Abstract
Let G be a separable unimodular locally compact group of type I, and let N be a unimodular closed normal subgroup of G of type I, such that G/N is compact. Let for \(1<p\le 2\), \(\Vert {\mathscr {F}}^p(G)\Vert \) and \( \Vert {\mathscr {F}}^p(N )\Vert \) denote the norms of the corresponding \(L^p\)-Fourier transforms. We show that \(\Vert {\mathscr {F}}^p(G)\Vert \le \Vert {\mathscr {F}}^p(N )\Vert \). In the particular case where \(G=K < imes N\) is defined by a semi-direct product of a separable unimodular locally compact group N of type I and a compact subgroup K of the automorphism group of N, we show that equality holds if N has a K-invariant sequence \((\varphi _j)_j\) of functions in \(L^1(N)\cap L^p(N)\) such that \({\Vert {\mathscr {F}}\varphi _j \Vert _q}/{\Vert \varphi _j \Vert _p}\) tends to \(\Vert {\mathscr {F}}^p(N )\Vert \) when j goes to infinity. We show further that in some cases, an extremal function of N extends to an extremal function of G.
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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:
Let \(\mu _G \) be the Plancherel measure on \(\widehat{G}\), which is uniquely determined by the abstract Plancherel formula: for \(\varphi \in L^1(G)\cap L^2(G)\),
The Hausdorff–Young inequality, generalized by Kunze in [15] for separable locally compact unimodular groups of type I, is the assertion
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 \(\pi (\varphi )\)
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
Then we can write (1.1) as
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
For the abelian group \(G={\mathbb {R}}^n\), it is proved that
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
- (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
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
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
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
Let dn and \(d\nu \) be Haar measures on N and G/N normalized by
Letting \(d\mu _N\) be the Plancherel measure on \(\widehat{N}\) normalized by
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
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 \(f\in L^1(G)\cap L^2(G)\)
Denoting \(\pi _\gamma ={{\,\mathrm{Ind}\,}}_N^G\gamma \) for \(\gamma \in \widehat{N}\), let us remark that by Proposition 2.1 we have
and that the equality
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.
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
Thus we also obtain that the norm \(\Vert \mathscr {F}f\Vert _q\) is computed by
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].):
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
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
for \(\phi \in L^2(X,\mathscr {H})\) and \(\mu \)-almost all \(x\in X\). Now we define
Then if \(1< p\le 2\) and \(q=\frac{p}{p-1}\), we have
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
and compactly supported modulo N. Let \(\mathscr {H}_{\pi _\gamma }\) be the completion of \(C(G/N,\gamma )\) with respect to
We realize the induced representation \(\pi _\gamma ={{\,\mathrm{Ind}\,}}_N^G\gamma \) on \(\mathscr {H}_{\pi _\gamma }\) by
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 \(x\in G\),
where
For any compactly supported continuous function \(f\in C_c(G)\), we have
Let
Then
We also have
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
and describe the induced representation \(\pi _\gamma \) by
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:
where
We remark that for \(f\in L^1(G)\cap L^p(G)\), the norm (2.5) is written with
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:
It follows that
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
This means that
We get therefore:
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
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,
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
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. \(\square \)
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The authors would like to thank the two Referees for having suggested many valuable comments to improve the final form of the paper.
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Communicated by Hartmut Führ.
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This work was completed with the support of D.G.R.S.R.T through the Research Laboratory LR 11ES52. This work was partially supported by JSPS KAKENHI Grant Numbers 25400115, JP17K05280.
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Baklouti, A., Inoue, J. The \(L^p\)-Fourier Transform Norm on Compact Extensions of Locally Compact Groups. J Fourier Anal Appl 26, 26 (2020). https://doi.org/10.1007/s00041-020-09739-5
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DOI: https://doi.org/10.1007/s00041-020-09739-5