# The Hausdorff–Young inequality on Lie groups

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## Abstract

We prove several results about the best constants in the Hausdorff–Young inequality for noncommutative groups. In particular, we establish a sharp local central version for compact Lie groups, and extend known results for the Heisenberg group. In addition, we prove a universal lower bound to the best constant for general Lie groups.

## Mathematics Subject Classification

22E30 43A15 43A30## 1 Introduction

*f*by

*p*, that is, \(1/p' = 1 - 1/p\). Then interpolation implies the Hausdorff–Young inequality, namely,

*C*, by \(H_p(\mathbb {R}^n)\). This was found many years after the original result. We define the Babenko–Beckner constant \(B_p\) by

### Theorem 1.1

Babenko treated the case where \(p' \in 2 \mathbb {Z}\), and Beckner proved the general case. The extremal functions are gaussians; see [46] for an alternative proof.

One can extend the Babenko–Beckner theorem to more general contexts than \(\mathbb {R}^n\), such as locally compact abelian groups *G*. For instance, the best constant \(H_p(G)\) for the inequality (1.1) when \(G = \mathbb {R}^a \times \mathbb {T}^b \times \mathbb {Z}^c\) is \((B_p)^a\). The extremal functions are of the form \(\gamma \otimes \chi \otimes \delta \), where \(\gamma \) is a gaussian on \(\mathbb {R}^a\), \(\chi \) is a character of \(\mathbb {T}^b\), and \(\delta \) is the characteristic function of a point in \(\mathbb {Z}^c\).

For nonabelian groups, matters are more complicated, in part because the interpretation of the \(L^{q}\) norm of the Fourier transform for \(q \in (2,\infty )\) is trickier. We refer the reader to Sect. 2 below for details. General versions of the Hausdorff–Young inequality (1.1) were obtained by Kunze [43] and Terp [63] for arbitrary locally compact groups *G*, and a number of works in the literature are devoted to the study of the corresponding best constants \(H_p(G)\). It is known, at least in the unimodular case, that \(H_p(G)<1\) for \(p \in (1,2)\) if and only if *G* has no compact open subgroups [25, 56]. On the other hand, when \(H_p(G)\) is not 1, its value is known only in few cases, and typically only for exponents *p* whose conjugate exponent is an even integer; in addition, as shown by Klein and Russo, extremal functions need not exist [38].

Recently, various authors considered local versions of the Hausdorff–Young inequality. Namely, for each neighbourhood *U* of the identity \(e \in G\), define \(H_p(G;U)\) as the best constant in the inequality (1.1) with the additional support constraint \({{\,\mathrm{supp}\,}}f \subseteq U\), and let \(H_p^\mathrm{loc}(G)\) be the infimum of the constants \(H_p(G;U)\). Clearly \(H_p^\mathrm{loc}(G) \le H_p(G)\), and equality holds whenever *G* has a contractive automorphism. For other groups, however, the inequality may be strict, which makes the study of \(H_p^\mathrm{loc}(G)\) interesting also for groups where \(H_p(G) = 1\), such as compact groups. Indeed, in the case of the torus \(G = \mathbb {T}^n\), the value of \(H_p^\mathrm{loc}(G)\) is known and is strictly less than 1 for \(p \in (1,2)\).

### Theorem 1.2

Here we are interested in analogues of the above result for noncommutative Lie groups *G*. We also study what happens when additional symmetries are imposed by restricting to functions *f* on *G* which are invariant under a compact group *K* of automorphisms of *G*. Let us denote by \(H_{p,K}(G)\) and \(H_{p,K}^\mathrm{loc}(G)\) the corresponding global and local best Hausdorff–Young constants. Note that the original constants \(H_p(G)\) and \(H_{p}^\mathrm{loc}(G)\) correspond to the case where *K* is trivial. When *K* is nontrivial, *a priori* the new constants \(H_{p,K}(G)\) and \(H_{p,K}^\mathrm{loc}(G)\) might be smaller. However we can prove a universal lower bound, which is independent of the symmetry group *K* and depends only on *p* and the dimension of *G*.

### Theorem 1.3

*G*be a Lie group and

*K*be a compact group of automorphisms of

*G*. For all \(p \in [1,2]\),

Recall that a function *f* on a group *G* is *central* if \(f(xy) = f(yx)\), that is, if *f* is invariant under the group \({{\,\mathrm{Inn}\,}}(G)\) of inner automorphisms of *G*. García-Cuerva, Marco and Parcet [28] and García-Cuerva and Parcet [29] studied the Hausdorff–Young inequality for compact semisimple Lie groups *G* restricted to central functions; in particular, they obtained the inequality \(H^\mathrm{loc}_{p,{{\,\mathrm{Inn}\,}}(G)}(G) > 0\), which they applied to answer questions about Fourier type and cotype of operator spaces (see also [52]). Theorem 1.3 gives a substantially more precise lower bound to \(H^\mathrm{loc}_{p,{{\,\mathrm{Inn}\,}}(G)}(G)\). As a matter of fact, in this case we can prove that equality holds.

### Theorem 1.4

*G*is a compact connected Lie group. Then, for all \(p \in [1,2]\),

Note on the one hand that, in the abelian case \(G = \mathbb {T}^n\), all functions are central, so Theorem 1.4 extends Theorem 1.2. On the other hand, it would be interesting to know whether the result holds also without the restriction to central functions.

*G*. As a matter of fact, the equality

*G*and

*K*whenever \(p' \in 2\mathbb {Z}\), as a consequence of a recent result of Bennett, Bez, Buschenhenke, Cowling and Flock [7] and the relation between the best constants for the Young and the Hausdorff–Young inequalities (see Proposition 2.2 below). In particular, by interpolation,

*G*and

*K*with \(\dim (G) > 0\). Moreover, the equality

*G*with a contractive automorphism (which are nilpotent—see [60]), and also for all solvable Lie groups

*G*admitting a chain of closed subgroups

*G*, the upper bound \(H_{p}(G) \le (B_p)^{\dim (G)}\) for \(p'\in 2\mathbb {Z}\) was proved in [38], but the question of the lower bound was left open there, except for the Heisenberg groups. Hence Theorem 1.3 proves the sharpness of a number of results in [38].

*q*th Schatten class of operators on the Hilbert space \(\mathcal {H}\), and \(C \le 1\). As above, we can define \(W_p(\mathbb {C}^n)\) as the best constant in (1.3), as well as corresponding local and symmetric versions \(W_{p}^\mathrm{loc}(\mathbb {C}^n), W_{p,K}(\mathbb {C}^n), W_{p,K}^\mathrm{loc}(\mathbb {C}^n)\). A scaling argument (see Proposition 5.1 below) then shows that, for all compact subgroups

*K*of the unitary group \({\text {U}}(n)\),

### Theorem 1.5

*K*be a compact subgroup of \(\mathrm{U}(n)\). Then, for all \(p \in [1,2]\),

*f*:

Both cases where we can prove equalities in Theorems 1.4 and 1.5 for general \(p \in [1,2]\) correspond to Gelfand pairs (see, for example, [12]): indeed, central functions on a compact group *G* and polyradial functions on the Heisenberg group \(\mathbb {H}_n\) form commutative subalgebras of the respective convolution algebras \(L^1(G)\) and \(L^1(\mathbb {H}_n)\). It seems a reasonable intermediate question to ask for best constants in Hausdorff–Young inequalities in the context of Gelfand pairs, since here the group Fourier transform reduces to the Gelfand transform for the corresponding commutative algebra of invariant functions, which makes the \(L^q\) norm of the Fourier transform in these settings more accessible. Indeed, in both the proofs of Theorems 1.4 and 1.5, this additional commutativity allows one to relate the group Fourier transform and the Weyl transform with the Euclidean Fourier transform, for which the Babenko–Beckner result is available. Regrettably, even in the case of polyradial functions on the Heisenberg group we are not able yet to fully answer the question. Indeed, as we discuss in Sect. 5, in this case it seems unlikely that the best Hausdorff–Young constant on the Heisenberg group can be obtained by a direct reduction to the corresponding sharp Euclidean estimate, and new ideas appear to be needed.

As for the universal lower bound of Theorem 1.3, the intuitive idea behind its proof is that, at smaller and smaller scales, the group structure of a Lie group *G* looks more and more like the abelian group structure of its Lie algebra \(\mathfrak {g}\), whence \(H_p^\mathrm{loc}(G)\) is likely to be related to \(H_p(\mathfrak {g}) = (B_p)^{\dim (G)}\). Indeed, a scaling argument based on this idea readily yields the analogue of Theorem 1.3 for Young’s convolution inequality (see the discussion in Sect. 2 below). This appears to have been overlooked in [38], where a number of upper bounds for Young constants on Lie groups are proved, which are actually equalities in view of this observation.

*G*approximates the commutative convolution on \(\mathfrak {g}\), the same is not so evident for the Fourier transform: indeed, if the group Fourier transform is defined, as it is common, in terms of irreducible unitary representations, then it is not immediately clear how to relate the representation theories of

*G*and \(\mathfrak {g}\) for an arbitrary Lie group

*G*, let alone the corresponding Fourier transforms and \(L^q\) norms thereof. Here we completely bypass the problem, by characterising the \(L^q\) norm of the Fourier transform in terms of an operator norm of a fractional power of an integral operator, acting on functions on

*G*:

*f*and \(\Delta \) is the operator of multiplication by the modular function of

*G*. A transplantation argument, not dissimilar from those in [36, 49, 51], allows us to relate the operator \(L_f \Delta ^{1/q}\) on

*G*to its counterpart on \(\mathfrak {g}\) and obtain the desired lower bound.

Although it might be evident to some experts in noncommutative integration, we are not aware of the characterisation (1.6) being explicitly observed before. What is interesting about (1.6) is that it allows one to access the \(L^q\) norm of the Fourier transform through properties of a more “geometric” convolution-multiplication operator on *G*, which appears to be more tractable. As a matter of fact, when dealing with convolution, one can use induction-on-scales methods to completely determine the best local constants for the Young convolution inequality on any Lie group *G*; this result has been recently proved in [7], as a corollary of a more general result for nonlinear Brascamp–Lieb inequalities. It would be interesting to know whether similar methods could be applied to the Hausdorff–Young inequality on noncommutative Lie groups as well.

### 1.1 Plan of the paper

In Sect. 2 we discuss the definition of the \(L^q\) norm of the Fourier transform for an arbitrary Lie group, by comparing a number of definitions available in the literature, and prove the characterisation (1.6); we also present a proof of the universal lower bound of Theorem 1.3, as well as its analogue for the Young convolution inequality, and discuss relations between best constants for Young and Hausdorff–Young inequalities. The sharp local central Hausdorff–Young inequality for arbitrary compact Lie groups (Theorem 1.4) is proved in Sect. 4; to better explain the underlying idea without delving into technicalities, the proof of the abelian case (Theorem 1.2) is briefly revisited in Sect. 3. Finally, in Sect. 5 we discuss the relations between Hausdorff–Young constants for the Heisenberg group and the Weyl transform and prove Theorem 1.5, together with the weighted inequality (1.5) for polyradial functions.

## 2 \(L^q\) norm of the Fourier transform

Let *G* be a Lie group (or, more generally, a separable locally compact group) with a fixed left Haar measure. In order to discuss best Hausdorff–Young constants in this generality, we first need to clarify what is meant by the “Fourier transform” in this setting and how Hausdorff–Young inequalities — even the endpoint ones, such as the Plancherel formula — can be stated in this context.

*G*(see, for example, [47] or [23, Chapter 7] for a survey). Namely, let \({\widehat{G}}_\mathrm{u}\) be the “unitary dual” of

*G*, that is, the set of (equivalence classes of) irreducible unitary representations of

*G*, endowed with the Fell topology and the Mackey Borel structure. The (unitary) Fourier transform \(\mathcal {F}_\mathrm{u}f\) of a function \(f \in L^1(G)\) is then defined as the operator-valued function on \({\widehat{G}}_\mathrm{u}\) given by

*G*is unimodular and type I (this includes the cases where

*G*is abelian or compact), the Plancherel formula can be stated in the form

*q*th Schatten class of operators on \(\mathcal {H}_\pi \), and the operator-valued \(L^q\)-spaces \(L^q_\mathrm{u}({\widehat{G}})\) are defined in terms of measurable fields of operators as in [47]. The fact that the spaces \(L^q_\mathrm{u}({\widehat{G}})\) constitute a complex interpolation family, that is,

In the case where *G* is not unimodular, under suitable type I assumptions it is possible to prove a Plancherel formula similar to (2.1), where the right-hand side is adjusted by means of “formal dimension operators” [19, 26, 39, 40, 62]. Analogous modifications of (2.2) lead to a version of the Hausdorff–Young inequality that has been studied in a number of works [4, 21, 27, 32, 57].

When *G* is not type I, the above approach to the Plancherel formula based on irreducible unitary representation theory does not work as neatly. This however does not prevent one from studying the Hausdorff–Young inequality. Indeed, what is possibly the first appearance in the literature of the Hausdorff–Young inequality in a noncommutative setting, that is, the work of Kunze [43] for arbitrary unimodular locally compact groups (not necessarily of type I), does not express the Fourier transform in terms of irreducible unitary representations, but uses instead the theory of noncommutative integration (the same theory was used in earlier works of Mautner [50] and Segal [58] to express the Plancherel formula). This point of view was subsequently developed by Terp [63] to cover the case of non-unimodular groups and more recently has been further extended to the context of locally compact quantum groups [13, 15].

*M*and its predual \(M_*\) (which play the role of \(L^\infty \) and \(L^1\) respectively) [33, 42, 54, 64]. In general this requires establishing a “compatibility” between

*M*and \(M_*\), which may involve a number of choices, but in our case there appears to be a natural way to proceed (see also [17, 24]). Namely, the von Neumann algebra \(\mathrm{VN}(G)\) of

*G*(that is, the weak\({}^*\)-closed \(*\)-subalgebra of \(\mathcal {L}(L^2(G))\) of the operators which commute with right translations) can be identified with the space \(\mathrm{Cv}^2(G)\) of left convolutors of \(L^2(G)\), that is, those distributions on

*G*which are left convolution kernels of \(L^2(G)\)-bounded operators. Moreover, the predual \(\mathrm{VN}(G)_*\) can be identified with the Fourier algebra

*A*(

*G*), an algebra of continuous functions on

*G*defined by Eymard [20] for arbitrary locally compact groups

*G*. Now

*A*(

*G*) and \(\mathrm{Cv}^2(G)\) are naturally compatible as spaces of distributions on

*G*(see [20, Propositions (3.26) and (3.27)]), so we can use complex interpolation to define Fourier–Lebesgue spaces of distributions on

*G*: for \(q\in [1,\infty ]\), we set

We then define the \(L^p\) Hausdorff–Young constant \(H_p(G)\) on the group *G* as the minimal constant *C* for which (2.4) holds for all \(f \in L^p(G)\). Similarly, if *U* is a neighbourhood of the identity in *G*, we let \(H_p(G;U)\) be the minimal constant *C* in (2.4) when *f* is constrained to have support in *U*, and define the local \(L^p\) Hausdorff–Young constant \(H_p^\mathrm{loc}(G)\) as the infimum of the constants \(H_p(G;U)\) where *U* ranges over the neighbourhoods of the identity of *G*.

The approach to Hausdorff–Young constants via \(\mathcal {F}L^q\) spaces is consistent with the unitary Fourier transformation approach described above, when the latter is applicable. Indeed, as discussed in [47, Theorems 2.1 and 3.1], in the case where *G* is unimodular and type I, the unitary Fourier transformation \(\mathcal {F}_\mathrm{u}\) induces isometric isomorphisms \(\mathrm{Cv}^2(G) \cong L^\infty _\mathrm{u}({\widehat{G}})\) and \(A(G) \cong L^1_\mathrm{u}({\widehat{G}})\), besides the Plancherel isomorphism \(L^2(G) \cong L^2_\mathrm{u}({\widehat{G}})\) (analogous results in the nonunimodular case can be found in [26, Theorems 3.48 and 4.12]); so by interpolation \(\mathcal {F}_u\) induces an isometric isomorphism between \(\mathcal {F}L^q(G)\) and \(L^q_\mathrm{u}({\widehat{G}})\) for all \(q \in [1,\infty ]\). Hence defining Hausdorff–Young constants in terms of the inequality (2.2) would lead to the same constants \(H_p(G)\) and \(H_p^\mathrm{loc}(G)\) as those we have defined in terms of \(\mathcal {F}L^q\) spaces. On the other hand, the approach via \(\mathcal {F}L^q\) spaces does not require type I assumptions, or even separability, and can be applied to every locally compact group *G*.

*T*belongs to \(L^q_{\mathrm{VN}}({\widehat{G}})\) for some \(q \in [1,\infty )\), then \(|T|^q = (T^* T)^{q/2}\) belongs to \(L^1_{\mathrm{VN}}({\widehat{G}})\) and

*f*, and we identify the modular function \(\Delta \) of

*G*with the corresponding multiplication operator (see [13, Proposition 2.21(ii)]). Recall that convolution on

*G*is given by

*f*and \(\phi \) are in \(C_c(G)\).

Note that, when \(q=p'\), (2.6) matches the definitions by Kunze and by Terp of the \(L^p\) Fourier transformation \(\mathcal {F}_p : L^p(G) \rightarrow L^{p'}_{\mathrm{VN}}({\widehat{G}})\) for \(p \in [1,2]\) [43, 63]. In other words, the \(L^p\) Fourier transformation \(\mathcal {F}_p : L^p(G) \rightarrow L^{p'}_{\mathrm{VN}}({\widehat{G}})\) factorises as the inclusion map \(L^p(G) \rightarrow \mathcal {F}L^{p'}(G)\) and the isometric isomorphism \(\mathcal {F}L^{p'}(G) \rightarrow L^{p'}_{\mathrm{VN}}({\widehat{G}})\), whence the compatibility with the Kunze–Terp approach of the above definition of the best Hausdorff–Young constants based on (2.4).

Another consequence of the above discussion is the following characterisation of the \(\mathcal {F}L^q(G)\) norm in terms of a more “concrete” operator norm.

### Proposition 2.1

### Proof

*g*must be a function of positive type (see, for example, [23, Section 3.3]), whence

*G*, the

*k*-linear version of Young’s inequality takes the following form: for all \(p_1,\dots ,p_k,r \in [1,\infty ]\) such that \(\sum _{j=1}^k 1/p_j' = 1/r'\),

*G*as the smallest constant

*C*for which (2.8) holds for all \(f_1 \in L^{p_1}(G), \dots , f_k \in L^{p_k}(G)\), as well as the localised versions \(Y_{p_1,\dots ,p_k}(G;U)\) for neighbourhoods

*U*of the identity of

*G*(corresponding to the constraint \({{\,\mathrm{supp}\,}}f_1, \dots , {{\,\mathrm{supp}\,}}f_k \subseteq U\)) and \(Y^\mathrm{loc}_{p_1,\dots ,p_k}(G)\).

### Proposition 2.2

*G*be a locally compact group.

- (i)For all \(p_1,\dots ,p_k,q \in [1,2]\) such that \(\sum _j 1/p_j' = 1/q\),$$\begin{aligned} Y_{p_1,\dots ,p_k}(G)&\le H_{q}(G) \, H_{p_1}(G) \cdots H_{p_k}(G), \\ Y_{p_1,\dots ,p_k}^\mathrm{loc}(G)&\le H_{q}^\mathrm{loc}(G) \, H_{p_1}^\mathrm{loc}(G) \cdots H_{p_k}^\mathrm{loc}(G). \end{aligned}$$
- (ii)For all \(p \in [1,2)\) such that \(p'=2k\), \(k \in \mathbb {Z}\), if \(p_1=\dots =p_k=p\), then$$\begin{aligned} H_{p}(G)&= Y_{p_1,\dots ,p_k}(G)^{1/k}, \\ H_{p}^\mathrm{loc}(G)&= Y_{p_1,\dots ,p_k}^\mathrm{loc}(G)^{1/k}. \end{aligned}$$
- (iii)If
*N*is a closed normal subgroup of*G*, then, for all \(p_1,\dots ,p_k \in [1,\infty ]\) such that \(\sum _{j=1}^k 1/p_j' \in [0,1]\),with equality when \(G \cong N \times (G/N)\).$$\begin{aligned} Y_{p_1,\dots ,p_k}(G)&\le Y_{p_1,\dots ,p_k}(N) \, Y_{p_1,\dots ,p_k}(G/N),\\ Y_{p_1,\dots ,p_k}^\mathrm{loc}(G)&\le Y_{p_1,\dots ,p_k}^\mathrm{loc}(N) \, Y_{p_1,\dots ,p_k}^\mathrm{loc}(G/N), \end{aligned}$$

### Proof

*U*of the identity, then Open image in new window is supported in \(U^k\) and, to estimate its \(L^{q'}\) norm, it is enough to test it against functions

*g*that are also supported in \(U^k\); the same argument as above then also gives

*f*, according to whether \(k-j\) is odd or even, and define Open image in new window . Then, since \(p'=2k\),

(iii). The inequalities are proved by a simple extension of Klein and Russo’s argument for the case of semidirect products [38, proof of Lemma 2.4], using the “measure disintegration” in [23, Theorem (2.49)]. In the case of direct products, equalities follow by testing on tensor product functions (see [6, Lemma 5]). \(\square \)

The next lemma contains the fundamental approximation results that allow us to relate Hausdorff–Young constants on a Lie group *G* and on its Lie algebra \(\mathfrak {g}\) by means of a “transplantation” or “blow-up” technique. The Lie algebra \(\mathfrak {g}\) will be considered as an abelian group with addition, and the Lebesgue measure on \(\mathfrak {g}\) is normalised so that the Jacobian determinant of the exponential map \(\exp : \mathfrak {g} \rightarrow G\) is equal to 1 at the origin. The context will make clear whether the notation for convolution, involution and convolution operators (\(f*g\), \(f^*\), \(L_f\)) refers to the group structure of *G* or the abelian group structure of \(\mathfrak {g}\).

Denote by \(C_{pg}([0,\infty ))\) the space of continuous functions \(\Phi : [0,\infty ) \rightarrow \mathbb {C}\) with at most polynomial growth, that is, \(|\Phi (u)| \le C(1+u)^N\) for some \(C,N \in (0,\infty )\) and all \(u \in [0,\infty )\).

### Lemma 2.3

*G*be a Lie group with Lie algebra \(\mathfrak {g}\) of dimension

*n*, and let \(\exp : \mathfrak {g} \rightarrow G\) be the exponential map. Let \(\Omega \) be an open neighbourhood of the origin in \(\mathfrak {g}\) such that \(\Omega = -\Omega \) and \(\exp |_\Omega : \Omega \rightarrow \exp (\Omega )\) is a diffeomorphism. For all \(f \in C_c(\mathfrak {g})\), \(\lambda \in (0,\infty )\), \(\alpha \in \mathbb {R}\) and \(p \in [1,\infty ]\), define \(f^{\lambda ,p,\alpha } : G \rightarrow \mathbb {C}\) by

- (i)For all \(f \in C_c(\mathfrak {g})\), \(\alpha \in \mathbb {R}\) and \(p \in [1,\infty ]\),for all \(\lambda \in (0,\infty )\), and$$\begin{aligned} \Vert f^{\lambda ,p,\alpha }\Vert _{L^p(G)} \le C_{\alpha ,p,\Omega } \, \Vert f\Vert _{L^p(\mathfrak {g})} \end{aligned}$$(2.13)as \(\lambda \rightarrow 0\).$$\begin{aligned} \Vert f^{\lambda ,p,\alpha }\Vert _{L^p(G)} \rightarrow \Vert f\Vert _{L^p(\mathfrak {g})} \end{aligned}$$(2.14)
- (ii)For all \(k \in \mathbb {N}\), \(\alpha _1,\dots ,\alpha _k,\beta \in \mathbb {R}\), \(f_1,\dots ,f_k,g \in C_c(\mathfrak {g})\),as \(\lambda \rightarrow 0\).$$\begin{aligned} \langle f_1^{\lambda ,1,\alpha _1} * \cdots * f_k^{\lambda ,1,\alpha _k} , g^{\lambda ,\infty ,\beta } \rangle _{L^2(G)} \rightarrow \langle f_1 * \cdots * f_k , g \rangle _{L^2(\mathfrak {g})} \end{aligned}$$(2.15)
- (iii)For all \(\alpha \in \mathbb {R}\), \(f,g,h \in C_c(\mathfrak {g})\), \(\Phi \in C_{pg}([0,\infty ))\),as \(\lambda \rightarrow 0\).$$\begin{aligned} \langle \Phi (\Delta ^\alpha L_{(f^{\lambda ,1})^* * f^{\lambda ,1}} \Delta ^\alpha ) g^{\lambda ,2}, h^{\lambda ,2}\rangle _{L^2(G)} \rightarrow \langle \Phi (L_{f^* * f}) g, h\rangle _{L^2(\mathfrak {g})} \end{aligned}$$(2.16)
- (iv)For all \(\alpha \in \mathbb {R}\), \(f,g,h \in C_c(\mathfrak {g})\) and \(q \in [0,\infty )\),as \(\lambda \rightarrow 0\).$$\begin{aligned} \lambda ^{-n(q-1)} \langle |L_{f^{\lambda ,\infty }} \Delta ^\alpha |^{q} g^{\lambda ,1}, h^{\lambda ,1}\rangle _{L^2(G)} \rightarrow \langle |L_{f}|^{q} g, h\rangle _{L^2(\mathfrak {g})} \end{aligned}$$

### Proof

Let \(J : \mathfrak {g} \rightarrow \mathbb {R}\) denote the modulus of the Jacobian determinant of \(\exp \), and define \(\Delta _e : \mathfrak {g} \rightarrow (0,\infty )\) to be \(\Delta \circ \exp \).

*f*is compactly supported and \(\lim _{X \rightarrow 0} (J \Delta _e^{-\alpha p})(X) = J(0) \Delta (e)^{-\alpha p} = 1\).

*m*; indeed we can find a sufficiently small neighbourhood \(\tilde{\Omega }\subseteq \Omega \) of the origin in \(\mathfrak {g}\) so that, if \(X_1,\dots ,X_k \in \tilde{\Omega }\), then \(X_1 + \dots + X_k + B(X_1,\dots ,X_k) \in \Omega \).

*G*is nonunimodular) and that, for all \(N \in \mathbb {N}\), the \(L^2\)-domain of \((\Delta ^\alpha L_{(f^{\lambda ,1})^* * f^{\lambda ,1}} \Delta ^\alpha )^N\) contains all compactly supported functions in \(L^2(G)\), so the left-hand side of (2.16) is well-defined. Note moreover that

*G*, and moreover the topological boundary of \(\exp (\Omega )\) has null Haar measure (indeed shrinking \(\Omega \) does not change the left-hand side of (2.16) for \(\lambda \) sufficiently small). As in [49, proof of Theorem 5.2], we can now extend the diffeomorphism \(\phi := \exp |_\Omega ^{-1} : \exp (\Omega ) \rightarrow \Omega \) to a diffeomorphism \(\phi _* : U \rightarrow V\), where

*U*and

*V*are open sets in

*G*and \(\mathfrak {g}\) containing \(\exp (\Omega )\) and \(\Omega \), and moreover \(G \setminus U\) has null Haar measure. Finally, let \(J_* : V \rightarrow (0,\infty )\) be the density of the push-forward via \(\phi _*\) of the Haar measure with respect to the Lebesgue measure (so \(J_* = J\) on \(\Omega \)), and define an isometric isomorphism \(\Psi : L^2(G) \rightarrow L^2(V)\) by

*A*is a bounded self-adjoint operator on \(L^2(\mathfrak {g})\), \(C_c(\mathfrak {g})\) is a core for

*A*and [67, Theorem 9.16] implies that

(iv). This is just a restatement of part (iii) in the case where \(\Phi (u) = u^{q/2}\). \(\square \)

We can finally prove the enunciated relation between Hausdorff–Young constants of a Lie group and its Lie algebra. We find it convenient to state the result together with its analogue for Young constants, since both follow by the approximation results of Lemma 2.3. Part (ii) of Proposition 2.4, together with the following Remark 2.5 and the Babenko–Beckner theorem for \(\mathbb {R}^n\), prove Theorem 1.3.

As in [60], we define a *contractive automorphism* of a locally compact group *G* as an automorphism \(\tau \) such that \(\lim _{k\rightarrow \infty } \tau ^k(x)=e\) for all \(x \in G\).

### Proposition 2.4

*G*be a locally compact group.

- (i)For all \(p_1,\dots ,p_k \in [1,\infty ]\) such that \(\sum _{j=1}^k 1/p_j' \in [0,1]\),with equality when$$\begin{aligned} Y_{p_1,\dots ,p_k}(G) \ge Y_{p_1,\dots ,p_k}^\mathrm{loc}(G), \end{aligned}$$(2.22)
*G*has a contractive automorphism; moreover, if*G*is a Lie group with Lie algebra \(\mathfrak {g}\),$$\begin{aligned} Y_{p_1,\dots ,p_k}^\mathrm{loc}(G) \ge Y_{p_1,\dots ,p_k}(\mathfrak {g}). \end{aligned}$$(2.23) - (ii)For all \(p \in [1,2]\),with equality if$$\begin{aligned} H_p(G) \ge H_p^\mathrm{loc}(G), \end{aligned}$$(2.24)
*G*has a contractive automorphism. Moreover, when*G*is an*n*-dimensional Lie group with Lie algebra \(\mathfrak {g}\),$$\begin{aligned} H_p^\mathrm{loc}(G) \ge H_p(\mathfrak {g}). \end{aligned}$$(2.25)

### Proof

*G*has a contractive automorphism, the reverse inequality follows from a scaling argument. Indeed, for all automorphisms \(\gamma \) of

*G*, there exists \(\kappa _\gamma \in (0,\infty )\) such that the push-forward via \(\gamma \) of the Haar measure on

*G*is \(\kappa _\gamma \) times the Haar measure. So, if \(R_\gamma f = f \circ \gamma ^{-1}\), thenwhence it is immediate that both sides of Young’s inequality (2.8) are scaled by the same factor when each \(f_j\) is replaced with \(R_\gamma f_j\). Now, by density, the value of the best constant \(Y_{p_1,\dots ,p_k}(G)\) may be determined by testing (2.8) on arbitrary \(f_1,\dots ,f_k \in C_c(G)\). Moreover, if \(\tau \) is a contractive automorphism of

*G*and

*U*is any neighbourhood of the identity, then, for all compact subsets \(K \subseteq G\), there exists \(N \in \mathbb {N}\) such that \(\tau ^N(K) \subseteq U\) [60, Lemma 1.4(iv)]; in particular, for all \(f_1,\dots ,f_k \in C_c(G)\), by taking \(\gamma = \tau ^N\) for sufficiently large \(N \in \mathbb {N}\), we see that \({{\,\mathrm{supp}\,}}R_\gamma f_j \subseteq U\). This shows that \(Y_{p_1,\dots ,p_k}(G) \le Y_{p_1,\dots ,p_k}(G;U)\) for all neighbourhoods

*U*of \(e \in G\), and consequently \(Y_{p_1,\dots ,p_k}(G) \le Y_{p_1,\dots ,p_k}^\mathrm{loc}(G)\).

As for the second inequality, let *U* be an arbitrary neighbourhood of \(e \in G\). To conclude, it is sufficient to show that \(Y_{p_1,\dots ,p_k}(\mathfrak {g}) \le Y_{p_1, \dots , p_k}(G;U)\).

*G*has a contractive automorphism, since

*G*.

*U*of \(e \in G\). Set \(q = p'\) and note that, by (2.7),

### Remark 2.5

The argument in Proposition 2.4 can be extended to the case of inequalities restricted to particular classes of functions on *G*. In particular, suppose that the class of functions is determined by invariance with respect to the action of a compact group *K* of automorphisms of *G*. Then it is possible to choose a positive inner product on \(\mathfrak {g}\) so that *K* acts on \(\mathfrak {g}\) by isometries (take any inner product on \(\mathfrak {g}\) and average it with respect to the action of *K*), and the correspondence (2.12) preserves *K*-invariance whenever \(\Omega \) is a ball centred at the origin. Moreover the class of functions on \(\mathfrak {g}\) under consideration contains all radial functions. Since the extremisers for Young and Hausdorff–Young constants on \(\mathfrak {g}\) are centred gaussians [6, 9], which may be assumed to be radial, the resulting lower bounds do not change. This observation completes the proof of Theorem 1.3.

### Remark 2.6

*G*(note that, when

*G*is compact, the global Young and Hausdorff–Young constants are equal to 1), it appears natural to ask whether the inequalities (2.23) and (2.25) are actually equalities. We are not aware of any counterexample. As a matter of fact, a particular case of a recent result of Bennett, Bez, Buschenhenke, Cowling and Flock about nonlinear Brascamp–Lieb inequalities [7] entails that equality

*always*holds in (2.23) for all Lie groups

*G*:

*all*\(p_1,\dots ,p_k \in [1,\infty ]\) such that \(\sum _{j=1}^k 1/p_j' \in [0,1]\). By Proposition 2.2(ii), this in turn implies that

*a fortiori*the same equality holds for the

*K*-invariant version of the constants for any compact group of automorphisms

*K*.

As a consequence of the above results, we strengthen some results of Klein and Russo [38, Corollaries 2.5’ and 2.8], where upper bounds for Young and Hausdorff–Young constants are obtained for particular solvable Lie groups. Klein and Russo explicitly remark that they are able to obtain equalities instead of upper bounds in the particular case of the Heisenberg groups and only for special exponents (through a different argument, involving the analysis of the Weyl transform) and seem to leave the general case open. Here instead we obtain equality for all the Young constants, as well as a lower bound for the Hausdorff–Young constants (which becomes an equality in the case of Babenko’s exponents).

### Corollary 2.7

*G*be a

*n*-dimensional solvable Lie group admitting a chain of closed subgroups

- (i)For all \(p_1,\dots ,p_k,r \in [1,\infty ]\) such that \(\sum _{j=1}^k 1/p_j' = 1/r'\),$$\begin{aligned} Y_{p_1,\dots ,p_k}(G) = Y^\mathrm{loc}_{p_1,\dots ,p_k}(G) = (B_{r'} B_{p_1} \cdots B_{p_k})^n. \end{aligned}$$
- (ii)For all \(p \in [1,2]\),with equalities if \(p' \in 2\mathbb {Z}\).$$\begin{aligned} H_p(G) \ge H_p^\mathrm{loc}(G) \ge (B_p)^n, \end{aligned}$$

### Proof

*q*is an even integer. On the other hand, by Proposition 2.4(ii),

## 3 The *n*-torus \(\mathbb {T}^n\) revisited

The proof of the central local Hausdorff–Young theorem on a compact Lie group mimics that of the local Hausdorff–Young theorem on \(\mathbb {T}^n\), and we present this case first to make the proof of the general case more evident.

### Proof of Theorem 1.2

There is no loss of generality in supposing functions smooth; this ensures that all the sums and integrals that occur in the proof below converge.

*f*is given by

*V*the open subset \((-1/2,1/2)^n\) of \(\mathbb {R}^n\). For any function \(f \in L^1(\mathbb {T}^n)\) such that \({{\,\mathrm{supp}\,}}f \subseteq V\), we define

*F*on \(\mathbb {R}^n\) by

*F*corresponds to

*f*. Clearly \(F \in L^1(\mathbb {R}^n)\) and \({\hat{F}}|_{\mathbb {Z}^n} = {\hat{f}}\); further, if

*f*is smooth, so is

*F*. We are going to transfer the sharp Hausdorff–Young theorem for

*F*to

*f*.

*not*interpolate, because (3.1) does not hold for all \({\hat{F}}\) in \(L^{2}(\mathbb {R}^n)\), or even for all \({\hat{F}}\) in a dense subspace of \(L^{2}(\mathbb {R}^n)\), but only for those \({\hat{F}}\) where \({{\,\mathrm{supp}\,}}F \subseteq V\);

*inter alia*, this ensures that \({\hat{F}}\) is smooth so that \({\hat{F}}|_{\mathbb {Z}^n}\) is well-defined. So we prove a variant of (3.2).

*U*be a small neighbourhood

*U*of 0 in \(\mathbb {T}^n\) such that \({\overline{U}} \subseteq V\), and take \(\phi \in A(\mathbb {R}^n)\) such that \({{\,\mathrm{supp}\,}}\phi \subseteq V \) and \(\phi (x) = 1\) for all \(x \in U\). We now define

*V*, whence

*q*is 2 or \(\infty \). The Riesz–Thorin interpolation theorem establishes (3.3) for all \(q \in [2, \infty ]\).

*F*correspond to

*f*. Then \({\hat{F}} \in L^1(\mathbb {R}^n) \cap L^{\infty }(\mathbb {R}^n)\) and \({\hat{\phi }} * {\hat{F}} = {\hat{F}}\). Thus

*U*small enough, we may make \(\Vert {\hat{\phi }} \Vert _{L^1(\mathbb {R}^n)}\) as close to 1 as we like (see [45]): indeed, we can take \(\phi = |K|^{-1} \mathbf{1 }_{U+K} * \mathbf{1 }_K\), where \(K=-K\) is a fixed small neighbourhood of the origin (here \(\mathbf{1 }_\Omega \) denotes the characteristic function of a measurable set \(\Omega \subseteq \mathbb {R}^n\) and \(|\Omega |\) its Lebesgue measure), so that \({{\,\mathrm{supp}\,}}\phi \subseteq U +2K\) and

## 4 Compact Lie groups

Before entering into the proof of Theorem 1.4, we present a summary of the theory of representations and characters of compact connected Lie groups *G*. For more details, the reader may consult, for example, [11, 41]. We assume throughout that *G* is not abelian, since the abelian case was treated in Theorem 1.2.

A compact connected Lie group *G* comes with a set \(\Lambda ^+\) of *dominant weights*, which parametrise the collection of irreducible unitary representations \(\pi _\lambda \) of *G* modulo equivalence. Each such representation \(\pi _\lambda \) is of finite dimension \(d_\lambda \) and has a character \(\chi _\lambda \) given by \({{\,\mathrm{trace}\,}}\pi _\lambda (\cdot )\).

*G*is normalised so as to have total mass 1. The Peter–Weyl theory gives us the Plancherel formula: if \(f \in L^2(G)\), then

*G*can be identified with the discrete measure on \(\Lambda ^+\) that assigns mass \(d_\lambda \) to the point \(\lambda \). From the discussion in Sect. 2, we deduce that

*f*is a central function, then \(\pi _\lambda (f)\) is a multiple of the identity and

*G*on itself determines the adjoint representation of

*G*on \(\mathfrak {g}\):

*G*is compact, there exists an \({\text {Ad}}(G)\)-invariant inner product on \(\mathfrak {g}\), which in turn determines a Lebesgue measure on \(\mathfrak {g}\); we scale the inner product so that the Jacobian determinant \(J : \mathfrak {g} \rightarrow \mathbb {R}\) of the exponential mapping is 1 at the origin. Clearly

*J*is an \({\text {Ad}}(G)\)-invariant function.

The group *G* contains a maximal torus *T*, that is, a maximal closed connected abelian subgroup, which is unique up to conjugacy; its Lie algebra \(\mathfrak {t}\) is a maximal abelian Lie subalgebra of \(\mathfrak {g}\). The set \(\Gamma \) of *X* in \(\mathfrak {t}\) such that \(\exp X = e\) is a lattice in \(\mathfrak {t}\), and *T* may be identified with \(\mathfrak {t} / \Gamma \). The *weight lattice*\(\Lambda \) is the dual lattice to \(\Gamma \), that is, the set of elements \(\lambda \) of the dual space \(\mathfrak {t}^*\) taking integer values on \(\Gamma \): equivalently, \(\Lambda \) is the set of the \(\lambda \in \mathfrak {t}^*\) such that \(X \mapsto e^{2\pi i \lambda (X)}\) descends to a character \(\kappa _\lambda \) of *T*. We say that a weight \(\lambda \in \Lambda \) occurs in a unitary representation \(\pi \) of *G* if the character \(\kappa _\lambda \) of *T* is contained in the restriction of \(\pi \) to *T*. Weights occurring in the (complexified) adjoint representation are called *roots*. A choice of an ordering splits roots into into *positive* and *negative* roots. We denote by \(\rho \) half the sum of the positive roots. The set \(\Lambda ^+\) of *dominant weights* is the set of the \(\lambda \in \Lambda \) having nonnegative inner product with all positive roots. The irreducible representation \(\pi _\lambda \) of *G* corresponding to \(\lambda \in \Lambda ^+\) is determined, up to equivalence, by the fact that \(\lambda \) is the highest weight occurring in \(\pi _\lambda \) (that is, \(\lambda \) occurs in \(\pi _\lambda \), while \(\lambda + \alpha \) does not occur in \(\pi _\lambda \) for any positive root \(\alpha \)).

*orbit*of \(\lambda \). Kirillov’s character formula [37, p. 459] states that, for all \(X \in \mathfrak {g}\) and all \(\lambda \in \Lambda ^+\),

### Proof of Theorem 1.4

Take a small connected conjugation-invariant neighbourhood *U* of the identity in *G* that is also symmetric, that is, \(U^{-1} = U\). Then \(U = \bigcup _{x \in G} x (U \cap T) x^{-1}\). Let *V* be the small connected neighbourhood of 0 in \(\mathfrak {g}\) such that \(U = \exp V\) and \(\exp \) is a diffeomorphism from a neighbourhood of \({\overline{V}}\) onto a neighbourhood of \({\overline{U}}\) in *G*.

*f*on

*G*supported in

*U*, we associate the function

*F*on \(\mathfrak {g}\) supported in

*V*by the formula

*F*as follows:

*f*is central on

*G*;

*F*is \({\text {Ad}}(G)\)-invariant on \(\mathfrak {g}\); and \({\hat{F}}\) is \({\text {Ad}}(G)^*\)-invariant on \(\mathfrak {g}^*\).

*f*is central and supported in

*U*, and let

*F*be the associated function on \(\mathfrak {g}\). From the character formula (4.1), a change of variables, and a change of order of integration,

*G*and for functions on \(\mathfrak {g}\), implies that

*H*on \(\mathfrak {g}^*\), we define

*V*and takes the value 1 on the open \({\text {Ad}}(G)\)-invariant subset

*W*of

*V*. For

*H*in \(L^1(\mathfrak {g}^*) + L^\infty (\mathfrak {g}^*)\), we define the function

*TH*by

*H*, the inverse Fourier transform

*F*of \({\hat{\phi }}*H^G\) is supported in

*V*and is \({\text {Ad}}(G)\)-invariant, so the corresponding function

*f*on

*G*is central and supported in

*U*. From our previous discussion,

*f*is a central function on

*G*supported in \(\exp (W) \subseteq U\), and

*F*is the \({\text {Ad}}(G)\)-invariant function on \(\mathfrak {g}\) corresponding to

*f*, then \(T{\hat{F}}(\lambda ) = {\hat{\phi }} * {\hat{F}}(\lambda +\rho ) = {\hat{F}}(\lambda +\rho )\) for all \(\lambda \in \Lambda ^+\). Hence, if \(n=\dim G\), from the Hausdorff–Young inequality on \(\mathbb {R}^n\) we deduce that

*W*small, we may make both \(\sup _{X \in W} J(X)^{1/2-1/p}\) and \(\Vert {\hat{\phi }} \Vert _1\) close to 1. So \(H_{p,{{\,\mathrm{Inn}\,}}(G)}^\mathrm{loc}(G) \le (B_p)^n\), and the converse inequality is given by Theorem 1.3. \(\square \)

## 5 The Weyl transform

*f*should rather be \(\rho ({\hat{f}})\), the pseudodifferential operator associated to the symbol

*f*in the Weyl calculus [22, Chapter 2]. Nevertheless, we shall use the definition of Weyl transform above.

*f*—this appears to be even more inappropriate, as \(\nu (f)\) is actually more closely related to the Kohn–Nirenberg calculus (see, for example, [22, (2.32)]). In any case, it is easily seen that the operators \(\nu (f)\) and \(\rho (f)\) are related by the identity

*f*, for instance Schwartz functions. In light of (5.1), we may work with \(\nu (f)\) in place of \(\rho (f)\) equally well. As discussed in the introduction, we denote by \(W_p(\mathbb {C}^n)\) the best constant

*C*in (5.2), and use the symbols \(W_p^\mathrm{loc}(\mathbb {C}^n)\), \(W_{p,K}(\mathbb {C}^n)\) and \(W_{p,K}^\mathrm{loc}(\mathbb {C}^n)\) for the corresponding local and

*K*-invariant variants.

*f*. This raises the question whether (5.4) holds for more general \(p\in [1,2]\).

### Proposition 5.1

*K*of \(\mathrm{U}(n)\) and all \(p \in [1,2]\),

### Proof

*f*on \(\mathbb {C}^n\),

*K*-invariant. Then \(Z_\lambda F^\lambda \) is also

*K*-invariant for all \(\lambda >0\). Hence, by (5.6) to (5.8),

*K*-invariant functions, a scaling, the Minkowski integral inequality (note that \(p'/p \ge 1\)) and the sharp Hausdorff–Young inequality on \(\mathbb {R}\). This shows that \(H_{p,K}(\mathbb {H}_n) \le B_p W_{p,K}(\mathbb {C}^n)\).

*K*-invariant and \(\phi : \mathbb {R}\rightarrow \mathbb {C}\) be in the Schwartz class, and let \(F = f \otimes \phi \). Then

*F*is also

*K*-invariant, and moreover \(F^\lambda = {\hat{\phi }}(\lambda ) f\). So, by applying the sharp Hausdorff–Young inequality on \(\mathbb {H}_n\) to

*F*we obtain that

*p*but restricts the class of functions

*f*and, regrettably, also requires a weight in the

*p*-norm, gives another indication that this might be true. Recall that a function

*f*on \(\mathbb {C}^{n}\) is

*polyradial*if

*f*is invariant under the

*n*-fold product group \({\text {U}}(1) \times \dots \times {\text {U}}(1)\).

### Proposition 5.2

As observed in the introduction, this inequality implies that \(W^\mathrm{loc}_{p,K}(\mathbb {C}^n) \le (B_p)^{2n}\) for \(K = {\text {U}}(1) \times \dots \times {\text {U}}(1)\), and *a fortiori* also for any larger group *K*.

### Proof

*g*on \(\mathbb {C}\cong \mathbb {R}^2\) can be written in polar coordinates as

*f*has compact support, and put \(F(z)=e^{(\pi /2) |z|^2} f(z)\). Since also \({\hat{F}}\) is radial, we may write \({\hat{F}}(\zeta )={\hat{F}}_0(|\zeta |).\) Combining (5.11) and (5.14), we obtain

*f*. For suitable functions \(\phi \) on the positive real line, let us write

*n*. The Laguerre functions must be replaced by the

*n*-fold tensor products

*n*-fold tensor products \(d\mu _k=d\mu _{k_1}\otimes \dots \otimes d\mu _{k_n}\), which are again probability measures, and so on. It then becomes evident that the proof carries over without any difficulty to this general case. \(\square \)

### Remark 5.3

There are indications that it may not be possible to establish (5.10) without the presence of the weight \(e^{(\pi /2) |\cdot |^2}\) by means of a reduction to the Euclidean Fourier transform and the Babenko–Beckner estimate, and that new techniques are required. Let us again restrict our discussion for simplicity to the case \(n=1\).

*f*and

*g*, that is,

*f*is radial and real-valued, then \(f = f^*\) and therefore

*m*factors

*f*. A reduction to the sharp estimate for the Euclidean Fourier transform \({\hat{f}}\) of

*f*would therefore require the validity of an estimate of the form

*f*(not even with some constant larger than one multiplying the right-hand side).\(\square \)

*K*of \({\text {U}}(n)\). As we will see, this can be done much as in Sect. 2. For a function \(f \in L^1(\mathbb {C}^n) + L^2(\mathbb {C}^n)\), let \(T_f\) denote the operator of twisted convolution on the left by

*f*, that is,

### Proposition 5.4

### Proof

### Remark 5.5

Given the noncommutative subject of this paper, it is natural to ask whether the best constants \(H_p(G), H^\mathrm{loc}_p(G),\dots \) are the same in the category of operator spaces (that is, quantized or noncommutative Banach spaces). To be more precise, let us equip the (commutative and noncommutative) \(L^q\)-spaces involved in the corresponding Hausdorff–Young inequality with their natural operator space structures [53]. Does the complete \(L^p \rightarrow L^{p'}\) norm of the Fourier transform coincide with the corresponding norm \(H_p(G)\) in the category of Banach spaces? In the Euclidean case of \(H_p(\mathbb {R}^n)\), this problem was asked by Pisier in 2002 to the fourth-named author, but it is still open. Éric Ricard recently noticed that such a result for the Euclidean Fourier transform (that is, its completely bounded norm is still given by the Babenko–Beckner constant raised to the dimension of the underlying space) would give the expected constants for the Weyl transform in CCR algebras and, therefore, also for the Fourier transform in the Heisenberg group. Unfortunately, Beckner’s original strategy crucially uses hypercontractivity, which has been recently proved to fail in the completely bounded setting [5]. In conclusion, the above discussion indicates one more time (see Remark 5.3) that some new ideas seem to be necessary to solve these questions.

## Notes

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