$

for $$u\in K$$.

If $$\varphi_{j}(k)$$ are the column vectors of $$\varphi\in L^{2}(K,\sigma )$$, then $$\varphi_{j}(uk)=\sigma(u)\varphi_{j}(k)$$ for all $$u \in M$$. Therefore, $$L^{2}(K,\sigma)$$ can be written as the direct sum of $$d_{\sigma}$$ copies of $$H(K,\sigma)$$, where

\begin{aligned} H(K,\sigma) =& \biggl\{ \varphi \Bigm| \varphi: K \rightarrow\mathbb{C} ^{d_{\sigma}}, \int{\bigl\| \varphi(k)\bigr\| ^{2}} \,dk < \infty, \\ &\varphi (uk)= \sigma(u)\varphi(k), \text{for } u \in M \text{ and } k\in K \biggr\} . \end{aligned}

It can be shown that $$\pi_{\lambda,\sigma}$$ restricted to $$H(K,\sigma)$$ is an irreducible unitary representation of $$M(n)$$. Moreover, any irreducible unitary representation of $$M(n)$$ which is infinite dimensional is unitarily equivalent to one and only one $$\pi _{\lambda,\sigma}$$.

The Fourier transform of $$f \in L^{2}(M(n))$$ is given by

$$\hat{f}(\lambda,\sigma) =\int_{M(n)}{f(z,k) \pi_{\lambda ,\sigma}(z,k)^{\ast}} \,dz \,dk.$$

$$\hat{f}(\lambda,\sigma)$$ is a Hilbert-Schmidt operator on $$H(K,\sigma)$$.

A solid harmonic of degree m is a polynomial which is homogeneous of degree m and whose Laplacian is zero. The set of all such polynomials will be denoted by $$\mathbb{H}_{m}$$ and the restriction of elements of $$\mathbb{H}_{m}$$ to $$S^{n-1}$$ is denoted by $$S_{m}$$. By choosing an orthonormal basis $$\{ g_{mj} : j = 1, 2,\ldots,d_{m}\}$$ of $$S_{m}$$ for each $$m = 0, 1, 2, \ldots$$ , we get an orthonormal basis for $$L^{2}(S^{n-1})$$.

The Haar measure on $$M(n)$$ is $$dg=dz \,dk$$, where dz is Lebesgue measure on $$\mathbb{R}^{n}$$ and dk is the normalized Haar measure on $$\operatorname{SO}(n)$$.

The Plancherel formula on $$M(n)$$ is given as follows (see ).

### Proposition 3.1

(Plancherel formula)

Let $$f \in L^{2}(M(n))$$, then

$$\int_{M(n)}{\bigl\vert f(z_{1},z_{2}, \ldots,z_{n},k)\bigr\vert ^{2}} \,dz_{1} \,dz_{2} \cdots \,dz_{n} \,dk =c_{n}\int _{0}^{\infty} \biggl(\sum_{\sigma\in \widehat{M}}{d_{\sigma}\bigl\Vert \hat{f}(\lambda,\sigma)\bigr\Vert _{\mathrm{HS}}^{2}} \biggr) \lambda^{n-1}\, d\lambda ,$$

where $$c_{n}=\frac{2}{2^{n/2} \Gamma (\frac{n}{2} )}$$.

We shall now state and prove the following generalized Heisenberg uncertainty inequality for a Fourier transform on $$M(n)$$.

### Theorem 3.2

For any $$f \in L^{2}(M(n))$$ and $$a,b \geq1$$, we have

\begin{aligned} \frac{\|f\|_{2}^{ (\frac{1}{a}+\frac{1}{b} )}}{2\sqrt {c_{n}}} \leq& \biggl(\int_{K}\int _{\mathbb{R}^{n}}{\|z\|^{2a} \bigl\vert f(z,k)\bigr\vert ^{2}} \,dz \,dk \biggr)^{\frac{1}{2a}} \\ &{}\times \biggl(\int_{0}^{\infty} \sum_{\sigma \in\widehat{M}}{d_{\sigma}\lambda^{2b} \bigl\Vert \hat{f}(\lambda ,\sigma)\bigr\Vert _{\mathrm{HS}}^{2}} \lambda^{n-1}\, d\lambda \biggr)^{\frac {1}{2b}}. \end{aligned}
(3.1)

### Proof

Consider the norm $$\|\cdot\|$$ on $$L^{2}(M(n))$$ defined by

\begin{aligned} \|f\| : =& \biggl(\int_{\mathbb{R}^{n}}\int_{K}{ \bigl(1+\|z\|^{2a}\bigr) \bigl\vert f(z,k)\bigr\vert ^{2}} \,dz \,dk \biggr)^{1/2} \\ &{}+ \biggl(\int_{0}^{\infty} \sum_{\sigma\in \widehat{M}}{d_{\sigma}\bigl(1+ \lambda^{2b}\bigr)\bigl\Vert \hat{f}(\lambda,\sigma )\bigr\Vert _{\mathrm{HS}}^{2}} \lambda^{n-1} \,d\lambda \biggr)^{1/2}. \end{aligned}

This gives us a Banach space $$B=\{f\in L^{2}(G):\|f\|<\infty\}$$, which is contained in $$L^{2}(M(n))$$ and the space $$\mathcal{S}(M(n))$$ of $$C^{\infty}$$-functions which are rapidly decreasing on $$M(n)$$ can be shown to be dense in B. It suffices to prove the inequality of Theorem 3.2 for functions in $$\mathcal{S}(M(n))$$; it is automatically valid for any $$f\in B$$. If $$0\neq f \in L^{2}(M(n))\setminus B$$, then the right-hand side of the inequality is always +∞ and the inequality is trivially valid.

Let $$f\in\mathcal{S}(M(n))$$. Assuming that both integrals on the right-hand side of (3.1) are finite, we have

$$\int_{\mathbb{R}^{n}}{\bigl\vert f(z,k)\bigr\vert ^{2}} \,dz < \infty, \quad \text {for all } k\in K.$$

For $$k \in K$$, we define $$f_{k}(z)=f(z,k)$$, for every $$z \in\mathbb {R}^{n}$$.

Clearly, $$f_{k} \in L^{2}(\mathbb{R}^{n})$$, for all $$k\in K$$.

Take $$z=(z_{1},z_{2},\ldots,z_{n})$$ and $$w=(w_{1},w_{2},\ldots,w_{n})$$.

By the Heisenberg inequality on $$\mathbb{R}^{n}$$, we have

\begin{aligned}& \frac{\|f_{k}\|_{2}^{2}}{4\pi} \leq \biggl(\int_{\mathbb {R}^{n}}{|z_{1}|^{2} \bigl\vert f_{k}(z)\bigr\vert ^{2}} \,dz \biggr)^{1/2} \biggl(\int_{\mathbb{R}^{n}}{|w_{1}|^{2} \bigl\vert \hat {f}_{k}(w)\bigr\vert ^{2}} \,dw \biggr)^{1/2} \\& \quad \Rightarrow \quad \frac{1}{4\pi}\int_{\mathbb{R}^{n}}{\bigl\vert f(z,k)\bigr\vert ^{2}} \,dz \leq \biggl(\int _{\mathbb{R}^{n}}{|z_{1}|^{2} \bigl\vert f(z,k)\bigr\vert ^{2}} \,dz \biggr)^{1/2} \biggl(\int _{\mathbb{R}^{n}}{|w_{1}|^{2} \bigl\vert \hat{f}_{k}(w)\bigr\vert ^{2}} \,dw \biggr)^{1/2} . \end{aligned}

Integrating both sides with respect to dk, we get

$$\frac{1}{4\pi}\int_{K}\int_{\mathbb{R}^{n}}{ \bigl\vert f(z,k)\bigr\vert ^{2}} \,dz \,dk \leq \int _{K} \biggl(\int_{\mathbb{R}^{n}}{|z_{1}|^{2} \bigl\vert f(z,k)\bigr\vert ^{2}} \,dz \biggr)^{1/2} \biggl(\int_{\mathbb{R}^{n}}{|w_{1}|^{2} \bigl\vert \hat{f}_{k}(w)\bigr\vert ^{2}} \,dw \biggr)^{1/2} \,dk ,$$

which implies

\begin{aligned} \frac{\|f\|_{2}^{2}}{4\pi} \leq&\int_{K} \biggl(\int _{\mathbb {R}^{n}}{|z_{1}|^{2} \bigl\vert f(z,k)\bigr\vert ^{2}} \,dz \biggr)^{1/2} \biggl(\int _{\mathbb{R}^{n}}{|w_{1}|^{2} \bigl\vert \hat {f}_{k}(w)\bigr\vert ^{2}} \,dw \biggr)^{1/2} \,dk \\ \leq& \biggl(\int_{K}\int_{\mathbb{R}^{n}}{|z_{1}|^{2} \bigl\vert f(z,k)\bigr\vert ^{2}} \,dz \,dk \biggr)^{1/2} \biggl(\int_{K}\int_{\mathbb{R}^{n}}{|w_{1}|^{2} \bigl\vert \hat {f}_{k}(w)\bigr\vert ^{2}} \,dw \,dk \biggr)^{1/2} \\ & (\text{by the Cauchy-Schwarz inequality}) \\ \leq& \biggl(\int_{K}\int_{\mathbb{R}^{n}}{\|z \|^{2} \bigl\vert f(z,k)\bigr\vert ^{2}} \,dz \,dk \biggr)^{1/2} \biggl(\int_{K}\int _{\mathbb{R}^{n}}{|w_{1}|^{2} \bigl\vert \hat {f}_{k}(w)\bigr\vert ^{2}} \,dw \,dk \biggr)^{1/2}. \end{aligned}
(3.2)

Now,

\begin{aligned}& \biggl(\int_{K}\int_{\mathbb{R}^{n}}{\|z \|^{2a} \bigl\vert f(z,k)\bigr\vert ^{2}} \,dz \,dk \biggr)^{\frac{1}{a}} \biggl(\int_{K}\int _{\mathbb{R}^{n}}{\bigl\vert f(z,k)\bigr\vert ^{2}} \,dz \,dk \biggr)^{1-\frac{1}{a}} \\& \quad = \biggl(\int_{K}\int_{\mathbb{R}^{n}}{ \bigl(\|z\|^{2} \bigl\vert f(z,k)\bigr\vert ^{\frac {2}{a}} \bigr)^{a}} \,dz \,dk \biggr)^{\frac{1}{a}} \biggl(\int _{K}\int_{\mathbb{R}^{n}}{ \bigl(\bigl\vert f(z,k) \bigr\vert ^{2 (1-\frac {1}{a} )} \bigr)^{\frac{1}{ (1-\frac{1}{a} )}}} \,dz \,dk \biggr)^{1-\frac{1}{a}} \\& \quad \geq\int_{K}\int_{\mathbb{R}^{n}}{\|z \|^{2} \bigl\vert f(z,k)\bigr\vert ^{\frac {2}{a}}\bigl\vert f(z,k)\bigr\vert ^{2 (1-\frac{1}{a} )}} \,dz \,dk \quad \bigl(\text{by H\"{o}lder's inequality}\bigr) \\& \quad = \int_{K}\int_{\mathbb{R}^{n}}{\|z \|^{2} \bigl\vert f(z,k)\bigr\vert ^{2}} \,dz \,dk . \end{aligned}
(3.3)

Combining (3.2) and (3.3), we get

\begin{aligned} \frac{\|f\|_{2}^{2}}{4\pi} \leq& \biggl(\int_{K}\int _{\mathbb {R}^{n}}{\|z\| ^{2a} \bigl\vert f(z,k)\bigr\vert ^{2}} \,dz \,dk \biggr)^{\frac{1}{2a}} \bigl(\|f\| _{2}^{2} \bigr)^{\frac{1}{2}-\frac{1}{2a}} \\ &{}\times\biggl(\int_{K}\int _{\mathbb{R} ^{n}}{|w_{1}|^{2} \bigl\vert \hat{f}_{k}(w)\bigr\vert ^{2}} \,dw \,dk \biggr)^{1/2}. \end{aligned}
(3.4)

Now, using the Plancherel formula on $$\mathbb{R}^{n}$$, we have

\begin{aligned}& \int_{K}\int_{\mathbb{R}^{n}}{|w_{1}|^{2} \bigl\vert \hat{f}_{k}(w)\bigr\vert ^{2}} \,dw \,dk \\& \quad =\int_{K}\int_{\mathbb{R}^{n}}{|w_{1}|^{2} \biggl\vert \int_{\mathbb {R}^{n}}{f(z,k) e^{-2\pi i\langle{z,w}\rangle}} \,dz\biggr\vert ^{2}} \,dw \,dk \\& \quad =\int_{K}\int_{\mathbb{R}^{n}}{|w_{1}|^{2} \bigl\vert \mathscr{F}_{1,2,\ldots, n} f(w_{1},w_{2},\ldots ,w_{n},k)\bigr\vert ^{2}} \,dw_{1} \,dw_{2} \cdots \,dw_{n} \,dk \\& \quad =\int_{K}\int_{\mathbb{R}^{n}}{|w_{1}|^{2} \bigl\vert \mathscr{F}_{1} f(w_{1},z_{2},\ldots ,z_{n},k)\bigr\vert ^{2}} \,dw_{1} \,dz_{2} \cdots \,dz_{n} \,dk. \end{aligned}
(3.5)

Since $$\frac{\partial f}{\partial z_{1}}\in\mathcal{S}(M(n))$$, we have

$$\int_{\mathbb{R}}{\biggl\vert \frac{\partial f}{\partial z_{1}}(z_{1},z_{2}, \ldots ,z_{n},k)\biggr\vert ^{2}} \,dz_{1} < \infty,$$

for all $$z_{i} \in\mathbb{R}$$ and $$k \in K$$.

Therefore, $$w_{1}\mathscr{F}_{1} f(w_{1},z_{2},\ldots,z_{n},k)\in L^{2}(\mathbb {R})$$ and

$$\biggl(\frac{\partial f}{\partial z_{1}}(z_{1},z_{2},\ldots,z_{n},k) \biggr)^{\land}(w_{1}) =2\pi i w_{1} \mathscr{F}_{1} f(w_{1},z_{2},\ldots,z_{n},k),$$

for all $$z_{i} \in\mathbb{R}$$ and $$k \in K$$. Then

\begin{aligned}& \int_{\mathbb{R}}{|w_{1}|^{2} \bigl\vert \mathscr{F}_{1} f(w_{1},z_{2},\ldots ,z_{n},k)\bigr\vert ^{2}} \,dw_{1} \\& \quad = \frac{1}{4\pi^{2}}\int_{\mathbb{R}}{\biggl\vert \frac{\partial f}{\partial z_{1}}(z_{1},z_{2},\ldots,z_{n},k)\biggr\vert ^{2}} \,dz_{1}, \end{aligned}

which implies

\begin{aligned}& \int_{K}\int_{\mathbb{R}^{n}}{|w_{1}|^{2} \bigl\vert \mathscr{F}_{1} f(w_{1},z_{2}, \ldots,z_{n},k)\bigr\vert ^{2}} \,dw_{1} \,dz_{2} \cdots \,dz_{n} \,dk \\& \quad =\frac{1}{4\pi^{2}}\int_{K}\int_{\mathbb{R}^{n}}{ \biggl\vert \frac {\partial f}{\partial z_{1}}(z_{1},z_{2}, \ldots,z_{n},k)\biggr\vert ^{2}} \,dz_{1} \,dz_{2} \cdots \,dz_{n} \,dk. \end{aligned}
(3.6)

By Proposition 3.1, we obtain

\begin{aligned}& \int_{K}\int_{\mathbb{R}^{n}}{\biggl\vert \frac{\partial f}{\partial z_{1}}(z_{1},z_{2},\ldots,z_{n},k)\biggr\vert ^{2}} \,dz_{1} \,dz_{2} \cdots \,dz_{n} \,dk \\& \quad =c_{n}\int_{0}^{\infty}\sum _{\sigma\in\widehat{M}}{d_{\sigma}\biggl\Vert \biggl( \frac{\partial f}{\partial z_{1}} \biggr)^{\land}(\lambda,\sigma)\biggr\Vert _{\mathrm{HS}}^{2}} \lambda^{n-1} \,d\lambda . \end{aligned}
(3.7)

Combining (3.4), (3.5), (3.6), and (3.7), we obtain

\begin{aligned} \frac{\|f\|_{2}^{2}}{2\sqrt{c_{n}}} \leq& \biggl(\int_{K}\int _{\mathbb{R} ^{n}}{\|z\|^{2a} \bigl\vert f(z,k)\bigr\vert ^{2}} \,dz \,dk \biggr)^{\frac{1}{2a}} \bigl(\|f\| _{2}^{2} \bigr)^{\frac{1}{2}-\frac{1}{2a}} \\ &{}\times \biggl(\int_{0}^{\infty}\sum _{\sigma\in\widehat {M}}{d_{\sigma}\biggl\Vert \biggl( \frac{\partial f}{\partial z_{1}} \biggr)^{\land}(\lambda,\sigma)\biggr\Vert _{\mathrm{HS}}^{2}} \lambda ^{n-1} \,d\lambda \biggr)^{1/2}. \end{aligned}
(3.8)

For each $$\lambda>0$$ and $$\sigma\in\widehat{M}$$, consider the representation $$\pi_{\lambda,\sigma}(z,k)$$ realized on $$L^{2}(K,\sigma )$$ as

$$\pi_{\lambda,\sigma}(z,k)g(u) =e^{i\lambda\langle{u^{-1}\cdot e_{1},z}\rangle} g(uk), \quad u \in \operatorname{SO}(n).$$

Denote $$u=[u_{ij}]_{n\times n}$$; we have

$$u^{-1}\cdot e_{1} =u^{T}\cdot e_{1}=[ \begin{array}{@{}c@{\quad}c@{\quad}c@{\quad}c@{}} u_{11}& u_{12}& \cdots &u_{1n} \end{array}]^{T}.$$

Therefore, $$\langle{u^{-1}\cdot e_{1},z}\rangle=\sum_{i=1}^{n}{u_{1i}z_{i}}$$.

Since $$f\in\mathcal{S}(M(n))$$,

\begin{aligned}& \biggl(\frac{\partial f}{\partial z_{1}} \biggr)^{\land}(\lambda,\sigma)g(u) \\& \quad =\int_{\mathbb{R}^{n}}\int_{K}{ \frac{\partial f}{\partial z_{1}}(z_{1},z_{2},\ldots,z_{n},k) \pi_{\lambda,\sigma}(z_{1},z_{2},\ldots ,z_{n},k)^{\ast}g(u)} \,dz_{1} \,dz_{2} \cdots \,dz_{n} \,dk \\& \quad =\int_{\mathbb{R}^{n}}\int_{K}\lim _{h\rightarrow0}{ \biggl[\frac {f(z_{1}+h,z_{2},\ldots,z_{n},k)-f(z_{1},z_{2},\ldots,z_{n},k)}{h} \biggr]} \\& \qquad {}\times\pi _{\lambda,\sigma}(z_{1},z_{2},\ldots,z_{n},k)^{\ast}g(u)\,dz_{1} \,dz_{2} \cdots \,dz_{n} \,dk \\& \quad =\lim_{h\rightarrow0}\frac{1}{h}\biggl[\int _{\mathbb{R}^{n}}\int_{K}{f(z_{1}+h,z_{2}, \ldots,z_{n},k) \pi_{\lambda,\sigma}(z_{1},z_{2}, \ldots ,z_{n},k)^{\ast}g(u)} \,dz_{1} \,dz_{2} \cdots \,dz_{n} \,dk \\& \qquad {}-\int_{\mathbb{R}^{n}}\int_{K}{f(z_{1},z_{2}, \ldots ,z_{n},k) \pi _{\lambda,\sigma}(z_{1},z_{2}, \ldots,z_{n},k)^{\ast}g(u)} \,dz_{1} \,dz_{2} \cdots \,dz_{n} \,dk\biggr] \\& \quad =\lim_{h\rightarrow0}\frac{1}{h}\biggl[\int _{\mathbb{R}^{n}}\int_{K}f(z_{1},z_{2}, \ldots,z_{n},k) e^{-i \lambda h u_{11}}\pi_{\lambda ,\sigma}(z_{1},z_{2}, \ldots,z_{n},k)^{\ast}\\& \qquad {}\times g(u) \,dz_{1} \,dz_{2} \cdots \,dz_{n} \,dk \\& \qquad {}-\int_{\mathbb{R}^{n}}\int_{K}{f(z_{1},z_{2}, \ldots ,z_{n},k) \pi _{\lambda,\sigma}(z_{1},z_{2}, \ldots,z_{n},k)^{\ast}g(u)} \,dz_{1} \,dz_{2} \cdots \,dz_{n} \,dk\biggr] \\& \quad =\lim_{h\rightarrow0} \biggl[\frac{e^{-i \lambda h u_{11}}-1}{h} \biggr]\int _{\mathbb{R}^{n}}\int_{K}f(z_{1},z_{2}, \ldots ,z_{n},k) \pi_{\lambda,\sigma}(z_{1},z_{2}, \ldots,z_{n},k)^{\ast}\\& \qquad {}\times g(u) \,dz_{1} \,dz_{2} \cdots \,dz_{n} \,dk \\& \quad =i\lambda u_{11}\int_{\mathbb{R}^{n}}\int _{K}{f(z_{1},z_{2},\ldots ,z_{n},k) \pi _{\lambda,\sigma}(z_{1},z_{2}, \ldots,z_{n},k)^{\ast}g(u)} \,dz_{1} \,dz_{2} \cdots \,dz_{n} \,dk \\& \quad =i\lambda u_{11} \hat{f}(\lambda,\sigma)g(u). \end{aligned}

Hence,

\begin{aligned} \biggl\Vert \biggl(\frac{\partial f}{\partial z_{1}} \biggr)^{\land}(\lambda,\sigma) \biggr\Vert _{\mathrm{HS}}^{2} =&\sum_{m=0}^{\infty } \sum_{j=1}^{d_{m}}\int_{K}{ \bigl\vert i\lambda u_{11} \hat{f}(\lambda ,\sigma)g_{mj}(u) \bigr\vert ^{2}} \, du \\ \leq&\lambda^{2}\sum_{m=0}^{\infty} \sum_{j=1}^{d_{m}}\int_{K}{ \bigl\vert \hat{f}(\lambda,\sigma)g_{mj}(u)\bigr\vert ^{2}}\, du = \lambda^{2}\bigl\Vert \hat{f}(\lambda,\sigma) \bigr\Vert _{\mathrm{HS}}^{2}. \end{aligned}

Therefore, (3.8) can be written as

\begin{aligned} \frac{\|f\|_{2}^{2}}{2\sqrt{c_{n}}} \leq& \biggl(\int_{K}\int _{\mathbb {R}^{n}}{\| z\|^{2a} \bigl\vert f(z,k)\bigr\vert ^{2}} \,dz \,dk \biggr)^{\frac{1}{2a}} \bigl(\|f\| _{2}^{2} \bigr)^{\frac{1}{2}-\frac{1}{2a}} \\ &{} \times \biggl(\int_{0}^{\infty}\sum _{\sigma\in\widehat {M}}{d_{\sigma}\lambda^{2}\bigl\Vert \hat{f}(\lambda,\sigma)\bigr\Vert _{\mathrm{HS}}^{2}} \lambda^{n-1} \,d\lambda \biggr)^{1/2}. \end{aligned}
(3.9)

Now, again using Hölder’s inequality, we have

\begin{aligned}& \biggl(\int_{0}^{\infty}\sum _{\sigma\in\widehat{M}}{d_{\sigma}\lambda^{2b}\bigl\Vert \hat{f}(\lambda,\sigma)\bigr\Vert _{\mathrm{HS}}^{2}} \lambda^{n-1} \,d\lambda \biggr)^{\frac{1}{b}} \biggl(\int _{0}^{\infty}\sum_{\sigma\in\widehat{M}}{d_{\sigma}\bigl\Vert \hat {f}(\lambda,\sigma)\bigr\Vert _{\mathrm{HS}}^{2}} \lambda^{n-1} \,d\lambda \biggr)^{1-\frac{1}{b}} \\& \quad \geq\int_{0}^{\infty}\sum _{\sigma\in\widehat{M}}{d_{\sigma}^{1/b} \lambda^{2} \bigl\Vert \hat{f}(\lambda,\sigma)\bigr\Vert _{\mathrm{HS}}^{\frac{2}{b}} d_{\sigma}^{ (1-\frac{1}{b} )}\bigl\Vert \hat{f}(\lambda,\sigma)\bigr\Vert _{\mathrm{HS}}^{2 (1-\frac {1}{b} )}} \lambda^{n-1} \,d\lambda \\& \quad = \int_{0}^{\infty}\sum _{\sigma\in\widehat{M}}{d_{\sigma}\lambda^{2} \bigl\Vert \hat{f}(\lambda,\sigma)\bigr\Vert _{\mathrm{HS}}^{2}} \lambda ^{n-1} \,d\lambda, \end{aligned}

which implies

\begin{aligned}& \int_{0}^{\infty}\sum_{\sigma\in\widehat{M}}{d_{\sigma}\lambda ^{2} \bigl\Vert \hat{f}(\lambda,\sigma)\bigr\Vert _{\mathrm{HS}}^{2}} \lambda^{n-1} \,d\lambda \\& \quad \leq \biggl(\int _{0}^{\infty}\sum_{\sigma\in\widehat {M}}{d_{\sigma}\lambda^{2b}\bigl\Vert \hat{f}(\lambda,\sigma)\bigr\Vert _{\mathrm{HS}}^{2}} \lambda^{n-1} \,d\lambda \biggr)^{\frac{1}{b}} \bigl(\|f\| _{2}^{2} \bigr)^{1-\frac{1}{b}}. \end{aligned}
(3.10)

Combining (3.9) and (3.10), we obtain

\begin{aligned} \frac{\|f\|_{2}^{ (\frac{1}{a}+\frac{1}{b} )}}{2\sqrt {c_{n}}} \leq& \biggl(\int_{K}\int _{\mathbb{R}^{n}}{\|z\|^{2a} \bigl\vert f(z,k)\bigr\vert ^{2}} \,dz \,dk \biggr)^{\frac{1}{2a}} \\ &{}\times \biggl(\int_{0}^{\infty} \sum_{\sigma \in\widehat{M}}{d_{\sigma}\lambda^{2b} \bigl\Vert \hat{f}(\lambda ,\sigma)\bigr\Vert _{\mathrm{HS}}^{2}} \lambda^{n-1} \,d\lambda \biggr)^{\frac{1}{2b}}. \end{aligned}

□

## A class of nilpotent Lie groups

In this section, we shall prove the Heisenberg uncertainty inequality for a class of connected, simply connected nilpotent Lie groups G for which the Hilbert-Schmidt norm of the group Fourier transform $$\pi_{\xi}(f)$$ of f attains a particular form.

Let $$\mathfrak{g}$$ be an n-dimensional real nilpotent Lie algebra, and let $$G=\exp\mathfrak{g}$$ be the associated connected and simply connected nilpotent Lie group . Let $$\mathcal{B}=\{ X_{1},X_{2},\ldots,X_{n}\}$$ be a strong Malcev basis of $$\mathfrak{g}$$ through the ascending central series of $$\mathfrak{g}$$. We introduce a ‘norm function’ on G by setting, for $$x=\exp(x_{1}X_{1}+x_{2}X_{2}+\cdots+x_{n}X_{n}) \in G$$, $$x_{j}\in\mathbb{R}$$,

$$\|x\| =\bigl(x_{1}^{2}+\cdots+x_{n}^{2} \bigr)^{1/2}.$$

The composed map

$$\mathbb{R}^{n}\rightarrow\mathfrak{g}\rightarrow G,$$

given as

$$(x_{1},\ldots,x_{n})\rightarrow\sum _{j=1}^{n}{x_{j}X_{j}} \rightarrow \exp \Biggl(\sum_{j=1}^{n}{x_{j}X_{j}} \Biggr),$$

is a diffeomorphism and maps a Lebesgue measure on $$\mathbb{R}^{n}$$ to a Haar measure on G. In this manner, we shall always identify $$\mathfrak {g}$$, and sometimes G, as sets with $$\mathbb{R}^{n}$$. Thus, measurable (integrable) functions on G can be viewed as such functions on $$\mathbb{R}^{n}$$.

Let $$\mathfrak{g}^{\ast}$$ denote the vector space dual of $$\mathfrak {g}$$ and $$\{X_{1}^{\ast},\ldots,X_{n}^{\ast}\}$$ the basis of $$\mathfrak{g}^{\ast}$$ which is dual to $$\{ X_{1},\ldots,X_{n}\}$$. Then $$\{X_{1}^{\ast},\ldots,X_{n}^{\ast}\}$$ is a Jordan-Hölder basis for the coadjoint action of G on $$\mathfrak {g}^{\ast}$$. We shall identify $$\mathfrak{g}^{\ast}$$ with $$\mathbb{R}^{n}$$ via the map

$$\xi=(\xi_{1},\ldots,\xi_{n})\rightarrow\sum _{j=1}^{n}{\xi _{j}X_{j}^{\ast}}$$

and on $$\mathfrak{g}^{\ast}$$ we introduce the Euclidean norm relative to the basis $$\{X_{1}^{\ast},\ldots,X_{n}^{\ast}\}$$, i.e.

$$\Biggl\Vert \sum_{j=1}^{n}{ \xi_{j}X_{j}^{\ast}}\Biggr\Vert =\bigl( \xi_{1}^{2}+\cdots +\xi_{n}^{2}\bigr)=\| \xi\|.$$

Let $$\mathfrak{g}_{j}=\mathbb{R}\mbox{-} \operatorname{span}\{X_{1},\ldots,X_{n}\}$$. For $$\xi \in\mathfrak{g}^{\ast}$$, $$\mathcal{O}_{\xi}$$ denotes the coadjoint orbit of ξ. An index $$j \in\{ 1,2,\ldots,n\}$$ is a jump index for ξ if

$$\mathfrak{g}(\xi)+\mathfrak{g}_{j} \neq\mathfrak{g}(\xi )+ \mathfrak{g}_{j-1}.$$

We consider

$$e(\xi)=\{j: j \text{ is a jump index for } \xi\}.$$

This set contains exactly $$\dim(\mathcal{O}_{l})$$ indices. Also, there are two disjoint sets S and T of indices with $$S \cup T =\{1,\ldots,n\}$$ and a G-invariant Zariski open set $$\mathcal{U}$$ of $$\mathfrak {g}^{\ast}$$ such that $$e(\xi)=S$$ for all $$\xi\in\mathcal{U}$$. We define the Pfaffian $$\operatorname{Pf} (\xi)$$ of the skew-symmetric matrix $$M_{S}(\xi)=(\xi([X_{i},X_{j}]))_{i,j\in S}$$ as

$$\bigl\vert \operatorname{Pf}(\xi)\bigr\vert ^{2}= \det{M_{S}(\xi)}.$$

Let $$V_{S}=\mathbb{R}\mbox{-} \operatorname{span}\{X_{i}^{\ast}: i \in S\}$$, $$V_{T}=\mathbb {R}\mbox{-} \operatorname{span}\{X_{i}^{\ast}: i \in T\}$$, and be the Lebesgue measure on $$V_{T}$$ such that the unit cube spanned by $$\{X_{i}^{\ast}:i \in T\}$$ has volume 1. Then $$\mathfrak{g} ^{\ast}=V_{T} \oplus V_{S}$$ and $$V_{T}$$ meets $$\mathcal{U}$$. Let $$\mathcal {W}=\mathcal{U}\cap V_{T}$$ be the cross section for the coadjoint orbits through the points in $$\mathcal{U}$$. If is the Lebesgue measure on $$\mathcal{W}$$, then $$d\mu(\xi )=|\operatorname{Pf} (\xi)| \, d\xi$$ is a Plancherel measure for $$\widehat{G}$$. The Plancherel formula is given by

$$\|f\|_{2}^{2}=\int_{\mathcal{W}}{\bigl\Vert \pi_{\xi}{(f)}\bigr\Vert _{\mathrm{HS}}^{2}}\, d\mu(\xi ), \quad f \in L^{1}\cap L^{2}(G),$$

where $$\|\pi_{\xi}{(f)}\|_{\mathrm{HS}}$$ denotes the Hilbert-Schmidt norm of $$\pi_{\xi}{(f)}$$ and dg is the Haar measure on G.

We shall consider the case in which $$\mathcal{W}$$ takes the following form:

$$\mathcal{W} =\bigl\{ \xi=(\xi_{1},\xi_{2},\ldots, \xi_{n})\in\mathfrak {g}^{\ast}: \xi_{j}=0, \text{for }(n-k)\text{ values of }j\text{ with } \bigl\vert \operatorname{Pf}(\xi)\bigr\vert \neq 0\bigr\} .$$

We denote the vanishing variables by $$\xi_{j_{1}},\xi_{j_{2}},\ldots,\xi _{j_{n-k}}$$.

We consider the class of groups for which for all $$\xi\in\mathcal {W}$$ and $$f \in L^{2}(G)$$ the Hilbert-Schmidt norm $$\|\pi_{\xi}(f)\|_{\mathrm{HS}}^{2}$$ has the following form:

$$\bigl\Vert \pi_{\xi}(f)\bigr\Vert _{\mathrm{HS}}^{2} = \bigl\vert h(\xi)\bigr\vert \int_{\mathbb {R}^{n-k}}{\bigl\vert \mathscr{F}(f\circ\exp) (\xi_{1},\xi_{2}+Q_{2}, \ldots,\xi _{n}+Q_{n} )\bigr\vert ^{2}}\, d \xi_{j_{1}}\, d\xi_{j_{2}} \cdots \, d\xi_{j_{n-k}},$$

where $$\mathscr{F}$$ denotes the Fourier transform on $$\mathbb {R}^{n-k}$$; h is a function from $$\mathcal{W}$$ to $$\mathbb{R}$$ which is nonzero on $$\mathcal{W}$$ and the functions $$Q_{m}=Q_{m}(\xi_{1},\xi_{2},\ldots,\xi_{m-1})$$ with $$2\leq m \leq n$$.

We have the following Heisenberg uncertainty inequality for such groups.

### Theorem 4.1

For any $$f \in L^{1}\cap L^{2}(G)$$ and $$a,b\geq1$$, we have

\begin{aligned} \frac{\|f\|_{2}^{ (\frac{1}{a}+\frac{1}{b} )}}{4\pi} \leq& \biggl(\int_{G}{\|x \|^{2a} \bigl\vert f(x)\bigr\vert ^{2}} \,dx \biggr)^{\frac {1}{2a}} \\ &{}\times\biggl(\int_{\mathcal{W}}{\|\xi \|^{2b} \bigl\Vert \pi_{\xi}(f)\bigr\Vert _{\mathrm{HS}}^{2}} \frac{1}{\vert h(\xi)\vert ^{b}\vert \operatorname{Pf}(\xi)\vert ^{b-1}}\, d\xi \biggr)^{\frac{1}{2b}}. \end{aligned}
(4.1)

### Proof

Assuming both integrals on the right-hand side of (4.1) to be finite, we have

\begin{aligned}& \biggl(\int_{G}{\|x\|^{2} \bigl\vert f(x)\bigr\vert ^{2}} \,dx \biggr)^{1/2} \biggl(\int _{\mathcal{W}}{\|\xi\|^{2} \bigl\Vert \pi_{\xi}(f)\bigr\Vert _{\mathrm{HS}}^{2}} \frac {1}{\vert h(\xi)\vert } \,d\xi \biggr)^{1/2} \\& \quad = \Biggl(\int_{\mathbb{R}^{n}}{\sum_{i=1}^{n}{|x_{i}|^{2}} \Biggl\vert (f\circ\exp) \Biggl(\sum_{i=1}^{n}{x_{i} X_{i}} \Biggr)\Biggr\vert ^{2}} \,dx_{1} \cdots \,dx_{n} \Biggr)^{1/2} \\& \qquad {}\times \Biggl(\int_{\mathbb{R}^{k}}\int_{\mathbb{R}^{n-k}}{ \sum_{i=1}^{n}{|\xi _{i}|^{2}} \bigl\vert \mathscr{F}(f\circ\exp) (\xi_{1},\xi _{2}+Q_{2}, \ldots,\xi _{n}+Q_{n} )\bigr\vert ^{2}} \,d \xi_{1} \cdots \,d\xi_{n} \Biggr)^{1/2} \\& \quad \geq \Biggl(\int_{\mathbb{R}^{n}}{|x_{1}|^{2} \Biggl\vert (f\circ\exp) \Biggl(\sum_{i=1}^{n}{x_{i} X_{i}} \Biggr)\Biggr\vert ^{2}} \,dx_{1} \cdots \,dx_{n} \Biggr)^{1/2} \\& \qquad {}\times \biggl(\int_{\mathbb{R}^{k}}\int_{\mathbb{R}^{n-k}}{| \xi _{1}|^{2} \bigl\vert \mathscr{F}(f\circ\exp) ( \xi_{1},\xi_{2}+Q_{2},\ldots,\xi_{n}+Q_{n} )\bigr\vert ^{2}} \,d\xi_{1} \cdots \,d\xi_{n} \biggr)^{1/2} \\& \quad = \biggl(\int_{\mathbb{R}^{n}}{|x_{1}|^{2} \bigl\vert F (x_{1},\ldots ,x_{n} )\bigr\vert ^{2}} \,dx_{1} \cdots \,dx_{n} \biggr)^{1/2} \\& \qquad {}\times \biggl(\int_{\mathbb{R}^{n}}{|\xi_{1}|^{2} \bigl\vert \widehat {F} (\xi _{1},\xi_{2},\ldots, \xi_{n} )\bigr\vert ^{2}} \,d\xi_{1} \,d \xi_{2} \cdots \,d\xi_{n} \biggr)^{1/2}, \end{aligned}
(4.2)

where $$F(x_{1},\ldots,x_{n})=(f\circ\exp) (\sum_{i=1}^{n}{x_{i} X_{i}} )$$ which is in $$L^{2}(R^{n})$$, $$\widehat{F}$$ being its Fourier transform.

By the Heisenberg inequality on $$\mathbb{R}^{n}$$, we have

\begin{aligned} \frac{\|F\|_{2}^{2}}{4\pi} \leq& \biggl(\int_{\mathbb{R}^{n}}{|x_{1}|^{2} \bigl\vert F (x_{1},\ldots,x_{n} )\bigr\vert ^{2}} \,dx_{1} \cdots \,dx_{n} \biggr)^{1/2} \\ & {}\times \biggl(\int_{\mathbb{R}^{n}}{|\xi_{1}|^{2} \bigl\vert \widehat {F} (\xi _{1},\xi_{2},\ldots, \xi_{n} )\bigr\vert ^{2}} \,d\xi_{1} \,d \xi_{2} \cdots \,d\xi_{n} \biggr)^{1/2}. \end{aligned}
(4.3)

But

\begin{aligned} \|F\|_{2}^{2} =&\int_{\mathbb{R}^{n}}{\bigl|F(x_{1}, \ldots,x_{n})\bigr|^{2}} \,dx_{1} \cdots \,dx_{n} \\ =&\int_{\mathbb{R}^{n}}{\Biggl\vert (f\circ\exp) \Biggl(\sum _{i=1}^{n}{x_{i} X_{i}} \Biggr)\Biggr\vert ^{2}} \,dx_{1} \cdots \,dx_{n}= \int_{G}{\bigl\vert f(x)\bigr\vert ^{2}} \,dx =\|f\|_{2}^{2}. \end{aligned}
(4.4)

Combining (4.2), (4.3), and (4.4), we get

$$\frac{\|f\|_{2}^{2}}{4\pi} \leq \biggl(\int_{G}{\|x\|^{2} \bigl\vert f(x)\bigr\vert ^{2}} \,dx \biggr)^{1/2} \biggl( \int_{\mathcal{W}}{\|\xi\|^{2} \bigl\Vert \pi_{\xi}(f)\bigr\Vert _{\mathrm{HS}}^{2}} \frac{1}{|h(\xi)|} \,d\xi \biggr)^{1/2}.$$
(4.5)

Now, as in the proof of Theorem 3.2, applications of Hölder’s inequality give

$$\int_{G}{\|x\|^{2} \bigl\vert f(x)\bigr\vert ^{2}} \,dx \leq \biggl(\int_{G}{\|x \|^{2a} \bigl\vert f(x)\bigr\vert ^{2}} \,dx \biggr)^{\frac{1}{a}} \bigl(\|f\|_{2}^{2} \bigr)^{1-\frac {1}{a}}$$
(4.6)

and

\begin{aligned}& \int_{\mathcal{W}}{\|\xi\|^{2} \bigl\Vert \pi_{\xi}(f)\bigr\Vert _{\mathrm{HS}}^{2}} \frac {1}{|h(\xi)|} \,d\xi \\& \quad \leq \biggl(\int_{\mathcal{W}}{\|\xi \|^{2b} \bigl\Vert \pi _{\xi}(f)\bigr\Vert _{\mathrm{HS}}^{2}} \frac{1}{|h(\xi)|^{b}|\operatorname {Pf}(\xi)|^{b-1}} \,d\xi \biggr)^{\frac{1}{b}} \bigl(\|f\|_{2}^{2} \bigr)^{1-\frac{1}{b}}. \end{aligned}
(4.7)

Combining (4.5), (4.6), and (4.7), we obtain

$$\frac{\|f\|_{2}^{ (\frac{1}{a}+\frac{1}{b} )}}{4\pi} \leq \biggl(\int_{G}{\|x \|^{2a} \bigl\vert f(x)\bigr\vert ^{2}} \,dx \biggr)^{\frac {1}{2a}} \biggl(\int_{\mathcal{W}}{\|\xi \|^{2b} \bigl\Vert \pi_{\xi}(f)\bigr\Vert _{\mathrm{HS}}^{2}} \frac{1}{|h(\xi)|^{b}|\operatorname{Pf}(\xi)|^{b-1}} \,d\xi \biggr)^{\frac{1}{2b}}.$$

□

### Example 4.2

We now list several classes that are included in the above general class.

1. For thread-like nilpotent Lie groups (for details, see ), we have $$\operatorname{Pf}(\xi)=\xi_{1}$$ and

$$\mathcal{W} =\bigl\{ \xi=(\xi_{1},0,\xi_{3},\ldots, \xi_{n-1},0):\xi_{j} \in\mathbb{R}, \xi_{1}\neq0 \bigr\} .$$

Also, $$\|\pi_{\xi}(f)\|_{\mathrm{HS}}$$ is given by

$$\bigl\Vert \pi_{\xi}(f)\bigr\Vert _{\mathrm{HS}}^{2} = \frac{1}{|\xi_{1}|}\int_{\mathbb{R} ^{2}}{\bigl\vert \mathscr{F} {(f\circ \exp)} (\xi_{1},t,\xi _{3}+Q_{3},\ldots,\xi _{n-1}+Q_{n-1},s )\bigr\vert ^{2}} \, ds\, dt,$$

where $$Q_{j}(\xi_{1},0,\xi_{3},\ldots,\xi_{j-1},t)=\sum_{k=1}^{j-1}{\frac{1}{k!} \frac{t^{k}}{\xi_{1}^{k}} \xi_{j-k}}$$, for $$3\leq j\leq n-1$$.

Thus, for $$h(\xi)=\frac{1}{|\xi_{1}|}=\frac{1}{|\operatorname {Pf}(\xi)|}$$, one obtains the Heisenberg uncertainty inequality

$$\frac{\|f\|_{2}^{ (\frac{1}{a}+\frac{1}{b} )}}{4\pi} \leq \biggl(\int_{G}{\|x \|^{2a} \bigl\vert f(x)\bigr\vert ^{2}} \,dx \biggr)^{\frac {1}{2a}} \biggl(\int_{\mathcal{W}}{\|\xi \|^{2b} \bigl\Vert \pi_{\xi}(f)\bigr\Vert _{\mathrm{HS}}^{2}} |\xi_{1}| \,d\xi \biggr)^{\frac{1}{2b}}.$$

2. For 2-NPC nilpotent Lie groups (for details, see ), let $$\{0\}=\mathfrak{g}_{0} \subset\mathfrak {g}_{1} \subset\cdots \subset\mathfrak{g}_{n} =\mathfrak{g}$$ be a Jordan-Hölder sequence in $$\mathfrak{g}$$ such that $$\mathfrak{g}_{m}=\mathfrak{z}(g)$$ and $$\mathfrak{h}=\mathfrak {g}_{n-2}$$. Let us consider the ideal $$[\mathfrak{g},\mathfrak{g} _{m+1}]$$ of $$\mathfrak{g}$$ which is one or two dimensional in $$\mathfrak{g}$$. We discuss the two cases separately:

(a) $$\dim{[\mathfrak{g},\mathfrak{g}_{m+1}]}=2$$.

In this case, for every basis $$\{X_{1},X_{2}\}$$ of $$\mathfrak{h}$$ in $$\mathfrak{g}$$ and every $$Y_{1} \in\mathfrak{g}_{m+1}\setminus\mathfrak{z}(\mathfrak {g})$$, the vectors $$Z_{1}=[X_{1},Y_{1}]$$ and $$Z_{2}=[X_{2},Y_{1}]$$ are linearly independent and lie in the center of $$\mathfrak{g}$$. Assume that $$\mathfrak{g}_{1}=\mathbb {R}\mbox{-} \operatorname{span}\{Z_{1}\}$$, $$\mathfrak{g}_{2}=\mathbb{R} \mbox{-} \operatorname{span}\{Z_{1},Z_{2}\}$$. Let $$Z_{3},\ldots,Z_{m}$$ be some vectors such that $$\mathfrak{z}(\mathfrak{g})=\mathbb{R}\mbox{-} \operatorname{span}\{Z_{1},\ldots,Z_{m}\}$$ and $$\mathcal{B}=\{Z_{1},\ldots,Z_{n}\}$$ a Jordan-Hölder basis of $$\mathfrak{g}$$ chosen as follows:

1. (i)

$$\mathfrak{z}(\mathfrak{g})=\mathbb{R}\mbox{-} \operatorname{span}\{Z_{1},\ldots ,Z_{m}\}$$;

2. (ii)

$$\mathfrak{h}=\mathbb{R}\mbox{-} \operatorname{span}\{Z_{1},\ldots,Z_{n-2}\}$$;

3. (iii)

$$\mathfrak{g}=\mathbb{R}\mbox{-} \operatorname{span}\{Z_{1},\ldots,Z_{n-2}, X_{1}=Z_{n-1},X_{2}=Z_{n}\}$$.

For $$m_{1}=m+1$$ and $$m+2\leq m_{2} \leq n-2$$, we denote $$Z_{m_{1}}=Z_{m+1}=Y_{1}$$, $$Z_{m_{2}}=Y_{2}$$. These vectors can be chosen such that $$\xi_{1}=\xi([X_{1},Y_{1}])\neq0$$, $$\xi_{2,2}=\xi([X_{2},Y_{2}])\neq 0$$, for all $$\xi\in\mathcal{W}$$, where

$$\mathcal{W} =\bigl\{ \xi=(\xi_{1},\xi_{2},\ldots, \xi_{m},0,0,\xi_{m+3},\xi _{m+4},\ldots, \xi_{n-2},0,0):\xi_{j} \in\mathbb{R}, \bigl\vert \operatorname {Pf}(\xi)\bigr\vert \neq 0\bigr\} .$$

Also, we have $$\operatorname{Pf}(\xi)=\xi(Z_{1}) \xi([X_{2},Y_{2}])-\xi ([X_{1},Y_{2}]) \xi(Z_{2})$$ and $$\|\pi_{\xi}(f)\|_{\mathrm{HS}}$$ is given by

\begin{aligned} \bigl\Vert \pi_{\xi}(f)\bigr\Vert _{\mathrm{HS}}^{2} =& \bigl\vert h(\xi)\bigr\vert \int_{\mathbb{R}^{4}} \biggl\vert \mathscr{F} {(f\circ\exp)}\biggl(s_{2},s_{1},P_{n-2} \biggl(\xi,-\frac{t_{1}}{\tilde {\xi}_{1,1}},-\frac{t_{2}}{\tilde{\xi}_{2,2}} \biggr),\ldots, \\ & P_{m+3} \biggl(\xi,-\frac{t_{1}}{\tilde{\xi }_{1,1}},-\frac{t_{2}}{\tilde{\xi}_{2,2}} \biggr),t_{2},t_{1},\xi _{m},\ldots, \xi_{1}\biggr)\biggr\vert ^{2} \, ds_{1}\, ds_{2}\, dt_{1}\, dt_{2}, \end{aligned}

where h is the function defined by

$$h(\xi) =\frac{|\xi_{1}\xi_{2,2}|^{2}}{|\xi_{1}\xi_{2,2}-\xi _{1,2}\xi_{2}|^{2}},$$

$$\xi_{i,j}=\xi([X_{i},Y_{j}])$$, $$\tilde{\xi}_{i,j}=\xi([X_{i}(\xi ),Y_{j}])$$, and $$P_{j}(\xi,t)$$ is a polynomial function with respect to the variables $$t=(t_{1},t_{2})$$ and $$\xi_{m+1},\ldots,\xi_{j}$$ and rational in the variables $$\xi_{1},\ldots,\xi_{m}$$. Thus, one obtains the Heisenberg uncertainty inequality

$$\frac{\|f\|_{2}^{ (\frac{1}{a}+\frac{1}{b} )}}{4\pi} \leq \biggl(\int_{G}{\|x \|^{2a} \bigl\vert f(x)\bigr\vert ^{2}} \,dx \biggr)^{\frac {1}{2a}} \biggl(\int_{\mathcal{W}}{\|\xi \|^{2b} \bigl\Vert \pi_{\xi}(f)\bigr\Vert _{\mathrm{HS}}^{2}} \frac{1}{|h(\xi)|^{b}|\operatorname{Pf}(\xi)|^{b-1}} \,d\xi \biggr)^{\frac{1}{2b}}.$$

(b) $$\dim{[\mathfrak{g},\mathfrak{g}_{m+1}]}=1$$.

In this case, we have $$\operatorname{Pf}(\xi)=\xi([X_{1},Y_{1}])\cdot \xi([X_{2},Y_{2}])$$ and

\begin{aligned} \mathcal{W} =&\bigl\{ \xi=(\xi_{1},\xi_{2},\ldots, \xi_{m},0,\xi _{m+2},\ldots ,\xi_{m+d+1},0, \xi_{m+d+3},\ldots,\xi_{n-2},0,0): \\ &\xi_{j} \in\mathbb{R}, \bigl\vert \operatorname {Pf}(\xi)\bigr\vert \neq 0\bigr\} . \end{aligned}

Also, $$\|\pi_{\xi}(f)\|_{\mathrm{HS}}$$ is given by

\begin{aligned} \bigl\Vert \pi_{\xi}(f)\bigr\Vert _{\mathrm{HS}}^{2} =&\frac{1}{|\operatorname{Pf}(\xi)|}\int_{\mathbb{R}^{4}}\biggl\vert \mathscr{F} {(f \circ\exp)}\biggl(s_{2},s_{1},P_{n-2} \biggl(\xi ,- \frac{t_{1}}{\xi_{1}},-\frac{t_{2}+R(-\frac{t_{1}}{\xi_{1}},\xi _{1},\ldots,\xi_{m+d})}{\xi_{2,2}}\biggr), \\ &\ldots,t_{2},\ldots,P_{m+2} \biggl(\xi,-\frac{t_{1}}{\xi_{1}} \biggr),t_{1},\xi_{m},\ldots,\xi_{1}\biggr)\biggr\vert ^{2}\, ds_{1}\, ds_{2}\, dt_{1}\, dt_{2}. \end{aligned}

Thus, for $$h(\xi)=\frac{1}{|\operatorname{Pf}(\xi)|}$$, one obtains the Heisenberg uncertainty inequality,

$$\frac{\|f\|_{2}^{ (\frac{1}{a}+\frac{1}{b} )}}{4\pi} \leq \biggl(\int_{G}{\|x \|^{2a} \bigl\vert f(x)\bigr\vert ^{2}} \,dx \biggr)^{\frac {1}{2a}} \biggl(\int_{\mathcal{W}}{\|\xi \|^{2b} \bigl\Vert \pi_{\xi}(f)\bigr\Vert _{\mathrm{HS}}^{2}} \bigl\vert \operatorname{Pf}(\xi)\bigr\vert \,d\xi \biggr)^{\frac{1}{2b}}.$$

3. For connected, simply connected nilpotent Lie groups $$G=\exp {\mathfrak{g} }$$ such that $$\mathfrak{g}(\xi) \subset[\mathfrak{g},\mathfrak {g}]$$ for all $$\xi\in\mathcal{U}$$ (for details, see ), we consider $$S=\{ j_{1}<\cdots<j_{d}\}$$ and $$T=\{t_{1}<\cdots<t_{r}\}$$ to be the collection of jump and non-jump indices, respectively, with respect to the basis $$\mathcal{B}$$. We have $$j_{d}=n$$ and

$$\mathcal{W} =\bigl\{ \xi=(\xi_{1},\xi_{2},\ldots, \xi_{n})\in\mathfrak {g}^{\ast}: \xi _{j_{i}}=0, \text{for }j_{i} \in S\text{ with } \bigl\vert \operatorname{Pf}(\xi) \bigr\vert \neq 0\bigr\} .$$

Also, $$\|\pi_{\xi}(f)\|_{\mathrm{HS}}$$ is given by

$$\bigl\Vert \pi_{\xi}(f)\bigr\Vert _{\mathrm{HS}}^{2} = \frac{|\xi ([X_{j_{1}},X_{n}])|}{|\operatorname{Pf} (\xi)|^{2}}\int_{\mathcal{W}}{\bigl\vert \mathscr{F} {(f\circ \exp )} (\xi,w )\bigr\vert ^{2}} \,dw,$$

where $$\xi=(\xi_{t_{i}})_{t_{i}\in T}$$ and $$w=(w_{j_{i}})_{j_{i}\in S}$$. Thus, for $$h(\xi)=\frac{|\xi([X_{j_{1}},X_{n}])|}{|\operatorname {Pf}(\xi)|^{2}}$$, one obtains the Heisenberg uncertainty inequality

$$\frac{\|f\|_{2}^{ (\frac{1}{a}+\frac{1}{b} )}}{4\pi} \leq \biggl(\int_{G}{\|x \|^{2a} \bigl\vert f(x)\bigr\vert ^{2}} \,dx \biggr)^{\frac {1}{2a}} \biggl(\int_{\mathcal{W}}{\|\xi \|^{2b} \bigl\Vert \pi_{\xi}(f)\bigr\Vert _{\mathrm{HS}}^{2}} \frac{|\operatorname{Pf}(\xi)|^{b+1}}{|\xi ([X_{j_{1}},X_{n}])|^{b}} \,d\xi \biggr)^{\frac{1}{2b}}.$$

4. For low-dimensional nilpotent Lie groups of dimension less than or equal to 6 (for details, see ) except for $$G_{6,8}$$, $$G_{6,12}$$, $$G_{6,14}$$, $$G_{6,15}$$, $$G_{6,17}$$, an explicit form of $$\|\pi_{\xi}(f)\|_{\mathrm{HS}}$$ can be obtained. Thus, an explicit Heisenberg uncertainty inequality can be written down.

5. The classes mentioned above are distinct. For instance, $$G_{5,5}$$ is thread-like nilpotent Lie group, but it does not belong to the class mentioned in item 3. above. Also, $$G_{5,3}$$ belongs to the class mentioned in item 3. above, but it is not a thread-like nilpotent Lie group.

## References

1. Folland, GB, Sitaram, A: The uncertainty principle: a mathematical survey. J. Fourier Anal. Appl. 3(3), 207-238 (1997)

2. Thangavelu, S: Some uncertainty inequalities. Proc. Indian Acad. Sci. Math. Sci. 100(2), 137-145 (1990)

3. Sitaram, A, Sundari, M, Thangavelu, S: Uncertainty principles on certain Lie groups. Proc. Indian Acad. Sci. Math. Sci. 105, 135-151 (1995)

4. Xiao, J, He, J: Uncertainty inequalities for the Heisenberg group. Proc. Indian Acad. Sci. Math. Sci. 122(4), 573-581 (2012)

5. Sarkar, RP, Thangavelu, S: On the theorems of Beurling and Hardy for the Euclidean motion group. Tohoku Math. J. 57, 335-351 (2005)

6. Kumahara, K, Okamoto, K: An analogue of the Paley-Wiener theorem for the Euclidean motion group. Osaka J. Math. 10, 77-92 (1973)

7. Corwin, L, Greenleaf, FP: Representations of Nilpotent Lie Groups and Their Applications: Part I. Basic Theory and Examples. Cambridge University Press, Cambridge (1990)

8. Kaniuth, E, Kumar, A: Hardy’s theorem for simply connected nilpotent Lie groups. Math. Proc. Camb. Philos. Soc. 131, 487-494 (2001)

9. Baklouti, A, Salah, NB: On theorems of Beurling and Cowling-Price for certain nilpotent Lie groups. Bull. Sci. Math. 132, 529-550 (2008)

10. Smaoui, K: Beurling’s theorem for nilpotent Lie groups. Osaka J. Math. 48, 127-147 (2011)

11. Nielson, OA: Unitary Representations and Coadjoint Orbits of Low-Dimensional Nilpotent Lie Groups. Queens Papers in Pure and Appl. Math. Queen’s University, Kingston (1983)

## Acknowledgements

The second author was supported by R & D grant of University of Delhi. The authors would like to thank the referees for many valuable suggestions which helped in improving the exposition.

## Author information

Authors

### Corresponding author

Correspondence to Ajay Kumar.

### Competing interests

The authors declare that they have no competing interests.

### Authors’ contributions

All authors contributed equally to this paper and they read and approved the final manuscript.

## Rights and permissions 