1 Introduction

Let G be a separable unimodular locally compact group of type I and \(\widehat{G}\) its unitary dual, the set of equivalence classes of irreducible unitary representations, endowed with the Mackey–Borel structure. Let dg be the Haar measure on G. The operator valued Fourier transform on G maps each \(\varphi \in L^1(G)\) to the field \({\mathscr {F}}\varphi =(\pi (\varphi ))_{\pi \in \widehat{G}}\) of bounded operators on \(\widehat{G}\), where \( \pi (\varphi ) \) is defined by:

$$\begin{aligned} \pi ( \varphi ) = \int _G \varphi ( g)\pi (g) dg . \end{aligned}$$

Let \(\mu _G \) be the Plancherel measure on \(\widehat{G}\), which is uniquely determined by the abstract Plancherel formula: for \(\varphi \in L^1(G)\cap L^2(G)\),

$$\begin{aligned} \int _G\vert \varphi (g)\vert ^2 dg=\int _{\widehat{G}}\hbox {tr}((\pi (\varphi ) ^{*}\pi (\varphi )) d\mu _G(\pi ). \end{aligned}$$

The Hausdorff–Young inequality, generalized by Kunze in [15] for separable locally compact unimodular groups of type I, is the assertion

$$\begin{aligned} \Big (\int _{\widehat{G}}\Vert \pi (\varphi )\Vert _{c_q}^qd\mu _G (\pi )\Big )^{1\over q}\le \Big ( \int _G\vert \varphi (g) \vert ^pdg\Big )^{1\over p}, \end{aligned}$$
(1.1)

where \(\varphi \in L^1(G)\cap L^p(G),\ 1<p\le 2,\ q \ \hbox {is the conjugate of}\ p,\ \hbox {i.e.,}\ {1\over p}+{1\over q}=1\), and \(\Vert \pi (\varphi )\Vert _{c_q}\) is the q-th Schatten norm of the operator \(\pi (\varphi )\)

$$\begin{aligned} \Vert \pi (\varphi )\Vert _{c_q}^q=\hbox {tr}\Big ( (\pi (\varphi )^*\pi (\varphi ))^{q\over 2}\Big ). \end{aligned}$$

Let \(L^q(\widehat{G})\) be the Banach space of \(\mu _G\)-measurable field of bounded operators F on \(\widehat{G}\) with the norm \(\Vert F\Vert _q<\infty \), where

$$\begin{aligned} \Vert F\Vert _q=\left( \int _{\widehat{G}}\Vert F(\pi ) \Vert ^q_{c_q}d\mu _G(\pi )\right) ^{{1\over q}}. \end{aligned}$$

Then we can write (1.1) as

$$\begin{aligned} \Vert { {\mathscr {F}}}\varphi \Vert _q\le \Vert \varphi \Vert _p. \end{aligned}$$

Thus the map \(\varphi \mapsto {{\mathscr {F}}}\varphi \) from \(L^1(G)\cap L^p(G)\) to \(L^q(\widehat{G})\) extends to a continuous operator \({{\mathscr {F}}}^p: L^p(G)\rightarrow L^q(\widehat{G})\) and its operator norm

$$\begin{aligned} \Vert { {\mathscr {F}}}^p(G)\Vert =\sup _{\Vert \varphi \Vert _p\le 1}\Vert { {\mathscr {F}}}\varphi \Vert _q\le 1. \end{aligned}$$

For the abelian group \(G={\mathbb {R}}^n\), it is proved that

$$\begin{aligned} \Vert { {\mathscr {F}}}^p({\mathbb {R}}^n)\Vert =A_p^n, \quad A_p=\sqrt{{p^{1\over p}}\over {q^{1\over q}}} \end{aligned}$$

by Babenko [1] for \(p=\frac{2k}{2k-1}\), where \(k\ge 2\) is an integer, and by Beckner [6] for all p\((1<p<2)\) .

On the other hand, it is shown by Fournier [8], that \(\Vert { {\mathscr {F}}}^p(G)\Vert =1\) if and only if G contains a compact open subgroup. For various classes of non-abelian, non-compact groups, further studies of Hausdorff–Young inequalities were made and many results concerning better estimates of \(\Vert { {\mathscr {F}}}^p(G) \Vert \) were obtained. See, e.g., [2,3,4,5, 7, 10, 12, 13, 16, 18,19,20,21,22].

Let G be a compact extension of a closed normal subgroup N, and suppose N is unimodular and type I. Then Russo obtained the estimate \(\Vert {\mathscr {F}}^p(G)\Vert \le \Vert {\mathscr {F}}^p(N)\Vert \) for \(p=\frac{2k}{2k-1}\), where \(k\ge 2\) is an integer [21, Theorem2]. In [13], Klein and Russo proved that if \(G=X < imes A\) is a semidirect product of locally compact unimodular groups X and A and if G is unimodular, then the inequality \(\Vert {\mathscr {F}}^p(G)\Vert \le \Vert {\mathscr {F}}^p(X) \Vert \Vert {\mathscr {F}}^p(A)\Vert \) holds for \(p=\frac{2k}{2k-1}\) with \(k\ge 2\) an integer. In particular, when G is the \(2n+1\)-dimensional Heisenberg group and \(p=\frac{2k}{2k-1}\) with \(k\ge 2\) an integer, they obtained \(\Vert {\mathscr {F}}^p(G)\Vert =A_p^{2n+1}\) and showed that there are no extremal functions [13, Theorem 3].

Let us mention that in [3], we treated the case \(G=K < imes {\mathbb {R}}^n\), a semidirect product of a vector group \(N={\mathbb {R}}^n\) and a compact group K, and obtained the norm \(\Vert {\mathscr {F}}^p(G)\Vert =A_p^n \ (=\Vert {\mathscr {F}}^p({\mathbb {R}}^n)\Vert )\) for all p satisfying \(1<p<2\). In this case, we have that an extension of the Gaussian function on \({\mathbb {R}}^n\) to G gives an extremal function.

We remark that the Hausdorff–Young theorem is generalized for non-unimodular groups by Terp [22], and there are several results treating some non-unimodular groups, e.g., in [4, 7, 10, 12, 13, 20].

Concerning group extensions, Führ [10] obtained the following estimate in a setting of non-unimodular groups: Let a group extension \(1\rightarrow N \rightarrow G \rightarrow H \rightarrow 1\) be given with N, H unimodular, N a type I group, and G has a type I regular representation. Letting \(\nu _N\) be the Plancherel measure on \(\widehat{N}\), assume that

  1. (A)

    There exists a Borel \(\nu _N\)-conull subset \(U\subset \widehat{N}\) with the following property: U is H-invariant with U/H standard. Moreover, for all \(\gamma \in U\), the induced representation \({{\,\mathrm{Ind}\,}}_N^G\gamma \in \widehat{G}\).

Then the inequality \(\Vert {\mathscr {F}}^p(G)\Vert \le \Vert {\mathscr {F}}^p(N)\Vert \) holds [10, Theorem 2].

In this paper, we assume that G is unimodular and we shall provide a generalization of Russo’s result [21, Theorem 2] of compact extensions for \(p=\frac{2k}{2k-1}\) to that for general exponents p satisfying \(1<p<2\). Our main result is the following:

Theorem 1.1

Let G be a separable unimodular locally compact group of type I, and let N be a closed normal subgroup of G. Suppose that N is unimodular and type I, and that G/N is compact. Then for \(1<p\le 2\), we have

$$\begin{aligned} \Vert {\mathscr {F}}^p(G)\Vert \le \Vert {\mathscr {F}}^p(N )\Vert . \end{aligned}$$
(1.2)

We next treat the case where G is a separable unimodular locally compact group of type I defined by \(G:=K < imes N\), where N is a separable unimodular locally compact group of type I and K stands for a compact subgroup of the automorphism group of N, equipped with the group law

$$\begin{aligned} (k_1,n_1).(k_2,n_2)=(k_1k_2,n_1k_1(n_2)),\ ((k_1,n_1),(k_2,n_2))\in G\times G. \end{aligned}$$

We shall provide a sufficient condition, under which the equality holds in (1.2). It is a generalization of our previous result [3, Theorem 3.1]. We have the following:

Theorem 1.2

Let \(G=K < imes N\) be a semidirect product of a separable unimodular locally compact group N of type I and a compact subgroup K of the automorphism group of N. Then for \(1<p\le 2\), we have the equality \(\Vert {\mathscr {F}}^p(G)\Vert = \Vert {\mathscr {F}}^p(N )\Vert \) if in addition N has a K-invariant sequence of functions \((\varphi _j)_{j\in \mathbb {N}}\) in \(L^1(N)\cap L^p(N)\) such that \({\Vert {\mathscr {F}}\varphi _j \Vert _q}\over {\Vert \varphi _j \Vert _p}\) tends to \(\Vert {\mathscr {F}}^p(N )\Vert \) when j goes to infinity.

Remark 1.3

In general, we may have \(\Vert {\mathscr {F}}^p(G)\Vert \ne \Vert {\mathscr {F}}^p(N )\Vert \): Let \(G={\mathbb {R}}\) and \(N={\mathbb {Z}}\). Then G/N is the torus group \(\mathbb T\), and \({\mathbb {R}}\) is a compact extension of \({\mathbb {Z}}\). The Hausdorff–Young inequality for Fourier series tells us that the \(L^p\)-Fourier transform on \({\mathbb {Z}}\) with the dual \(\widehat{\mathbb Z}\simeq \mathbb T\) is described by

$$\begin{aligned}&\mathscr {F}f(\gamma )=\sum _{n\in {\mathbb {Z}}}f(n)\gamma ^n, \quad \gamma \in \mathbb T, \quad f\in L^p({\mathbb {Z}})\cap L^1({\mathbb {Z}}). \end{aligned}$$
(1.3)

We have that \(\Vert \mathscr {F}^p({\mathbb {Z}})\Vert =1\): In fact, let f be a function on \({\mathbb {Z}}\) defined by \(f(0)=1\), \(f(m)=0\) for \(m\ne 0\). Then \(\Vert f\Vert _p=1\) and \(\Vert \mathscr {F}f\Vert _q=1\). Therefore, we have \(\Vert {\mathscr {F}}^p({\mathbb {R}})\Vert =A_p<\Vert {\mathscr {F}}^p({\mathbb {Z}})\Vert =1\) for \(1<p<2\).

In this example, the group \(G/N={\mathbb {R}}/{\mathbb {Z}}\) trivially acts on \(N={\mathbb {Z}}\) by conjugation, and thus the extremal function f for \(\mathscr {F}^p(N)\) is invariant under the action. Nevertheless, the equality does not hold for \(1<p<2\).

Remark 1.4

1. Remark first that in [21, Lemma 4.1], Russo obtained the inequality \(\Vert \mathscr {F}^p(G/K)\Vert \le \Vert \mathscr {F}^p(G)\Vert \) for \(1<p<2\) when G is a unimodular locally compact group of type I and K is a compact normal subgroup of G. Concerning the case of a direct product group \(G=K\times N\), the inequality \(\Vert \mathscr {F}^p(N)\Vert \le \Vert \mathscr {F}^p(G)\Vert \) for \(1<p<2\) follows from this result. Hence, for \(p=\frac{2k}{2k-1}\) with \(k\ge 2\) an integer, Russo’s results [21, Theorem 2] and [21, Lemma 4.1] give the equality \(\Vert {\mathscr {F}}^p(G)\Vert = \Vert {\mathscr {F}}^p(N)\Vert \).

For arbitrary \(1<p\le 2\), let \(G=K\times N\) be a direct product of a separable unimodular locally compact group N of type I and a compact group K. Then, since K trivially acts on N by conjugation, we can apply Theorem 1.2 and obtain the equality \(\Vert {\mathscr {F}}^p(K\times N)\Vert = \Vert {\mathscr {F}}^p(N )\Vert \).

2. Now, we also remark that in the proof of [18, Theorem 2], Russo obtained the inequality \(\Vert \mathscr {F}^p({\mathbb {R}}\times H)\Vert \le A_p\Vert \mathscr {F}^p(H)\Vert \) for a direct product of \({\mathbb {R}}\) and an arbitrary unimodular locally compact group H and \(1<p<2\). For the case that \(H:=K\) is a compact group, this inequality and [21, Lemma 4.1] concerning a direct product \(G=K\times {\mathbb {R}}\) give the equality \(\Vert \mathscr {F}^p(K\times {\mathbb {R}})\Vert =A_p\)\((=\Vert \mathscr {F}^p({\mathbb {R}})\Vert )\) for \(1<p<2\).

2 Backgrounds and Notations

2.1 Compact Extensions of Unimodular Locally Compact Groups

Let G be a separable unimodular locally compact group of type I, and let N be a closed normal subgroup. We assume that N is unimodular and type I, and that G/N is compact.

We recall the Plancherel formula for group extensions obtained by Kleppner and Lipsman [14] and sketch it for our case of compact extensions. For the complete description, we refer the reader to [14].

The unitary dual of G is described by Mackey with the little group method. (See e.g., [17].) For an irreducible representation \(\gamma \) of N, we denote by \(G_\gamma \) the stabilizer of \(\gamma \) under the action of G on the unitary dual \(\widehat{N}\) of N, given by \(( g\cdot \gamma )(n)=\gamma (g^{-1}n g)\) for \(g\in G\) and \(n\in N\). Then there exists a multiplier \(\sigma _\gamma \) on \(G_\gamma /N\) such that \(\gamma \) extends to a \(\sigma '_\gamma \)-representation \(\gamma '\) of \(G_\gamma \), where \(\sigma '_\gamma \) is the lift of \(\sigma _\gamma \) to \(G_\gamma \). Denote by \((\widehat{G_\gamma /N})^{\overline{\sigma }_\gamma }\) the set of irreducible \(\overline{\sigma }_\gamma \)-representations of \(G_\gamma /N\). Let \(\rho \in (\widehat{G_\gamma /N})^{\overline{\sigma }_\gamma }\) and \(\rho '\) be its lift to \(G_\gamma \). Then \(\pi _{\gamma , \rho }:={{\,\mathrm{Ind}\,}}_{G_\gamma }^G \gamma '\otimes \rho '\) is irreducible and every irreducible unitary representation of G is equivalent to some \(\pi _{\gamma ,\rho }\) by Mackey’s theory. Let \(\check{G}_\gamma \) be the set of \(\tau \in \widehat{G}_\gamma \) such that \(\tau |_{N}\) is a multiple of \(\gamma \). Then every \(\tau \in \check{G}_\gamma \) is described by \(\tau =\gamma '\otimes \rho '\) and the multiplicity \(n_\gamma (\tau )=\dim (\rho )\) of \(\gamma \) in \(\tau |_N\) is finite.

Under the assumptions above, we have the following Proposition:

Proposition 2.1

[14, Lemma 4.2] Let \(\gamma \in \widehat{N}\). Then we have

$$\begin{aligned} {{\,\mathrm{Ind}\,}}_N^G \gamma \simeq \underset{\tau \in \check{G}_\gamma }{\oplus } n_\gamma (\tau ){{\,\mathrm{Ind}\,}}_{G_\gamma }^G{\tau }. \end{aligned}$$

Let dn and \(d\nu \) be Haar measures on N and G/N normalized by

$$\begin{aligned} \int _G f(g)\> dg=\int _{G/N}\int _N f(gn)\> dn \> d\nu (g) \text{ for } f\in L^1(G). \end{aligned}$$

Letting \(d\mu _N\) be the Plancherel measure on \(\widehat{N}\) normalized by

$$\begin{aligned} \int _{N}|\psi (n)|^2 \> dn =\int _{\widehat{N}}\mathrm {tr}\>(\gamma (\psi )^*\gamma (\psi )) d\mu _N(\gamma ) \text{ for } \psi \in L^1(N)\cap L^2(N), \end{aligned}$$

Let \(\overline{\mu }_N\) be the image of the Plancherel measure \(\mu _N\) on \(\widehat{N}\) by the canonical projection \(\widehat{N}\ni \gamma \mapsto \bar{\gamma }:=G\cdot \gamma \in \widehat{N}/G\) so that

$$\begin{aligned} \int _{\widehat{N}}\varphi (\gamma ) d\mu _N(\gamma )=\int _{\widehat{N}/G}\int _{G/N}\varphi (g\cdot \gamma ) d\nu (g)d\bar{\mu }_N(\bar{\gamma }) \text{ for } \varphi \in L^1(\widehat{N}). \end{aligned}$$
(2.1)

For any \(f \in L^1(G)\cap L^2(G)\), \(\gamma \in \widehat{N}\) and \(\rho \in (\widehat{G_\gamma /N})^{\overline{\sigma }_\gamma }\), the operator \(\pi _{\gamma ,\rho }(f)\) is a Hilbert–Schmidt operator. We denote by \(\Vert \cdot \Vert _{\mathrm {HS}}\) the Hilbert–Schmidt norm.

The Plancherel measure on \(\widehat{G}\) is described by the following formula:

Theorem 2.2

[14, Theorem 4.4] For any \(f\in L^1(G)\cap L^2(G)\)

$$\begin{aligned} \int _{G}|f(g)|^2dg=\int _{\widehat{N}/G} \underset{\rho \in (\widehat{G_\gamma /N})^{\bar{\sigma }_\gamma }}{\sum }\Vert \pi _{\gamma ,\rho }(f)\Vert ^2_{\mathrm {HS}} \dim (\rho ) \> d\bar{\mu }_N(\bar{\gamma }). \end{aligned}$$
(2.2)

Denoting \(\pi _\gamma ={{\,\mathrm{Ind}\,}}_N^G\gamma \) for \(\gamma \in \widehat{N}\), let us remark that by Proposition 2.1 we have

$$\begin{aligned} \pi _\gamma \simeq \underset{\tau \in \check{G}_\gamma }{\oplus } n_\gamma (\tau ){{\,\mathrm{Ind}\,}}_{G_\gamma }^G{\tau } =\underset{\rho \in (\widehat{G_\gamma /N})^{\overline{\sigma }_\gamma }}{\oplus } \dim (\rho ) \pi _{\gamma ,\rho } \end{aligned}$$

and that the equality

$$\begin{aligned} \int _{\widehat{N}/G}\Vert \pi _\gamma (f)\Vert _{\mathrm {HS}}^2d\bar{\mu }_N(\bar{\gamma }) =\int _{\widehat{N}/G} \underset{\rho \in (\widehat{G_\gamma /N})^{\bar{\sigma }_\gamma }}{\sum }\Vert \pi _{\gamma ,\rho }(f)\Vert ^2_{\mathrm {HS}}\dim (\rho ) \> d\bar{\mu }_N(\bar{\gamma }). \end{aligned}$$
(2.3)

is obtained in the course of the proof of [14, Theorem 4.4].

2.2 \(L^p\)-Fourier Transforms

Let \(1<p\le 2\) and \(f\in L^1(G)\cap L^p(G)\). Our description of the \(L^p\)-Fourier transform is based on the Plancherel theorem given in Theorem 2.2.

$$\begin{aligned} \Vert \mathscr {F}f\Vert _q^q:= \int _{\widehat{N}/G} \underset{\rho \in (\widehat{G_\gamma /N})^{\overline{\sigma }_\gamma }}{\sum }\Vert \pi _{\gamma ,\rho }(f)\Vert ^q_{c_q} \dim (\rho ) \> d\bar{\mu }_N(\bar{\gamma }). \end{aligned}$$
(2.4)

We have an analogue of the computation of (2.3) above as follows: By Proposition 2.1 we have that the Schatten norm of \(\pi _\gamma (f)\) for \(f\in L^1(G)\cap L^p(G)\) is described by

$$\begin{aligned} \Vert \pi _\gamma (f)\Vert _{c_q}^q=\sum _{\rho \in (\widehat{G_\gamma /N})^{\overline{\sigma }_\gamma }} \Vert \pi _{\gamma ,\rho }(f)\Vert _{c_q}^q\dim (\rho ). \end{aligned}$$

Thus we also obtain that the norm \(\Vert \mathscr {F}f\Vert _q\) is computed by

$$\begin{aligned} \Vert \mathscr {F}f\Vert _q^q= \int _{\widehat{N}/G}\Vert \pi _\gamma (f)\Vert _{c_q}^q \> d\bar{\mu }_N(\bar{\gamma }). \end{aligned}$$
(2.5)

2.3 Minkowski’s Inequality for Integrals

Let \((X,\mu )\) and \((Y,\nu )\) be two measure spaces and f a measurable function on \(X\times Y\). Then for any \(r > 1\), we have (See, e.g., [11].):

$$\begin{aligned} \left( \int _{X}\left\{ \int _{Y}|f(x,y)|d\nu (y)\right\} ^rd\mu (x)\right) ^{\frac{1}{r}} \le \int _Y\left( \int _X|f(x,y)|^rd\mu (x)\right) ^{\frac{1}{r}}d\nu (y). \end{aligned}$$
(2.6)

2.4 The Hausdorff–Young Inequality for Integral Operators

In order to estimate the norm, we shall use the Hausdorff–Young inequality of integral operators due to Fournier and Russo [9]. Let X be a separable locally compact Hausdorff space and \(\mu \) a positive, \(\sigma \)-finite regular Borel measure on X. Let also \(\mathscr {H}=(\mathscr {H}, \langle \cdot ,\cdot \rangle )\) be a separable Hilbert space and \(L^2(X,\mathscr {H})\) the Hilbert space of \(\mathscr {H}\)-valued square integrable functions on X. We denote by \(\Vert \cdot \Vert \) the norm on \(L^2(X, \mathscr {H})\) associated with the inner product \(\langle \cdot ,\cdot \rangle \) of \(\mathscr {H}\). Let \(\mathscr {K}\) be a measurable \(\mathscr {B}(\mathscr {H})\)-valued function on \(X\times X\), where \(\mathscr {B}(\mathscr {H})\) is the space of bounded linear operators on \(\mathscr {H}\). If there exists a constant c such that

$$\begin{aligned} \iint _{X\times X} |\langle \mathscr {K}(x,y)\phi _1(y),\phi _2(x)\rangle | d\mu (y)d\mu (x) \le c\Vert \phi _1\Vert \Vert \phi _2\Vert \end{aligned}$$

for any \(\phi _1,\phi _2\in L^2(X,\mathscr {H})\), then we define an operator K on \(L^2(X,\mathscr {H})\) by

$$\begin{aligned} K\phi (x)=\int _X \mathscr {K}(x,y)\phi (y) d\mu (y) \end{aligned}$$

for \(\phi \in L^2(X,\mathscr {H})\) and \(\mu \)-almost all \(x\in X\). Now we define

$$\begin{aligned} \Vert \mathscr {K}\Vert _{q,p,q}=\left( \int _X \left\{ \int _X(\Vert \mathscr {K}(x,y)\Vert _{c_q})^p d\mu (x)\right\} ^{\frac{q}{p}}d\mu (y) \right) ^{\frac{1}{q}}. \end{aligned}$$

Then if \(1< p\le 2\) and \(q=\frac{p}{p-1}\), we have

$$\begin{aligned} \Vert K\Vert _{c_q}\le \Vert \mathscr {K}\Vert _{q,p,q}^{\frac{1}{2}} \Vert \mathscr {K}^*\Vert _{q,p,q}^{\frac{1}{2}}, \end{aligned}$$
(2.7)

where \(\mathscr {K}^*(x,y)=\mathscr {K}(y,x)^*\). (See [9, Corollary 2].)

3 Proof of the Results

3.1 Proof of Theorem 1.1

Let \(\gamma \) be an irreducible unitary representation of N in a Hilbert space \((\mathscr {H}_\gamma , \Vert \cdot \Vert _\gamma )\). Let \(C(G/N, \gamma )\) be the space of \(\mathscr {H}_{\gamma }\)-valued continuous functions \(\xi \) of G such that

$$\begin{aligned} \xi (gn)=\gamma (n)^{-1}\xi (g), \quad \forall g\in G, \forall n\in N \end{aligned}$$

and compactly supported modulo N. Let \(\mathscr {H}_{\pi _\gamma }\) be the completion of \(C(G/N,\gamma )\) with respect to

$$\begin{aligned} \Vert \xi \Vert ^2:=\int _{G/N}\Vert \xi (g)\Vert _\gamma ^2 \> d\nu (g). \end{aligned}$$

We realize the induced representation \(\pi _\gamma ={{\,\mathrm{Ind}\,}}_N^G\gamma \) on \(\mathscr {H}_{\pi _\gamma }\) by

$$\begin{aligned} \pi _{\gamma }(g)\xi (x)=\xi (g^{-1}x), \quad \xi \in \mathscr {H}_{\pi _\gamma }, g\in G, x \in G. \end{aligned}$$

Let \(f\in L^1(G)\). Since G and N are unimodular and G/N is compact, we have that the operator \(\pi _\gamma (f)\) is described by the integral operator as follows: For \(\xi \in \mathscr {H}_{\pi _\gamma }\) and \(x\in G\),

$$\begin{aligned} \pi _\gamma (f)\xi (x)&=\int _G f(g) \xi (g^{-1}x) \> dg \\&=\int _G f(xg) \xi (g^{-1}) \> dg \\&=\int _G f(xg^{-1})\xi (g)dg\\&=\int _{G/N} \int _N f(x(gn)^{-1})\xi (gn)\> dn \> d\nu (g)\\&=\int _{G/N} \int _N f(xn^{-1}g^{-1})\gamma (n)^{-1}\xi (g)\> dn \> d\nu (g)\\&=\int _{G/N} \int _N f(xng^{-1})\gamma (n)\xi (g)\> dn \> d\nu (g)\\&=: \int _{G/N} \mathscr {K}_\gamma ^f(x,g)\xi (g)\> d\nu (g), \end{aligned}$$

where

$$\begin{aligned} \mathscr {K}_\gamma ^f(x,g)&:=\int _N f(xng^{-1})\gamma (n)\> dn\nonumber \\&=\int _N f(xg^{-1}n)\gamma (g^{-1}ng)\>dn =:(g\cdot \gamma )(f(xg^{-1}(\cdot ))). \end{aligned}$$
(3.1)

For any compactly supported continuous function \(f\in C_c(G)\), we have

$$\begin{aligned} \Vert {\mathscr {F}} f\Vert _q&=\left( \int _{\widehat{N}/G}\Vert \pi _\gamma (f)\Vert _{c_q}^q \> d\bar{\mu }_N(\gamma )\right) ^{\frac{1}{q}}\\&\le \left( \int _{\widehat{N}/G}\Vert \mathscr {K}_\gamma ^f\Vert _{q, p, q}^{\frac{q}{2}} \Vert \mathscr {K}_\gamma ^{f^{*}}\Vert _{q, p, q}^{\frac{q}{2}} d\bar{\mu }_N(\bar{\gamma })\right) ^{\frac{1}{q}} \quad (\hbox {using} (2.7)) \\&\le \left( \int _{\widehat{N}/G}\Vert \mathscr {K}_\gamma ^f\Vert _{q, p, q}^{q} d\bar{\mu }_N(\bar{\gamma })\right) ^{\frac{1}{2q}} \left( \int _{\widehat{N}/G} \Vert \mathscr {K}_\gamma ^{f^{*}}\Vert _{q, p, q}^{q} d\bar{\mu }_N(\bar{\gamma })\right) ^{\frac{1}{2q}}. \end{aligned}$$

Let

$$\begin{aligned} I_{p,q}(f)= \int _{\widehat{N}/G}\Vert \mathscr {K}_\gamma ^f\Vert _{q, p, q}^{q} d\bar{\mu }_N(\bar{\gamma }). \end{aligned}$$

Then

$$\begin{aligned} I_{p,q}(f)&=\int _{\widehat{N}/G}\int _{G/N}\left( \int _{G/N} \Vert \mathscr {K}_\gamma ^f(x,g)\Vert _{c_q}^p d\nu (x)\right) ^{\frac{q}{p}} d\nu (g) d\bar{\mu }_N(\bar{\gamma })\\&=\int _{\widehat{N}/G}\int _{G/N}\left( \int _{G/N} \Vert \mathscr {K}_\gamma ^f(xg,g)\Vert _{c_q}^p d\nu (x)\right) ^{\frac{q}{p}} d\nu (g) d\bar{\mu }_N(\bar{\gamma }) \\&=\int _{\widehat{N}/G}\int _{G/N}\left( \int _{G/N} \Vert (g\cdot \gamma )f(xgg^{-1}(\cdot ))\Vert _{c_q}^p d\nu (x)\right) ^{\frac{q}{p}} d\nu (g) d\bar{\mu }_N(\bar{\gamma }) \quad (\hbox {by} (3.1))\\&=\int _{\widehat{N}/G}\int _{G/N}\left( \int _{G/N} \Vert (g\cdot \gamma )f(x(\cdot ))\Vert _{c_q}^p d\nu (x)\right) ^{\frac{q}{p}} d\nu (g) d\bar{\mu }_N(\bar{\gamma })\\&=\int _{\widehat{N}}\left( \int _{G/N} \Vert \gamma (f(x(\cdot ))) \Vert _{c_q}^p d\nu (x)\right) ^{\frac{q}{p}} d\mu _N(\gamma )\quad (\hbox {by} (2.1)) \\&\le \left( \int _{G/N}\left( \int _{\widehat{N}}\Vert \gamma (f(x(\cdot )))\Vert _{c_q}^q d\mu _N(\gamma )\right) ^{{\frac{p}{q}}} d\nu (x)\right) ^{\frac{q}{p}} \quad (\hbox {by} (2.6)) \\&\le \left( \int _{G/N}\Vert \mathscr {F}^p(N)\Vert ^p\Vert f(x(\cdot ))\Vert _{L^p(N)}^pd\nu (x)\right) ^{\frac{q}{p}} =\Vert {\mathscr {F}}^p(N)\Vert ^q\Vert f\Vert _p^q. \end{aligned}$$

We also have

$$\begin{aligned} I_{p,q}(f^*)=\int _{\widehat{N}/G}\Vert \mathscr {K}_\gamma ^{f^*}\Vert _{q, p, q}^{q} d\bar{\mu }_N(\bar{\gamma })\le \Vert {\mathscr {F}}^p(N)\Vert ^q\Vert f^*\Vert _p^q =\Vert {\mathscr {F}}^p(N)\Vert ^q\Vert f\Vert _p^q. \end{aligned}$$

Thus the inequality \(\Vert \mathscr {F}f\Vert _q\le I_{p,q}(f)^{\frac{1}{2q}}I_{p,q}(f^*)^{\frac{1}{2q}} \le \Vert {\mathscr {F}}^p(N)\Vert \Vert f\Vert _p\) holds and it concludes that \(\Vert {\mathscr {F}}^p(G)\Vert \le \Vert {\mathscr {F}}^p(N)\Vert \). \(\square \)

3.2 Case of a Semi-direct Product \(G:=K < imes N\).

As above, we now assume that G is a separable unimodular locally compact group of type I defined by \(G:=K < imes N\), where N is a separable unimodular locally compact group of type I and K stands for a compact subgroup of the automorphism group of N. We fix once for all a Haar measure \(dg:=d\nu \otimes dn\) of G, where \(d\nu \) and dn denote respectively the normalized Haar measure of K and N.

Let \((\gamma ,\mathscr {H}_\gamma )\) be an irreducible unitary representation of N. We realize the induced representation \(\pi _\gamma ={{\,\mathrm{Ind}\,}}_N^G\gamma \) on the space \(\mathscr {H}_{\pi _\gamma }\simeq L^2(K,\gamma )\) of the \(\mathscr {H}_\gamma \)-valued \(L^2\)-functions \(\xi :K\rightarrow \mathscr {H}_\gamma \) with respect to

$$\begin{aligned} \Vert \xi \Vert ^2:=\int _K\Vert \xi (k)\Vert _{\gamma }^2 \> d\nu (k)<\infty , \end{aligned}$$

and describe the induced representation \(\pi _\gamma \) by

$$\begin{aligned} \pi _\gamma (k,n)\xi (x):=\gamma (x^{-1}(n))\xi (k^{-1}x), \end{aligned}$$

for \(\xi \in \mathscr {H}_{\pi _\gamma }\), \(k,x\in K\), \(n\in N\).

Let \(f\in L^1(G)\), the operator \(\pi _\gamma (f)\) is described for \(\xi \in \mathscr {H}_{\pi _\gamma }\) and \(x\in K\) as follows:

$$\begin{aligned} \pi _\gamma (f)\xi (x)&=\int _G f(k,n) \gamma (x^{-1}(n))\xi (k^{-1}x) \> d\nu (k) dn \\&=\int _G f(xk^{-1}, n)\gamma (x^{-1}(n))\xi (k) \> d\nu (k) dn\\&=: \int _K \mathscr {K}_\gamma ^f(x,k)\xi (k)\> d\nu (k), \end{aligned}$$

where

$$\begin{aligned} \mathscr {K}_\gamma ^f(x,k):=\int _N \gamma (x^{-1}(n))f(xk^{-1}, n)\> dn =:(x\cdot \gamma )(f(xk^{-1},\cdot )). \end{aligned}$$

We remark that for \(f\in L^1(G)\cap L^p(G)\), the norm (2.5) is written with

$$\begin{aligned} \Vert \mathscr {F}f\Vert _q^q= \int _{\widehat{N}/K}\Vert \pi _\gamma (f)\Vert _{c_q}^q \> d\bar{\mu }_N(\bar{\gamma }). \end{aligned}$$
(3.2)

3.3 Proof of Theorem 1.2

Let now \((\varphi _j)_{j\in \mathbb {N}}\) be a K-invariant sequence in \(L^1(N)\cap L^p(N)\) such that \({\Vert {\mathscr {F}}\varphi _j \Vert _q}\over {\Vert \varphi _j \Vert _p}\) tends to \(\Vert {\mathscr {F}}^p(N )\Vert \), when j goes to infinity. Set for \(j\in \mathbb {N}\), \(f_j:=1\otimes \varphi _j\in L^1(G)\cap L^p(G)\). Then we have that:

$$\begin{aligned} \mathscr {K}_{\gamma }^{f_j}(k,x)&=\int _{N} \gamma (x^{-1}(n)) f_j(xk^{-1},n)dn \\&=\int _{N} \gamma (x^{-1}(n)) \varphi _j(n)dn\\&=\int _{N} \gamma (n) \varphi _j(x(n))dn\\&= \gamma (\varphi _j). \end{aligned}$$

It follows that

$$\begin{aligned} \pi _\gamma ({f_j})\xi (x)&=\int _{K} \mathscr {K}_{\gamma }^{f_j}(k,x)\xi (k)dk =\int _{K} {\gamma }({\varphi _j})\xi (k)dk, \end{aligned}$$

for \(\xi \in L^2(K,\gamma )\). Let then \(\lambda > 0\) and \(0\ne \xi \in L^2(K,\gamma )\) such that \(\pi _\gamma ({f_j})\pi _\gamma ({f_j^*})\xi (x)=\lambda \xi (x)\) for any \(x\in K\). Then \(\xi (x)=\xi _0\) does not depend upon x and therefore

$$\begin{aligned} \pi _\gamma ({f_j})\pi _\gamma ({f_j^*})\xi (x)&=\pi _\gamma ({f_j})\pi _\gamma ({f_j^*})\xi _0\\&=\int _{K} {\gamma }({\varphi _j})\gamma ({\varphi _j^*})\xi _0dk\\&=\gamma ({\varphi _j})\gamma ({\varphi _j^*})\xi _0=\lambda \xi _0. \end{aligned}$$

This means that

$$\begin{aligned} \Vert \pi _\gamma ({f_j})\Vert _{c_q}^q&=\Vert \gamma ({\varphi _j})\Vert _{c_q}^q. \end{aligned}$$

We get therefore:

$$\begin{aligned} \frac{1}{\Vert f_j\Vert _p^q}\int _{ \widehat{N}/ K} \Vert \pi _\gamma ({f_j})\Vert _{c_q}^q d\bar{\mu }_N(\bar{\gamma })&=\frac{1}{\Vert \varphi _j\Vert _{L^p(N)}^q}\int _{ \widehat{N}/ K} \Vert \gamma ({\varphi _j})\Vert _{c_q}^q d\bar{\mu }_N(\bar{\gamma }) \\&=\frac{\Vert \mathscr {F}\varphi _j\Vert _q^q}{\Vert \varphi _j\Vert _{L^p(N)}^q} \rightarrow \Vert {\mathscr {F}}^p(N )\Vert \quad (j \rightarrow \infty ), \end{aligned}$$

which is enough to conclude. \(\square \)

3.4 An Example

Let N designate the Heisenberg group of dimension \(2n+1\), \(n\ge 1\). The multiplication law of N is given by

$$\begin{aligned} (x,y,t)(x',y',t')=(x+x',y+y',t+t'+xy'), \quad x, y, x', y' \in {\mathbb {R}}^n, t, t' \in {\mathbb {R}}, \end{aligned}$$

where \(xy'=\sum _{i=1}^n x_iy'_i\) for \(x=(x_i)_i, y'=(y'_i)_i \in {\mathbb {R}}^n\). For \(p\in (1,2)\) of the form \(p=\frac{2k}{2k-1}\) for some integer \(k\ge 2\), we know from [13] that \(\Vert {\mathscr {F}}^p(N)\Vert =A_p^{2n+1}\) and that a sequence of the form \(\exp \{-a_m\Vert x\Vert ^2 -b_m\Vert y\Vert ^2 -\pi t^2\}, m\in \mathbb {N}\) for some real sequences \(a_m\) and \(b_m\) fits to the norm. In this case, an extremal function of N does not exist. Hence for any compact subgroup K of the orthogonal group \(O_n(\mathbb {R})\), we have \(\Vert {\mathscr {F}}^p(G)\Vert =A_p^{2n+1}\) for \(p=\frac{2k}{2k-1}\)\((k\ge 2, k\in \mathbb N)\), where \(G=K < imes N\) and the action of \(k\in K\) on N is defined by \(N\ni ((x_i)_i, (y_i)_i, t)\mapsto (k(x_i)_i, k(y_i)_i, t)\).

3.5 .

We now write some conclusions from the last theorem. It is easy to see that the set of extremal functions is invariant under the multiplication of scalars of the unit circle and under translations. One says that an extremal function is essentially unique if it is unique up to the multiplication of scalars of the unit circle and translations. We can now prove the following result:

Proposition 3.1

Assume that N has an essentially unique extremal function \(\varphi \) which satisfies that \(\varphi (e)\ne 0\) and \(\vert \varphi (e)\vert \ne \vert \varphi (n)\vert \) for all \(n\ne e\). Then \(\Vert {\mathscr {F}}^p(G)\Vert = \Vert {\mathscr {F}}^p(N )\Vert \) and \(1\otimes \varphi \) stands for an extremal function for G.

Proof

We first show that for any \(k\in K\), \(\varphi ^k\) is also an extremal function, where \(\varphi ^k(n)= \varphi (k^{-1}(n))\) for \(k\in K\). An easy computation shows that \(\varphi ^k*(\varphi ^k)^*= (\varphi *\varphi ^*)^k\) for any \(k\in K\). On the other hand,

$$\begin{aligned} \Vert {\mathscr {F}}^p(\varphi ^k)\Vert _q^q&={\displaystyle \int _{ \widehat{N}} \Vert \gamma ({\varphi ^k})\Vert _{c_q}^q d\mu _N(\gamma )}\\&={\displaystyle \int _{ \widehat{N}} \Vert k\cdot \gamma (\varphi )\Vert _{c_q}^q d\mu _N(\gamma )} \\&= {\displaystyle \int _{ \widehat{N}/ K}\int _{K} \Vert kh\cdot \gamma (\varphi )\Vert _{c_q}^q d\bar{\mu }_N(\bar{\gamma })} d\nu (h)\\&= {\displaystyle \int _{ \widehat{N}/ K}\int _{K} \Vert h\cdot \gamma (\varphi )\Vert _{c_q}^q d\bar{\mu }_N(\bar{\gamma })} d\nu (h)\\&= {\displaystyle \int _{ \widehat{N}}\Vert \gamma (\varphi )\Vert _{c_q}^q d\mu _N(\gamma )} \\&=\Vert {\mathscr {F}}^p(N )\Vert ^q \Vert \varphi \Vert _p^q\\&=\Vert {\mathscr {F}}^p(N )\Vert ^q \Vert \varphi ^k\Vert _p^q. \end{aligned}$$

By unicity, there exist for any \(k\in K\), \(\lambda _k\in \mathbb {C}\) such that \(\vert \lambda _k\vert =1\) and \(g_k\in N\) for which we have \( \varphi ^k(g)=\lambda _k\varphi (g_kg)\) for any \(g\in N\) and therefore

$$\begin{aligned} \varphi ^k(e)=\varphi (e)=\lambda _k\varphi (g_k). \end{aligned}$$
(3.3)

Since \(|\varphi (e)|=|\varphi (g_k)|\), we have \(g_k=e\) for all \(k\in K\). As \(\varphi (e)\not =0\), Eq. (3.3) says that \(\lambda _k=1\) for any \(k\in K\). This means that \(\varphi \) is an extremal K-invariant function of N and Theorem 1.2 allows us to conclude. \(\square \)