Sharp

We study inequalities in harmonic analysis in the context of non-commutative non-compact locally compact groups. Our main result is the determination of the best constant in the Hausdorff-Young inequality for Heisenberg groups. We also obtain the somewhat surprising fact that the resulting sharp inequality does not admit any extremal functions. These results are obtained after a detailed study of the operators which occur in the Fourier decomposition of the regular representation of the Heisenberg groups. These are called Weyl operators and are of independent interest. We also obtain bounds for the best constants in the Hausdorff-Young inequality and in Young's inequality on semi-direct product groups, including non-unimodular groups. In particular, for real nilpotent groups of dimension n those best constants are shown to be dominated by the corresponding best constants for IR n. Although some of our preliminary lemmas are valid for all values ofp~(1, 2) the methods we use for our main results require that p belong to the sequence 4/3, 6/5, 8/7 ..... i.e. that p', the conjugate index, be an even integer. The contents of this paper are as follows. In Section 1 we discuss Weyl operators and determine, for p' even, the best constant in a Hausdorff-Young type inequality (Theorem 1). We also show the non-existence of extremal functions for this inequality. In Section 2 we prove some general results for locally compact groups which includes a form of Young's inequality for convolution appropriate for non-unimodular groups. This is applied to arbitrary semi-direct products. Then using a duality argument which relates the inequalities of Young and of Hausdorff-Young we obtain bounds for the Hausdorff-Young inequality (Theorem 2) on unimodular semi-direct product groups (for p' even). An interesting consequence of these results is that for a connected simply connected real nilpotent Lie group of dimension n, the best constants in the inequalities of Young and Hausdorff-Young are dominated by the corresponding best constants for IR n. In Section 3 we show (Theorem 3), using the theory of Weyl operators developed


Introduction
We study inequalities in harmonic analysis in the context of non-commutative non-compact locally compact groups.Our main result is the determination of the best constant in the Hausdorff-Young inequality for Heisenberg groups.We also obtain the somewhat surprising fact that the resulting sharp inequality does not admit any extremal functions.These results are obtained after a detailed study of the operators which occur in the Fourier decomposition of the regular representation of the Heisenberg groups.These are called Weyl operators and are of independent interest.We also obtain bounds for the best constants in the Hausdorff-Young inequality and in Young's inequality on semi-direct product groups, including non-unimodular groups.In particular, for real nilpotent groups of dimension n those best constants are shown to be dominated by the corresponding best constants for IR n.Although some of our preliminary lemmas are valid for all values ofp~(1, 2) the methods we use for our main results require that p belong to the sequence 4/3, 6/5, 8/7 ..... i.e. that p', the conjugate index, be an even integer.
The contents of this paper are as follows.In Section 1 we discuss Weyl operators and determine, for p' even, the best constant in a Hausdorff-Young type inequality (Theorem 1).We also show the non-existence of extremal functions for this inequality.In Section 2 we prove some general results for locally compact groups which includes a form of Young's inequality for convolution appropriate for non-unimodular groups.This is applied to arbitrary semi-direct products.Then using a duality argument which relates the inequalities of Young and of Hausdorff-Young we obtain bounds for the Hausdorff-Young inequality (Theorem 2) on unimodular semi-direct product groups (for p' even).An interesting consequence of these results is that for a connected simply connected real nilpotent Lie group of dimension n, the best constants in the inequalities of Young and Hausdorff-Young are dominated by the corresponding best constants for IR n.In Section 3 we show (Theorem 3), using the theory of Weyl operators developed * Partially supported by the N.S.F.under grant MCS-76 06332 ** Partially supported by the N.S.F.under grant MCS-76 07219 0025 -5 831/78/023 5/017 5/$04.00 in Section 1, that for the Heisenberg groups, the best constant for the Hausdorff-Young inequality is the same as the corresponding one for ~," (p' even) but that there are no extremal functions.This can be contrasted with the classical case IR" where Gaussian functions are extremal functions.In Section 4 we use the methods of the present paper to improve on a previous estimate for the "ax + b" group (Theorem 4).
We now set down some of the notation which will be used throughout.If ~ is a complex Hilbert space, ~(~)f~) will denote the Banach space of bounded linear operators on ~, with the operator norm.Our inequalities will be stated in terms of the Banach spaces Cr(5~), 1 < r < oo, consisting of elements T of ~(Jt ~) for which II TII, = II Tllc,t~r) = [trace(T* T) r/z] 1/r < O0 . (0.1) We let j be the Fourier transform on IR" defined by where dx denotes n-dimensional Lebesgue measure and x,y is the Euclidean inner product.We also denote by ¢¢ the extension of the Fourier transform to the Lebesgue spaces LP(lR"), 1 <p__<2, given by the Plancherel theorem and the Hausdorff-Young theorem.Thus by results of Babenko (for p' an even integer) [1] and Beckner (in general) [2], iijfllLp,(~,) <A.pllfllL,(~,) ' 1 l Also, by results of Beckner [2] and Brascamp and Lieb [3], II f* 9 Jl cr(R,) < (Ar,AvAq)" II Nil L,(m-)II g II L,(,-), (0.4) In (0.3) and (0.4) and throughout this paper As we are concerned with the best possible constants in our inequalities, our methods, like those used in the proofs of (0.3) and (0.4) do not involve the Riesz Convexity theorem.
Finally, if X is a locally compact topological space, o~(X) denotes the continuous complex valued functions on X with compact support.

Wey| Operators
For xelR", consider the unitary operators U(x) and V(x) on LZ(IR ") defined by f~LZ(IR"), zEIR", (1.2) 1"77 It is well known and easy to verify that U and V are each n-parameter unitary groups which are unitarily equivalent (as groups) to each other via the Fourier Plancheret transform, i.e.JV(x)j-1 = U(x), x~IR", ( and and which satisfy the following commutation relation: For a measurable function F on 11t 2", the Weyl operator corresponding to F is the operator on LE(IR ") The operator K v certainly exists as an element of ~(L2(IR")) if F~ L 1 (R2,) and in fact A routine calculation shows that K v is an integral operator (1.11) states that we," ~ 1.It has been pointed out (Russo [ 14]) that wv, .< 1 for all p and n.The following theorem gives a complete analysis of the inequality 11KF]I v, < wv,,]l Flip, (1.12) for certain values of p.
(1.31) Our aim is to show that F = 0 a.e.For this purpose let S v be the set where F is not zero.We shall show that S v is a null set.We can write F = IFI U where I UI = 1 and then cancel all non-zero terms in (1,31).The result is for (x, y)s IR 2n -N, and (z, w)~ (IR 2n -N<x.y))c~Svn((x,y) + St).
Consider now the function/~(x, y)= F(y,-x), x, ye IR ".Using the properties of Weyl operators it is easily seen that (JKFJ-1).=K:.This shows that if F is an extremal function, then so is/~.We note that z(S:)= Sv where z(x, y)= (y, -x) and that we can write/~= [FIV where IV[ = 1 and V can be chosen such that
This completes the proof of Theorem 1.

Inequalities on Locally Compact Groups
Let G be a locally compact group and denote a right Haar measure on G by dx or dgx.Convolution and norms are defined with respect to dRx as follows: f .g(x) = i f(xy-1)g(y)dRy " (2.1) The symbol A, or A~ when necessary, denotes the modular function of G.It follows from a careful application of HOlder's inequality that tlfA-1fq',gltq<tlflllllgllq, l <q<oo , (2.3) where as usual 1 1 -+ =1.Thus for fixed gELq(G), the map f--,fA-11q', O on q simple functions is of type (1,q) with norm ~llgll~ and of type (q', ~) of norm __< Ilgll~.By the Riesz-Thorin Theorem this map is of type (Pt, q,) where 0< t< 1, 1

-t t 1-t t
Pt-1 + q"qt-q +--'oo If we setr=qtandp=ptweobtainageneralization of Young's inequality which we state as a lemma.(2.5) Remark 2.2.There is a corresponding inequality which uses left Haar measure and which takes the form tlf, gA t/p' ti, < llfllplIgllq, + -= t + 1.This can be proved in q r the same way and in this case the convolution is given by f*g(x) = ~f(y)g(y-lx)dLy where dLx is a left Haar measure.In this paper we shall always G use right Haar measures.
For any locally compact group G and p,q, re[1, oo] we define (using right Haar measure) when -+ -= 1 + -, p q r Thus y(O(~p,q,_j, which we shall sometimes write as Y,,q is the best constant in (2.5) and is =< 1.By repeated application of (2.5) we can obtain We shall now consider Young's inequality (Lemma 2.1) for semi-direct products.For direct products, the inequality we obtain in Lemma 2.4 in easily seen to be an equality.We conjecture this to be also the case for semi-direct products.
For certain indices and certain groups this is a consequence of Lemma 2.6 and Theorem 3.
Consider now a pairX, A of locally compact groups together with a homomorphism a-~% of A into the group of automorphisms of X.We suppose that (x, a)-~%(x) is continuous from X × A to X so that the semi-direct product group G =X ® A becomes a locally compact topological group with the product topology and multiplication (x, a)(y, b)=(x%(y), ab).We shall write a(x) instead of %(x).We recall (Hewitt and Ross [5, (15.29)]) that a right Haar measure on G is the product of the right Haar measures dRX on X and dRa on A and that the modular functions of the groups X,A, and G=X®A are related by A6(x, a) = 6(a)AA(a)Ax(X), (x, a)e G, (2.9) where 6 is (a homomorphism of A) defined by
We turn now to a discussion of the Hausdorff Young theorem for a locally compact group.The generalization to all locally compact Abelian groups of the inequality (0.3) with constant 1 is well known [5, (31.20)].So is the corresponding inequality for compact non-Abelian groups [5,(31.22)].In 1958 Kunze [9] extended the Riesz-Thorin Theorem to the setting of operator algebras and thereby was able to prove a Hausdorff-Young theorem for any locally compact unimodular group G.This theorem took the form IIz~llp,<llfll v, l<p<2,-+1 1-7=1 , yeLP(G), (2.13) P P where L I is the operator of convolution by f on the left in the Hilbert space L2(G) and the norm llLfllp, is defined by using a generalized trace cononically constructed from the group.This result was new even for compact groups and subsumed the usual Hausdorff-Young theorem in case G was Abelian.A concrete realization of (2.13) can be given if the unimodular group G is separable and of Type I (Lipsman [10,Theorem 22]).In this case letting G denote the space of unitary equivalence classes of continuous irreducible unitary representations of G, there is a measure/L~ on G such that The Hausdorff-Young inequality (2,13) now asserts We shall identify L I with the family { U~(f)}2~ 6 and consider it to be the Fourier transform of f.We define Ae(G ), for 1 < p < 2 to be the best constant in inequality (2.13)   Remark 2.7.In the proof of Lemma 2.6 we have used the following facts from Kunze [9]: (i) equality holds in (2.13) for p = 2.
Theorem 2. If G =X@ A is a semi-direct product of locally compact unimodular 2k groups X and A and if G is unimodular, then for p= (2k-1)' k an integer >2, we have

Ap(G) k = Yp(G) <= Yp(X) Yp(A)= Ap(X)kAp(A) k .
We now give some examples to which Theorem 2 applies.First let G = IR x H be a direct product where H is an arbitrary unimodular group.By Theorem 2 we have Ap(IR x H)< Ap(IR)Ap(H) for p' an even integer.As noted previously, since this is a direct product we have equality.On the other hand it is proved in Russo [12, Theorem 2] that equality holds here for all pc(l, 2).Next consider a semi-direct product IR"@ K where K is compact.By Theorem 2, for p' even Ap(IR"@ K) NAp(N") since Ap(K)= 1.For this example the proof of [12,Theorem 4] shows that equality holds for p' an even integer.Therefore Theorem 2 gives no new information for these examples.Consider next a connected simply connected real nilpotent Lie group F. A consequence of [14, Proposition 12] is that for all p~(1,2), Ap(F)<Atp where f is the dimension of the center of F. This can be improved using Theorem 2. Corollary 2.8.Let F be a connected simply connected real nilpotent Lie group of dimension n.Then if p=2k/(2k-1) for some integer k > 2, Ap(F)< A~.
Proof If n= 1 (or 2) F is the Abelian group IR (or IR 2) and Ap(IR)=Ap (and Ap(IR 2) = A2).If n > 2, write F = F'Q A where F' has dimension n-1 and A-~IK Then Theorem 2 and the induction hypothesis gives the corollary.
We shall show in the next section that if for example, n = 3, then equality holds in Corollary 2.8.Two other interesting groups for which Theorem 2 gives a specific bound less than 1 are the inhomogeneous Lorentz groups and the Oscillator group (see Kleppner and Lipsman [8]).Other examples of important semi-direct products can be found in Wolf [15].We note that, according to Fournier [4], Ap(G)< 1 if G has no compact open subgroups.

Helsenberg Groups
In this section F will denote the Heisenberg group of dimension 2n + 1, n > 1.Thus the points of F are triples 7=(x,y,t) with x,y~lR" and t~lR and the group multiplication is
Using the theory of Weyl operators developed in Section 1 we will prove, for p' an even integer, the best possible inequality of the type (3.5).Theorem 3. Let pc(t, 2) be of the form p =2k/(2k-t) for some integer k >2 and let F be the Heisenberg group of dimension 2n + 1.Then for (x, y)eIR z".

Lemma 2 . 1 .
Let G be a locally compact group with modular function A. If -+ -P q = 1 + _1 and p, q, re [1, oo] and if norms and convolution are defined relative to a r right Haar measure, then f A -~*g r < Il ftlpllg [Iq .