FOURIER TRANSFORMATION ON NON-UNIMODULAR LOCALLY COMPACT GROUPS MARIANNE TERP. Communicated by M. S. Moslehian

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1 Adv. Oper. Theory 2 (207), no. 4, ISSN: X (electronic) L p FOURIER TRANSFORMATION ON NON-UNIMODULAR LOCALLY COMPACT GROUPS MARIANNE TERP Communicated by M. S. Moslehian Abstract. Let G be a locally compact group with modular function and left regular representation λ. We define the L p Fourier transform of a function f L p (G), p 2, to be essentially the operator λ(f) q on L 2 (G) (where p + q = ) and show that a generalized Hausdorff Young theorem holds. To do this, we first treat in detail the spatial L p spaces L p (ψ 0 ), p, associated with the von Neumann algebra M = λ(g) on L 2 (G) and the canonical weight ψ 0 on its commutant. In particular, we discuss isometric isomorphisms of L 2 (ψ 0 ) onto L 2 (G) and of L (ψ 0 ) onto the Fourier algebra A(G). Also, we give a characterization of positive definite functions belonging to A(G) among all continuous positive definite functions. Introduction Suppose that G is an abelian locally compact group with dual group Ĝ. Then the Hausdorff Young theorem states that if f L p (G), where p 2, then its Fourier transform F(f) belongs to L q (Ĝ), where + = (cf. [23, p. 7]). p q In the case of Fourier series, i.e. when G is the circle group and Ĝ the integers, this is a classical result due to F. Hausdorff and W. H. Young. [24, p. 0]. An extension of this theorem to all unimodular locally compact groups was given by R. A. Kunze [4]. In this paper we shall treat the case of general, i.e. not necessarily unimodular, locally compact group. Copyright 206 by the Tusi Mathematical Research Group. Date: Received: Aug. 9, 207; Accepted: Sep. 9, Mathematics Subject Classification. Primary 39B82; Secondary 44B20, 46C05. Key words and phrases. L p Fourier transformation, locally compact group, Fourier algebra, positive definite function. 547

2 548 M. TERP In order to describe our results, we first briefly recall those of [4]. Suppose that f is an integrable function on a unimodular group G. Then we consider the Fourier transform F(f) to be the operator λ(f) of left convolution by f on L 2 (G). (As pointed out by Kunze [4], this point of view is justified by the fact that in the 2 abelian case λ(f) is unitarily equivalent to the operator on L (Ĝ) of multiplication by the (ordinary) Fourier transform ˆf. The Fourier transformation maps L (G) into the space L (G ), defined as the von Neumann algebra M generated by λ(l (G)). More generally, one can define λ(f) as an (unbounded) operator on L 2 (G) even for functions f not in L (G). It then turns out that λ maps each L p (G), p 2, norm-decreasingly into a certain space L q (G ) of closed densely defined operators on L 2 (G) (where + = ). This is the Hausdorff Young p q theorem. Kunze introduced the spaces L q (G ) as spaces of measurable operators (in the sense of [2]) with respect to the canonical gage on M [4, p. 533]. An equivalent but simpler way of introducing the L q (G ) is to consider the trace ϕ 0 on M characterized by ϕ 0 (λ(h) λ(h)) = h 2 2 for certain functions h, and then take L q (G ) to be L q (M, ϕ 0 ) as defined by E. Nelson [5], viewing it as a space of ϕ 0 -measurable operators [5, Theorem 5]. (In either case, the L q spaces obtained are isomorphic to the abstract L q spaces of J. Dixmier [5] associated with a trace on a von Neumann algebra.) In the general (non-unimodular) case, ϕ 0 is no longer a trace, and the lack of adequate spaces L q into which the L p (G) were to be mapped for a long time prevented the formulation of a Hausdorff Young theorem, except for some special cases ([7, 8], [20, Proposition 5]). In [0], however, U. Haagerup constructed abstract L p spaces corresponding to an arbitrary von Neumann algebra, and combining methods from [0] with the recent theory of spatial derivatives by A. Connes [2], M. Hilsum has developed a spatial theory of L p spaces [2]. If M is a von Neumann algebra acting on a Hilbert space H and ψ is a weight on its commutant M, then the elements of L p (M, H, ψ) are (in general unbounded) operators on H satisfying a certain homogeneity property with respect to ψ. We shall see that when using these spaces (in the particular case of M = λ(g), H = L 2 (G), and ψ = the canonical weight on M ) and when defining the L p Fourier transform of an L p function f to be the operator ξ f q ξ on L 2 (G) (where is the modular function of the group), one gets a nice L p Fourier transformation theory and in particular a Hausdorff Young theorem. The paper is organized as follows. In Section we fix the notations and describe our set-up. In Section 2, we study the L p spaces of [2] in our particular case; we give a reformulation of the α-homogeneity property appearing in [2] that does not involve modular automorphism groups and we characterize L p (ψ 0 ) operators among all ( )-homogeneous operators. In Section 3, we treat the case p = 2 p and obtain explicit expressions for the L 2 Fourier transformation F 2 = P, called the Plancherel transformation, as well as for its inverse. Next, in Section 4, we deal with the case of a general p [, 2] ; we define the L p Fourier transformation F p, and using interpolation (specifically, the three lines theorem) we prove our version of the Hausdorff Young theorem.

3 L p FOURIER TRANSFORMATION 549 Finally, in Section 5, we define an L p Fourier cotransformation F p taking L p (ψ 0 ), p 2, into L q (G) and we investigate the relations between cotransformation and Fourier inversion. A detailed study of the p = case gives a new characterization of A(G) + functions among all continuous positive definite functions on G.. Preliminaries and notation Let G be a locally compact group with left Haar measure dx. We denote by K(G) the set of continuous functions on G with compact support and by L p (G), p, the ordinary Lebesgue spaces with respect to dx. The modular function on G is given by f(xa )dx = (a) f(x)dx for all f K(G) and a G. For functions f on G we put ˇf(x) = f(x ), f(x) = f(x ), f (x) = (x)f(x ) and (Jf)(x) = 2 (x)f(x ) for all x G. More generally, for each p [, ], we define (J p f)(x) = /p (x)f(x ), x G. Then in particular J f = f, J 2 f = Jf, J f = f. Note that for each p [, ], the operation J p is a conjugate linear isometric involution of L p (G). We shall often make use of the following non-unimodular version of Young s inequalities for convolution: Lemma.. (Young s convolution inequalities.) Let p, p 2, p [, ] and p + =. Assume that p + p 2 =. Then for all f p L p (G) and f 2 L p 2 (G) q the convolution product f q f 2 exists and belongs to L p (G), and f q f 2 p f p f 2 p2. This theorem is well-known in the unimodular case as well as in the special cases (p, p 2, p) = (p, q, ) (where it follows from Hölder s inequality), (p, p 2, p) = (, p, p) or (p, p 2, p) = (p,, p) [, (20.4)], The general case has also been noted [3, Remark 2.2]. It can be proved by modifying the proof of [, (20.8)] or by interpolation from the special cases mentioned above. For operators T on the Hilbert space L 2 (G) we use the notation D(T ) (domain of T ), R(T ) (range of T ), N(T ) (kernel of T ). If T is preclosed, we denote by [T ] the closure of T. If T is a positive self-adjoint operator and P the projection onto N(T ), then by definition T it, t R, is the partial isometry coinciding with the unitary (T P ) it on N(T ) and O on N(T ). By convention, when speaking of operators, bounded always means bounded and everywhere defined. We denote by λ and ρ the left and right regular representations of G on L 2 (G), i.e. the unitary representations given by (λ(x)f)(y) = f(x y),

4 550 M. TERP (ρ(x)f)(y) = 2 (x)f(yx), for all x, y G and f L 2 (G). The corresponding representations of the algebra L (G) (as in [4, 3.3]) are given by λ(h)f = h f and ρ(h)f = f 2 ȟ for all h L (G) and f L 2 (G). We denote by M the von Neumann algebra of operators on L 2 (G) generated by λ(g) (or λ(k(g)), or λ(l (G))). In other words, M is the left von Neumann algebra of K(G), where K(G) is considered as a left Hilbert algebra [3, Definition 2.] with convolution, involution *, and the ordinary inner product in L 2 (G). The commutant M of M is the von Neumann algebra generated by ρ(g), and M = JMJ. A function ξ L 2 (G) is called left (resp. right) bounded if left (resp. right) convolution with ξ on K(G) extends to a bounded operator on L 2 (G), i.e. if there exists a bounded operator λ(ξ) (resp. λ (ξ)) such that k K(G) : λ(ξ)k = ξ k (resp. λ (ξ)k = k ξ). The set of left (resp. right) bounded L 2 (G)-functions is denoted A l (resp. A r ). Obviously, K(G) A l, K(G) A r, and for ξ K(G) we have λ (ξ) = ρ( 2 ˇξ). Note that ξ L 2 (G) is left bounded if and only if the operator η λ (η)ξ : A r L 2 (G) extends to a bounded operator on L 2 (G); if this is the case, we have λ(ξ)η = λ (η)ξ for all η A r. (Our definition of left-boundedness therefore agrees with [, Définition 2.]). If ξ A l and T M, then T ξ A l and λ(t ξ) = T λ(ξ). We denote by ϕ 0 the canonical weight on M [, Définition 2.2]. Then the weight ψ 0 on M given by ψ 0 (y) = ϕ 0 (JyJ) for all y (M ) + is called the canonical weight on M. The corresponding modular automorphism groups are given by σ ϕ 0 t (x) = it x it, x M, σ ψ 0 t (y) = it y it, y M, for all t R. Here, denotes the multiplication operator on L 2 (G) by the function (note that we shall not distinguish in our notation between the function and the corresponding multiplication operator). With this definition, is in fact the modular operator of K(G) (as defined in [3, Lemma 2.2]). It follows from the defining property of ϕ 0 [, Théorème 2.] that for all y M we have { η 2 ψ 0 (y y) = 2 if y = λ (η) for some η A r, otherwise We identify the Hilbert space completion H ψ0 of n ψ0 = {y M ψ 0 (y y) < } with L 2 (G) via η λ (η). Now recall that by definition [2, Definition ], D(L 2 (G), ψ 0 ) is the set of ξ L 2 (G) such that y yξ : n ψ0 L 2 (G) extends to a bounded operator R ψ 0 (ξ) : H ψ0 L 2 (G), i.e., in view of the identification of H ψ0 with L 2 (G), such that η λ (η)ξ : A r L 2 (G) extends to a bounded operator on L 2 (G). Thus D(L 2 (G), ψ 0 ) = A l, and for all ξ D(L 2 (G), ψ 0 ) we have R ψ 0 (ξ) = λ(ξ).

5 L p FOURIER TRANSFORMATION 55 If ϕ is a normal semi-finite weight on M, then by definition [2], unique positive self-adjoint operator T satisfying { ξ A l : ϕ(λ(ξ)λ(ξ) T 2 ξ ) = 2 if ξ D(T 2 ) otherwise and In particular, we have T 2 = [T 2 Al D(T 2 ) ]. dϕ 0 = dϕ is the (cf. [2, Lemma 0 (b)] together with the proof of [2, Lemma 0 (a)]. ( ) dϕ If ϕ is a functional, then by the definition of we have A l D ( dϕ ) 2 [ ] and ( dϕ ) 2 = ( dϕ ) 2 Al. Finally, we note that the predual space M of the von Neumann algebra M may be viewed as a space of functions on the group in the following manner: for each ϕ M, define u : G C by u(x) = ϕ(λ(x)), x G. Then u is a continuous function on the group determining ϕ completely. The linear space of such functions, normed by u = ϕ, is exactly the Fourier algebra A(G) of G introduced by P. Eymard [6] (this follows from [6, Théorème (3.0)]). The identification of A(G) with M is such that ϕ, λ(f) = ϕ(x)f(x)dx for all ϕ M A(G) and all f L (G). Recall that by [4, 3.4.4] a continuous function ϕ on G is positive definite if and only if ξ K(G) : ϕ(x)(ξ ξ )(x)dx 0, i.e. if and only if ξ K(G) : ϕ(yx )ξ(y)ξ(x)dydx 0. If ϕ A(G), then ϕ is positive definite if and only if the corresponding functional ϕ M is positive. We denote by A(G) + the set of positive definite ϕ A(G). 2. Homogeneous operators on L 2 (G) and the spaces L p (ψ 0 ) Definition 2.. Let α R. An operator T on L 2 (G) is called α-homogeneous if x G : ρ(x)t α (x)t ρ(x).

6 552 M. TERP Remark 2.2. () The O-homogeneous operators are precisely the operators affiliated with M. (2) If T is α-homogeneous, then actually ρ(x)t = α (x)t ρ(x) for all x G (to see this, replace x by x in the definition). (3) If T and S are both α-homogeneous, then T + S is α-homogeneous. If T is α-homogeneous and S is β-homogeneous, then T S is (α + β)-homogeneous. If T is densely defined and α-homogeneous, then T is also α-homogeneous. If T is positive self-adjoint and α-homogeneous and β R +, then T β is (αβ)- homogeneous (use ρ(x)t β ρ(x ) = (ρ(x)t ρ(x )) β ). (4) If T is α-homogeneous for some α R, then the projection onto N(T ) belongs to M (since N(T ) is invariant under all ρ(x), x G). (5) If a preclosed operator T is α-homogeneous, then its closure [T ] is also α-homogeneous. (6) For each α R, α is α-homogeneous. Lemma 2.3. Let T be a closed densely defined operator on L 2 (G) with polar decomposition T = U T. Let α R. Then T is α-homogeneous if and only if U M and T is α-homogeneous. Proof. If T is α-homogeneous, then, by Remark 2.2(3), T = (T T ) 2 is also α- homogeneous. Then for all x G and ξ D( T ) we have ρ(x)u T ξ = ρ(x)t ξ = α (x)t ρ(x)ξ = α (x)u T ρ(x)ξ = Uρ(x) T ξ, i.e. ρ(x)u Uρ(x) on R( T ). Since the projection onto R( T ) = N( T ) belongs to M, we conclude that U commutes with all ρ(x); thus U M. The if -part follows directly from Remarks 2.2(3) and 2.2(). Lemma 2.4. Let T be a closed densely defined operator on L 2 (G) and α C. If then x G : ρ(x)t α (x)t ρ(x), f K(G) : λ (f)t T λ ( α f). Proof. Let f K(G) and ξ D(T ). Then for all η D(T ) we have (ρ(f)t ξ η) = f(x)(ρ(x)t ξ η)dx = f(x) α (x)(t ρ(x)ξ η)dx = α (x)f(x)(ρ(x)ξ T η)dx = (ρ( α f)ξ T η). This shows that ρ( α f)ξ D(T ) = D(T ), and T ρ( α f)ξ = ρ(f)t ξ for all ξ D(T ), i.e. ρ(f)t T ρ( α f). Hence for all f K(G) we have λ (f)t = ρ( 2 ˇf) T ρ( α 2 ˇf) = T λ ( α f).

7 L p FOURIER TRANSFORMATION 553 Lemma 2.5. Let T be a closed densely defined operator on L 2 (G), α-homogeneous for some α R. Let ξ A l. Then for all t R we have T it ξ A l and λ( T it ξ) λ(ξ). Proof. By Lemma 2.3, we have ρ(x) T ρ(x ) = α (x) T for all x G, whence ρ(x) T it ρ(x ) = iαt (x) T it for all x G and all t R. Then, applying the preceding lemma to T it, we obtain for all η K(G) that T it ξ η = λ (η) T it ξ = T it λ ( iαt η)ξ = T it λ(ξ) iαt η and thus T it ξ η 2 T it λ(ξ) iαt η 2 λ(ξ) η 2. We conclude that T it ξ is left bounded and that λ( T it ξ) λ(ξ). Remark 2.6. In particular, it ξ A l with λ( it ξ) λ(ξ) for all ξ A l and t R Our next lemma shows that α-homogeneity as defined here is equivalent to homogeneity of degree α with respect to ψ 0 as defined in [2, Definition 7]. Lemma 2.7. Let α R, and let T be a closed densely defined operator on L 2 (G) with polar decomposition T = U T. Then the following conditions are equivalent: (i) T is α-homogeneous, (ii) U M and y M t R : σ ψ 0 αt (y) T it = T it y. Proof. By Lemma 2.3, we may assume that T is positive self-adjoint. Denote by P the projection onto N(T ). If either (i) or (ii) holds, then P is in M, and thus the subspace P L 2 (G) is invariant under all operators considered. Therefore, we may suppose that P M, and the lemma is proved when we have shown the equivalence of x G : ρ(x)t ρ(x )P = α (x)t P (2.) and t R y M : σ ψ 0 αt (y)p = T it yt it P. (2.2) Now for all x G we have since σ ψ 0 αt (ρ(x)) = iαt ρ(x) iαt = iαt (x)ρ(x) ( iαt ρ(x) iαt f)(z) = it (z) 2 (x) it (zx)f(zx) = it (x)(ρ(x)f)(z) for all f L 2 (G) and all x, z G. Then, since M is generated by the ρ(x), the condition (2.2) is equivalent to x G or (changing t into t) t R : iαt (x)ρ(x)p = T it ρ(x)t it P x G t R : ρ(x)t it ρ(x)p = iαt (x)t it P, which in turn is equivalent to (2.).

8 554 M. TERP Now, by [2, Theorem 3] a positive self-adjoint operator on L 2 (G) is ( )- homogeneous if and only if it has the form dϕ for a (necessarily unique) normal semi-finite weight ϕ on M. We define the integral with respect to ψ 0 of a positive self-adjoint ( )- homogeneous operator T as T = ϕ() [0, ], where T = dϕ. If T <, i.e. if ϕ is a functional, we shall say that T is integrable. (These definitions agree with those given in [2, remarks following Corollary 8].) For each p [, ), we denote by L p (ψ 0 ) the set of closed densely defined ( p )-homogeneous operators T on L2 (G) satisfying T p <. (Note that T p is ( )-homogeneous, so that T p is defined.) We put L (ψ 0 ) = M. The spaces L p (ψ 0 ) introduced here are special cases of the spatial L p -spaces of M. Hilsum [2]. We recall their main properties (note, however, that our notation differs from that of [2] in that we maintain throughout the distinction between operators and their closures): If T, S L p (ψ 0 ), then T + S is densely defined and preclosed, and the closure [T + S] belongs to L p (ψ 0 ). With the obvious scalar multiplication and the sum (T, S) [T +S], L p (ψ 0 ) is a linear space, and even a Banach space with the norm. p defined by T p = ( T p ) /p if p [, ) and T p = T (operator norm) if p =. The operation T T is an isometry of L p (ψ 0 ) onto L p (ψ 0 ). We denote L p (ψ 0 ) + the set of positive self-adjoint operators belonging to L p (ψ 0 ) By linearity, T T defined on L (ψ 0 ) + extends to a linear form on the whole of L (ψ 0 ) satisfying T = T and T T for all T L (ψ 0 ). Let p, p 2, p [, ] such that p + p 2 = p. If T Lp (ψ 0 ) and S L p 2 (ψ 0 ), then the operator T S is densely defined and preclosed, its closure [T S] belongs to L p (ψ 0 ), and [T S] p T p S p2. In particular, if T L p (ψ 0 ) and S L q (ψ 0 ), where + =, then [T S] p q L (ψ 0 ) and [T S] T p S q (Hölder s inequality); furthermore, [T S] = [ST ]dψ0. If p [, ) and + =, then we identify p q Lq (ψ 0 ) with the dual space of L p (ψ 0 ) by means of the form (T, S) [T S], T L p (ψ 0 ). S L q (ψ 0 ). In particular, L (ψ 0 ) is the predual of M = L (ψ 0 ). The space L 2 (ψ 0 ) is a Hilbert space with the inner product (T S) L 2 (ψ 0 ) = [S T ]. Remark 2.8. Suppose that G is unimodular. Then the α homogeneous operators for any α are simply the operators affiliated with M and the canonical weight

9 L p FOURIER TRANSFORMATION 555 ϕ 0 on M is a trace. We claim that T = ϕ 0 (T ) for all positive self-adjoint operators T affiliated with M, where we have written ϕ 0 (T ) for the value of ϕ = ϕ 0 (T.) at (with ϕ 0 (T.) defined as in [7, 4]). To see this, recall that dϕ 0 = =, so that using [2, Theorem 9, (2)], we have ( ) it ( ) it ( ) it dϕ T it dϕ0 dϕ = (Dϕ : Dϕ 0 ) t = = for all t R. Thus T = dϕ, and T = ϕ() = ϕ 0 (T ). (When proving T = dϕ, we implicitly assumed that T is injective so that ϕ = ϕ 0 (T.) is faithful. In the general case, denote by Q M the projection onto N(T ), note that T + Q is positive self-adjoint, affiliated with M, and injective, and verify that T + Q = dϕ 0((T + Q).) = dϕ 0(T.) + dϕ 0(Q.). Since the supports of dϕ 0(T.) and dϕ 0(Q.) are Q and Q, respectively, we conclude that T = dϕ 0(T.) as desired.) It follows that in this case the spaces L p (ψ 0 ) reduce the ordinary L p (M, ϕ 0 ) (discussed in the introduction). Returning to the general case, we now proceed to a more detailed study of the spaces L p (ψ 0 ). For this, we shall need the following slightly generalized version of [2, II, Proposition 2]. Lemma 2.9. Suppose that T is a positive self-adjoint operator on L 2 (G) and α homogeneous for some α R. Let ξ A l. Then for each n N there exists ξ n A l ( β R+ D(T β ) ) such that (i) n N : λ(ξ n ) λ(ξ), (ii) ξ n ξ as n, (iii) T β ξ n T β ξ as n whenever ξ and β R + satisfy ξ D(T β ). Proof. For each n N, define f n : [0, ) C by { f n (x) = π e t2 x it n dt if x > 0 if x = 0. Since for all x [0, ) we have f n (x) π e t2 dt =, the operators f n (T ) are bounded. For each η N, put ξ n = f n (T )ξ. To prove that the ξ n belong to A l and satisfy (i), denote by P the projection onto N(T ) and observe that for all η K(G) we have f n (T )P ξ η = λ (η)f n (T )P ξ = π = π = π e t2 λ (η)t it n ξdt e t2 T it n λ ( iαt n η)ξdt e t2 T it n (ξ iαt n η)dt,

10 556 M. TERP where we have used Lemma 2.4. It follows that f n (T )P ξ η 2 e t2 λ(ξ) iαt n η 2 dt λ(ξ) η 2. π On the other hand, ( P )ξ η 2 λ(( P )ξ) η 2 λ(ξ) η 2, since P M. In all, f n (T )ξ = f n (T )P ξ + ( P )ξ belongs to A l and λ(f n (T )ξ) λ(ξ). Now, to see that ξ n D(T β ) for all β R +, note that f n (x) = π e t2 e it n log x dt = e (log x)2 4n π = e (log x)2 4n e (t i 2 n log x)2 dt for all x > 0. Then x x β (β log x f n (x) = e 4n (log x)2) is bounded, so that T β f n (T ) is a bounded operator, and thus f n (T )ξ D(T β ). Since f n is bounded and f n (x) as n for all x [0, ), we have f n (T )ζ ζ as n for all ζ. From this, we immediately get (ii) and (iii). Indeed, ξ n = f n (T )ξ ξ, and if ξ D(T β ), then T β ξ n = T β f n (T )ξ = f n (T )T β ξ T β ξ. Proposition 2.0. Let T be a closed densely defined ( ) homogeneous operator on L 2 (G). Then the following conditions are equivalent: (i) T L (ψ 0 ), (ii) there exists a constant C 0 such that ξ A l D(T ) (iii) there exists a constant C 0 such that η A l : (T ξ η) C λ(ξ) λ(η), ξ A l D( T 2 ) : T 2 ξ 2 C λ(ξ) 2, (iv) there exists an approximate identity (ξ i ) i I in K(G) + such that all ξ i D( T 2 ) and lim inf i I T 2 ξi <. If T L (ψ 0 ), then A l D( T 2 ), and for any approximate identity (ξ i ) i I in K(G) + we have T = lim i I T 2 ξi 2. Furthermore, T is the smallest C satisfying (ii) and the smallest C satisfying (iii).

11 L p FOURIER TRANSFORMATION 557 Proof. Let T = U T be the polar decomposition of T. First, suppose that T L (ψ 0 ). Then T L (ψ 0 ) +, and therefore T = dϕ for some positive functional ϕ on M. Recall that A l D( T 2 ). Thus for all ξ A l D(T ) and η A l we have (T ξ η) = ( T 2 ξ T 2 U η) = ϕ(λ(ξ)λ(u η)) ϕ λ(ξ) λ(u η) T λ(ξ) λ(η), i.e. (ii) holds. Next, suppose that T satisfies (ii). Then for all ξ A l D( T ) we have T 2 ξ 2 = (T ξ Uξ) C λ(ξ) λ(uξ) C λ(ξ) 2. Now if ξ A l D( T 2 ), there exist (by Lemma 2.9) ξ n A l D( T ) such that T 2 ξ n T 2 ξ and λ(ξ n ) λ(ξ). Since T 2 ξn 2 C λ(ξ n ) 2 C λ(ξ) 2, we conclude that T 2 ξ 2 C λ(ξ) 2. Thus (iii) is proved. Now suppose that T satisfies (iii). First we show that this implies A l D( T 2 ). Let ξ A l. Then by Lemma 2.9 there exist ξ n A l D( T 2 ) such that ξ n ξ and λ(ξ n ) λ(ξ). Then for all η D( T 2 ) we have ( T 2 ξn η) T 2 ξn η and We conclude that C /2 λ(ξ n ) η C /2 λ(ξ) η ( T 2 ξn η) = (ξ n T 2 η) (ξ T 2 η). η D( T 2 ) : (ξ T 2 η) C /2 λ(ξ) η. Thus ξ D( T 2 ) as wanted. Now, still assuming (iii), let us prove (iv). Let (ξ i ) i I be any approximate identity in K(G) +. Then automatically all ξ i K(G) A l D( T 2 ), and λ(ξ i ) ξ i = so that T 2 ξi 2 C λ(ξ i ) 2 C, whence lim inf i I T 2 ξ i C 2 <. Finally, suppose that T satisfies (iv) for some (ξ i ) i I. Note that since (ξ i ξ i )(x)dx =, (ξ i ξ i ) i I is again an approximate identity in K(G) +. Therefore,

12 558 M. TERP λ(ξ i )λ(ξ i ) = λ(ξ i ξ i ) convergence strongly, and hence weakly, to in M. Since all λ(ξ i )λ(ξ i ), this convergence is also σ-weak, and by the σ-weak lower semicontinuity of ϕ, this implies ϕ() lim inf i I ϕ(λ(ξ i )λ(ξ i ) ) = lim inf T 2 ξi 2 i I C lim inf λ(ξ i ) 2 i I C <. Since ϕ() = T <, we have T L (ψ 0 ), i.e. (i) holds. Note that once ϕ() < is established, ϕ is known to be σ-weakly lower continuous and thus ϕ() = lim i I ϕ(λ(ξ i )λ(ξ i ) ) = lim i I T 2 ξi 2 for any approximate identity (ξ i ) i I, i.e. T = lim i I T 2 ξi 2. In the course of the proof we observed that T may be used as the constant C in (ii), that every constant C satisfying (ii) also satisfies (iii), and that any C satisfying (iii) is bigger than lim i I T 2 ξ i 2, i.e. bigger than T. This proves the remarks that end Proposition 2.0 As an immediate corollary, we have: Proposition 2.. Let T be a closed densely defined ( )-homogeneous operator 2 on L 2 (G). Then the following conditions are equivalent: (i) T L 2 (ψ 0 ), (ii) there exists a constant C 0 such that ξ A l D(T ) : T ξ C λ(ξ), (iii) there exists an approximate identity (ξ i ) i I in K(G) + such that all ξ i D(T ) and lim inf T ξ i <. i I If T L 2 (ψ 0 ), then A l D(T ), and for any approximate identity (ξ i ) i I in K(G) + we have T 2 = lim T ξ i ; i I furthermore, T 2 is the smallest constant C satisfying (ii). We now come to the case of a general p [, ). Suppose that T L P (ψ 0 ) and S L q (ψ 0 ), where + =. Then by [2, II, Proposition 5, ], we have p q (T ξ Sη) = [S T ], λ(ξ)λ(η) for all ξ A l D(T ) and η A l D(S). (Here,.,. denotes the form giving the duality of L (ψ 0 ) and M.) Using Hölder s inequality, we get (T ξ Sη) [S T ] λ(ξ)λ(η) T p S q λ(ξ) λ(η)

13 L p FOURIER TRANSFORMATION 559 for all such ξ and η. This kind of inequality in fact characterizes L p (ψ 0 )-operators among all ( ) homogeneous operators: p Proposition 2.2. Let p [, ] and define q by p + q =. Let T be a closed densely defined ( p ) homogeneous operator on L2 (G). Then the following conditions are equivalent: (i) T L p (ψ 0 ), (ii) there exists a constant C 0 such that S L q (ψ 0 ) ξ A l D(T ) η A l D(S) : (T ξ Sη) C S q λ(ξ) λ(η). If T L p (ψ 0 ), then T p is the smallest C satisfying (ii). Proof. In view of the remarks preceding this proposition, we just have to show that if T satisfies (ii) for some constant C, then T L p (ψ 0 ), and T p C. Therefore suppose that T with polar decomposition T = U T satisfies (ii). Then also ( T ξ Sη) = (T ξ U Sη) C [U S] q λ(ξ) λ(η) C S q λ(ξ) λ(η) for all S, ξ, and η chosen as in (ii). Then we may assume that T is positive self-adjoint. Let S L q (ψ 0 ) and η A l D(T 2 S). We claim that for all ξ A l D(T 2 ) we have (T 2 ξ (T 2 Sη) C S q λ(ξ) λ(η). (2.3) If ξ A l D(T ), this follows directly from the hypothesis. In case of a general ξ A l D(T 2 ), choose (by Lemma 2.9 ) ξ n A l D(T ) such that T 2 ξ n T 2 ξ and λ(ξ n ) λ(ξ). Then (2.3) follows by passing to the limit. Now since T is -homogeneous, there exist T p i L p (ψ 0 ) + satisfying T p i T p and T p = sup Ti P. (To see this, recall that T p = dϕ normal semi-finite weight ϕ on M; put T i = ( dϕ i functionals such that ϕ i ϕ; then dϕ i for some ) /p where ϕ i are positive normal dϕ by [2, Proposition 8], and T p = ϕ() = sup ϕ i () = sup T p i.) Since the function t t /p is operator monotone on [0, ) (by [6, Proposition.3.8]), we have T i T, i.e. D(T 2 i ) D(T 2 ) and ξ D(T 2 ) : T 2 i ξ T 2 ξ, for each i I (cf. also the remark following this proof). For each i, let B i be the bounded operator characterized by B i T 2 ξ = T 2 i ξ for all ξ D(T 2 ) and B i ξ = 0 for all ξ R(T 2 ). Then B i. Since B i T 2 T 2 i, and since T 2 and T 2 i are ( )-homogeneous, B p i is 0-homogeneous, i.e. B i M. Put A i = Bi. Then A i M, A i, and T 2 i T 2 Ai.

14 560 M. TERP Using this, the fact that T p i and (2.3), we find that for all ξ A l p q = Ti L q (ψ 0 ) with T p i q = T i p p, ( β R+ D(T β i ), ) we have T p 2 i ξ 2 = (T 2 i ξ T 2 i T p i ξ) = (T 2 Ai ξ T 2 Ai T p i ξ) C [A i T p i ] q λ(a i ξ) λ(ξ) C A i T p i q A i λ(ξ) 2 = C T i p p λ(ξ) 2. By means of Lemma 2.9, we conclude that the estimate T p 2 i 2 C T i p p λ(ξ) 2 holds for all ξ A l D(T p/2 i ). Thus by Proposition 2.0, i.e. Since this holds for all i, we have T p = sup thus T L p (ψ 0 ) and T i p C. T i p p = T p i C T i p p, T i p C. T p i C p < ; Remark 2.3. we have used the fact that if a continuous function f on [0, ) is operator monotone in the sense that R S implies f(r) f(s) for all positive bounded operators R and S, then the same is true for all - possibly unbounded - positive self-adjoint R and S. To see this, suppose that R S. Then for all ε R +, we have R( + εr) S( + εs) by [7, 4], and hence f(r( + εr) ) f(s( + εs) ). Now if ξ D(f(S) 2 ), we have by the spectral theory (f(r( + εr) )ξ ξ) (f(s + ( + εs) )ξ ξ) f(s) 2 ξ 2 as ε 0. Again by the spectral theory, we conclude that ξ D(f(R) 2 ) and that f(r) 2 ξ 2 = lim ε 0 (f(r( + εr) )ξ ξ) f(s) 2 ξ 2. In all, we have proved that f(r) f(s). Recall from [2,, Théorème 4, )], that if T and T 2 belong to some L p (ψ 0 ), p <, and if T 2 T, then T = T 2. Actually, a stronger result holds:

15 L p FOURIER TRANSFORMATION 56 Lemma 2.4. Let p [, ]. Let T L p (ψ 0 ) and let T 2 be a closed densely defined ( p )-homogeneous operator on L2 (G). If T 2 T or T T 2, then T = T 2. Proof. First suppose that T 2 T. If p =, the result is well-known (a closed densely defined operator having a bounded and everywhere defined extension is equal to that extension). If p [, ), we conclude by Proposition 2.2 that also T 2 L p (ψ 0 ), and thus by [2,, Théorème 4, )], T = T 2. (Alternatively, this can be proved directly, i.e. without using Proposition 2.2, by the methods of the proof of [2,, Théorème 4, )].) If T T 2, apply the first part of the proof to T2 T. A specific form of this lemma will be crucial to much of the following: Proposition 2.5. Let p [, ]. ) Let T and S be closed densely defined ( )-homogeneous operators on p L 2 (G) with K(G) D(T ) and K(G) D(S). Suppose that T ξ = Sξ for all ξ K(G). If one of the operators, say T, belongs to L p (ψ 0 ), we may conclude that T = S. 2) If T L p (ψ 0 ) and K(G) D(T ), then T = [T K(G) ]. Proof. (of both parts). Suppose that T L p (ψ 0 ). Then T K(G), being a restriction of a ( )-homogeneous operator to a right invariant subspace, is itself ( )- p p homogeneous. Therefore also [T K(G) ] is ( )-homogeneous. Since [T p K(G)] T, we conclude by the above lemma that T = [T K(G) ]. This proves 2). As for ), note that S S K(G) = T K(G), and thus S [T K(G) ] = T. Again we conclude S = T. Finally, for later reference, we summarize in a lemma some remarks of Hilsum [2]: Lemma 2.6. Let q [2, ). Let T L q (ψ 0 ). Then A l D(T ), and for all ξ A l we have T ξ T q λ(ξ) 2/q ξ 2/q. Proof. Since T q 2 L 2 (ψ 0 ), we have A l D( T q 2 ). Now let ξ A l. Then by the spectral theory ξ D( T ) and T ξ 2 ( T q 2 ξ 2 ) 2/q.( ξ 2 ) 2/q ( T q λ(ξ) 2 ) 2/q. ξ 2( 2/q) = ( T q λ(ξ) 2/q ξ 2/q ) The Plancherel transformation Given any functions f L 2 (G) and ξ L 2 (G), the convolution product f 2 ξ exists and belongs to L (G). Thus the following definition makes sense:

16 562 M. TERP Definition 3.. Let f L 2 (G). operator on L 2 (G) given by The Plancherel transform P(f) of f is the P(f)ξ = f 2 ξ, ξ D(P(f)), where D(P(f)) = {ξ L 2 (G) f 2 ξ L 2 (G)}. Theorem 3.2. (Plancherel). () Let f L 2 (G). Then P(f) belongs to L 2 (ψ 0 ), and P(f) 2 = f 2. (2) The Plancherel transformation P : L 2 (G) L 2 (ψ 0 ) is a unitary transformation of L 2 (G) onto L 2 (ψ 0 ). Proof. () First note that P(f) is ( )-homogeneous: for all x, y G and ξ 2 D(P(f)), we have ρ(x)(p(f)ξ)(y) = 2 (x)(f 2 ξ)(yx) = 2 (x) f(z) 2 (z yx)ξ(z yx)dz = 2 (x) f(z) 2 (z y)(ρ(x)ξ)(z y)dz = 2 (x)(f 2 ρ(x)ξ)(y), i.e. ρ(x)p(f) 2 P(f)ρ(x). We next show that P(f) is closed. Suppose that ξ n ξ in L 2 (G) and P(f)ξ n η in L 2 (G), where all the ξ n D(P(f)). Then f 2 ξ n f 2 ξ uniformly (by a simple case of Lemma.). Since f 2 ξ n η in L 2 (G), we conclude that η = f 2 ξ. Thus ξ D(P(f)) and P(f)ξ = η, so that P(f) is closed. Obviously, K(G) D(P(f)). In all, we have shown that P(f) is closed, densely defined, and ( )-homogeneous, so that we are now in a position to apply Proposition Let (ξ i ) i I be an approximate identity in K(G) +. Then P(f)ξ i = f 2 ξi f in L 2 (G). Thus P(f)ξ i f 2. By Proposition 2. we conclude that P(f) L 2 (ψ 0 ) and that P(f) 2 = f 2. (2) the map P is linear: if f, f 2 L 2 (G), then [P(f ) + P(f 2 )] and P(f + f 2 ) obviously agree on K(G) and therefore by Proposition 2.5, we have P(f + f 2 ) = [P(f ) + P(f 2 )]. Now, to prove that P is surjective, let T L 2 (ψ 0 ). We shall show that there exists a function f L 2 (G) such that T = P(f). Let (ε i ) i T be an approximation

17 L p FOURIER TRANSFORMATION 563 identity in K(G) +. Then for all η, ζ K(G) we have (η 2 ζ T ξi ) = (η (T ξ i ) 2 ζ) = (η T (ξ i ζ)) = (T η ξ i ζ) (T η ζ) = (η T ζ) where we have used the ( 2 )-homogeneity of T and the fact that K(G) D(T ) since T L 2 (ψ 0 ). Thus we can define a linear functional F on the dense subspace K(G) K(G) of L 2 (G) by F (ξ) = lim i (ξ T ξ i ). Since (ξ T ξ i ) ξ 2 T ξ i 2 ξ 2 T 2 λ(ξ i ) T 2 ξ 2, this functional is bounded and therefore is given by some f L 2 (G): In particular, we have for all η, ζ K(G). Since ξ K(G) K(G) : F (ξ) = (ξ f). (η T ζ) = F (η 2 ζ) = (η 2 ζ f) (η 2 ζ f) = (η f 2 ζ) = (η P(f)ζ), this implies ζ K(G) : T ζ = P(f)ζ, and we conclude, by Proposition 2.5, that T = P(f). Proposition 3.3. ) For all T M and all f L 2 (G), we have 2) For all f L 2 (G), we have P(T f) = [T P(f)]. P(Jf) = P(f). Proof. ) Let f L 2 (G) and T M. Then [T P(f)] and P(T f) both belong to L 2 (ψ 0 ), and for all ξ K(G) we have P(T f)ξ = (T f) 2 ξ = T (f 2 ξ) = [T P(f)]ξ, since T commutes with right convolution. By Proposition 2.5 we conclude that P(T f) = [T P(f)].

18 564 M. TERP 2) Let f L 2 (G). Then for all ξ, η K(G) we have (P(Jf)ξ η) = (Jf 2 ξ η) = (Jf η 2 ξ) = (J(η 2 ζ) f) = (ξ 2 η f) = (ξ f 2 η) = (ξ P(f)η), so that P(Jf) K(G) (P(f) K(G) ) = [P(f) K(G) ] = P(f) (since P(f) = [P(f) K(G) ]). We conclude by Proposition 2.5 that P(Jf) = P(f). Proposition 3.4. Let f L 2 (G). Then P(f) 0 if and only if f(x)(ξ Jξ)(x)dx 0 for all ξ K(G). Proof.. For all ξ K(G) we have f(x)(ξ Jξ)(x)dx = (f ξ 2 ˇξ) = (f ξ ξ) = (P(f)ξ ξ). Since P(f) = [P(f) K(G) ], we have P(f) 0 if and only if (P(f)η η) 0 for all η K(G), and the result follows. By [0, Theorem.2 (3)] (or, to be precise, its spatial analogue obtained by the methods of [2, ] connecting abstract [0] and spatial [2] L p spaces), L 2 (ψ 0 ) + is a selfdual cone in L 2 (ψ 0 ). By Proposition 3.4 and the unitarity of P we conclude that P 0 = {f L 2 (G) ξ K(G) : f(x)(ξ Jξ)(x) 0} is a selfdual cone in L 2 (G). Denote by P the ordinary selfdual cone in L 2 (G) associated with the achieved left Hilbert algebra A l A l, i.e. let P be the closure in L 2 (G) of the set {λ(ξ)(jξ) ξ A l A l } (see [8, ]). Since P is selfdual, we have P = {f L 2 (G) ξ A l A l : (f λ(ξ)(jξ)) 0}. Thus P P 0. Since P and P 0 are both selfdual, this implies that P = P 0. We have proved Corollary 3.5. A function f L 2 (G) belongs to the positive selfdual cone of L 2 (G) if and only if ξ K(G) : f(x)(ξ Jξ)(x)dx 0. Remark 3.6. This result is similar to the characterization of the cone P given in [8, p. 392] and proved in general in [9, Corollary 8]. The methods of [9] would also apply for our result. Our proof is based on the fact that P(f) = [P(f) K(G) ].

19 L p FOURIER TRANSFORMATION 565 Note 3.7. We have proved that P : L 2 (G) L 2 (ψ 0 ) carries the left regular representation on L 2 (G) into left multiplication on L 2 (ψ 0 ), takes J into, and maps the positive selfdual cone of L 2 (G) onto L 2 (ψ 0 ) +. That a unitary transformation L 2 (G) L 2 (ψ 0 ) having these properties exists (and is unique) also follows from [8, Theorem 2.3], since both representations of M are standard (by the spatial analogue of [0, Theorem.2, (3)]). In our approach, we have given a simple and direct definition of P. We can give an explicit description of the inverse of P: Proposition 3.8. Let T L 2 (ψ 0 ), and let (ξ i ) i I be an approximate identity in K(G) +. Then P (T ) = lim i I T ξ i. Proof. Let f P (T ). Then in L 2 (G). T ξ i = P(f)ξ i = f 2 ξi f Remark 3.9. From Proposition 2. we already knew that for any approximate identity (ξ i ) i I, the T ξ i tend to a limit and that this limit is independent of the choice of (ξ i ) i I. Now, using that L 2 (ψ 0 ) = P(L 2 (G)), we have proved that the same holds for the T ξ i themselves. As a corollary, we have the following characterization of the inner product in L 2 (ψ 0 ), generalizing the formula for T 2 given in Proposition 2.: Corollary 3.0. Let T, S L 2 (ψ 0 ). Then (T S) L 2 (ψ 0 ) = lim i I (T ξ i Sξ i ) for any approximate identity (ξ i ) i I in K(G) +. Proof. Since P is unitary, we have (T S) L 2 (ψ 0 ) = (P (T ) P (S)) L 2 (G) = lim i I (T ξ i Sξ i ) L 2 (G). 4. The L p Fourier transformations Let p [, 2] and define q [2, ] by p + q =. Definition 4.. Let f L p (G). The L p Fourier transform of f is the operator F p (f) on L 2 (G) given by F p (f)ξ = f q ξ, ξ D(Fp (f)), where D(F p (f)) = {ξ L 2 (G) f q ξ L 2 (G)}. Note that by Lemma. the convolution product f q ξ exists and belongs to L r (G), where r [2, ] is given by p + 2 r =, whenever f Lp (G) and ξ L 2 (G), so that the definition of D(F p (f)) makes sense.

20 566 M. TERP Remark 4.2. For p =, we write F = F; we have F(f)ξ = f ξ and D(F(f)) = L 2 (G), so that F(f) is simply λ(f). For p = 2, we have F 2 (f) = P(f). Now again let p [, 2]. Let f L p (G). Then the operator F p (f) is closed. To see this, suppose that ξ i D(F p (f)) converges in L 2 (G) to some ξ L 2 (G) and F p (f)ξ i converges in L 2 (G) to some η L 2 (G). Now by Lemma. we have F p (f)ξ i = f q ξi f q ξ in L r (G) (where p + 2 = ). Therefore r f q ξ = η, so that f q ξ L 2 (G), i.e. ξ D(F p (f)) and F p (f)ξ = η as wanted. Next we show that F p (f) is ( q )-homogeneous. For all ξ D(F p(f)) and all x, y G we have i.e. ρ(x)(f p (f)ξ)(y) = 2 (x)(f q ξ)(yx) = 2 (x) f(z) q (z yx)ξ(z yx)dz = q (x) f(z) q (z y) 2 (x)ξ(z yx)dz = q (x) f(z) q (z y)(ρ(x)ξ)(z y)dz for all x G as wanted. = q (x)(f q ρ(x)ξ)(y) = q (x)(fp (f)ρ(x)ξ)(y), ρ(x)f p (f) q (x)fp (f)ρ(x) Finally, note that if ξ L 2 (G) L s (G) where s [, 2] is given by p + s 2 =, then ξ D(F p (f)) by Lemma.. In particular, K(G) D(F p (f)). In all, we have proved that for all f L p (G), F p (f) is closed, densely defined, and ( )-homogeneous. We shall see, using the criterion from Proposition 2.2, that q actually F p (f) L q (ψ 0 ). The proof is based on interpolation from the special cases F : L (G) L (ψ 0 ) and P : L 2 (G) L 2 (ψ 0 ). First we restrict our attention to f K(G) Lemma 4.3. Let p [, 2]. Denote by A the closed strip {α C 2 Re(α) }. Let f K(G) and ξ A l. Then: (i) for each α A, the convolution product exists, and ξ α L 2 (G); ξ α = sg(f) f pα α ξ

21 L p FOURIER TRANSFORMATION 567 (ii) the function α ξ α, α A, with values in L 2 (G) is bounded; (iii) for each η L 2 (G), the scalar function α (ξ α η), α A, is continuous on A and analytic in the interior of A. Proof. Write g = /p ˇf. Then α A : sg(f) f pα = α (sg(g) g pα ). Note that g as well as all sg(g) g pα, α A, belong to K(G). For each η K(G), we define H η (α) = ξ(x)(sg(g) g pα α η)(x)dx, α A, (4.) i.e. H η (α) = ξ(x)(sg(g) g pα )(y) α (y x)η(y x)dydx (4.2) (later we shall recognize H η (α) as simply (ξ α η)). Note that α A : sg(g) g pα α η 2 g pre(α) Re(α) η 2 (4.3) K <, where K is a constant independent of α A. In particular, this allows us to apply Fubini s theorem to the double integral (4.2). We find H η (α) = ξ(x)(sg(g) g pα )(y ) α (yx)η(yx) (y)dydx = ξ(y x)(sg(g) g pα )(y ) α (x)η(x) (y)dxdy = (sg(f) f pα )(y) α (y x)ξ(y x)η(x)dydx; it also follows that the convolution integral ξ α (x) = (sg(f) f pα )(y) α (y x)ξ(y x)dy exists, so that we can write H η (α) = ξ α (x)η(x)dx. Now we shall prove that there exists a constant C 0 independent of α such that η K(G) : ξ α (x)η(x)dx C η 2. (4.4) This will imply that each ξ α, α A, is in L 2 (G) with ξ α 2 C, i.e. (i) and (ii) will be proved.

22 568 M. TERP Let us prove (4.4). Without loss of generality, we may assume that f p =. We want to show then that η K(G) : H η (α) ( λ(ξ) + ξ 2 ) η 2. (4.5) To do this, we shall apply the Phragmen Lindelöf principle [24, p.93]. Fix η K(G). By (4.2), H η is continuous on A and analytic in the interior of A (the integrand in (4.2) can be majorized by an integrable function that is independent of α). Furthermore, H η is bounded (use (4.3) and (4.)). Finally, we shall estimate H η on the boundaries of A. Let t R. Then it ξ A l and λ( it ξ) λ(ξ). Now so that ξ 2 +it L2 (G) with P(sg(f) f p( 2 +it) )( it ξ) = sg(f) f p( 2 +it) ( 2 +it) ξ = ξ 2 +it, ξ 2 +it 2 P(sg(f) f p( 2 +it) ) 2 λ( it ξ) sg(f) f p( 2 +it) 2 λ(ξ) = f p 2 2 λ(ξ) = λ(ξ) (where we have used Proposition 2., the fact that P is unitary, and the hypothesis f p = ). Similarly, F(sg(f) f p(+it) )( it ξ) = sg(f) f p(+it) (+it) ξ = ξ +it, so that ξ +it L 2 (G) with ξ +it 2 F(sg(f) f p(+it) ) it ξ 2 sg(f) f p(+it) ξ 2 = f p ξ 2 = ξ 2 (where we have used that F : L (G) L (ψ 0 ) is norm-decreasing). It follows that t R : H η ( 2 + it) = ξ +it(x)η(x)dx 2 and ξ 2 +it 2 η 2 λ(ξ) η 2 t R : H η ( + it) = ξ +it (x)η(x)dx ξ +it 2 η 2 ξ 2 η 2. Then by the Phragmen Lindelöf principle, we have established (4.5) and thus (i) and (ii).

23 L p FOURIER TRANSFORMATION 569 Finally, (iii) is easy. Indeed, since α ξ α as is bounded, each α (ξ α η), where η L 2 (G), can be uniformly approximated by functions α (ξ α ζ) with ζ K(G), so we just have to prove (iii) in the case of η K(G). This is already done since (ξ α η) = H η (α). Lemma 4.4. Let p [, 2]. Let f K(G) and S L p (ψ 0 ). Then for all ξ A l and η A l D(S) we have Note that ξ D(F p (f)) by Lemma 4.3. (F p (f)ξ Sη) f p S p λ(ξ) λ(η). Proof. We may assume that f p = and S p =. Furthermore, by Lemma 2.9, we need only consider η A l D( S p ). Let ξ A l and η A l D( S p ). For each α in the closed strip A = {α C 2 Re(α) }, put ξ α = sg(f) f pα α ξ as in Lemma 4.3. Note that for all α A we have (by the spectral theory ) η D(U S pα ) and U S pα η 2 2 S p 2 η S p η 2 2, where S = U S is the polar decomposition of S. For each α A, put η α = U S pα η. Then the function α η α with values in L 2 (G) is bounded on A. Furthermore, by [22, 9, 5], it is continuous on A and analytic in the interior of A. Now for each α A, let H(α) = (ξ α η α ). Then obviously H is bounded on A (by Lemma 4.3 (ii), α ξ α is bounded). Furthermore, H is continuous on A. To see this, note that α, α 0 A : (ξ α η α ) (ξ α0 η α0 ) = (ξ α η α η α0 ) + (ξ α ξ α0 η α0 ), the continuity follows since α ξ α is bounded and weakly continuous (Lemma 4.3 (iii)). Finally, we claim that H is analytic in the interior of A. First note that for each ζ L 2 (G) the function α (ζ η α ), being equal to α (η α ζ), is analytic. Next, recall that α ξ α is actually analytic as a function with values is L 2 (G) (by Lemma 4.3 (iii) and [9, Theorem 3.3]). Then, writing (ξ α η α ) (ξ α0 η α0 ) = ( (ξ α ξ α0 ) η α ) + (ξ α 0 η α ) (ξ α0 η α0 ), α α 0 α α 0 α α 0 we find that H has a derivative at each point α 0 in the interior of A. Now suppose that and t R : H( + it) λ(ξ) λ(η) (4.6) 2 t R : H( + it) λ(ξ) λ(η). (4.7) Then by the Phragmen Lindelöf principle [24, p. 93] we infer that α A : H(α) λ(ξ) λ(η),

24 570 M. TERP in particular, as desired, since (F p (f)ξ Sη) λ(ξ) λ(η) H( p ) = (f p ξ U S η) = (Fp (f) Sη). So we just have to prove (4.6) and (4.7). Since S L p (ψ 0 ) with S p = we have U S p 2 L 2 (ψ 0 ) with U S p 2 2 = (4.8) and U S p L (ψ 0 ) with U S p =. (4.9) Now let t R. Then by Lemma 2.5, we have S pit η A l with λ( S pit η) λ(η). (4.0) Using this, Proposition 2., the estimate ξ 2 +it 2 λ(ξ) given in the proof of Lemma 4.3, and (4.8), we get H( 2 + it) = (ξ 2 +it U S p 2 S pit η) ξ 2 +it 2 U S p 2 S pit η 2 λ(ξ) U S p 2 2 λ( S pit η) 2 λ(ξ) λ(η), i.e. (4.6) is proved. To prove (4.7), note that and ξ +it = sg(f) f p(+it) (+it) ξ = λ(sg(f) f p(+it) ) it ξ A l λ(ξ +it ) λ(sg(f) f p(+it) ) λ( it ξ) sg(f) f p(+it) λ(ξ) λ(ξ), since sg(f) f p(+it) = f p =. Using this together with (4.0), Proposition 2.0, and (4.9), we find so that (4.7) is proved. H( + it) = (ξ +it U S p S pit η) λ(ξ +it ) U S p λ( S pit η) λ(ξ) λ(η), In the formulation of the following theorem we include the case p = 2. Note however that the proof is based on the results for this special case (they were used for the preceding lemmas).

25 L p FOURIER TRANSFORMATION 57 Theorem 4.5 (Hausdorff Young). Let p ], 2] and p + q =. ) Let f L p (G). Then F p (f) L q (ψ 0 ) and 2) The mapping F p (f) q f p. F p : L p (G) L q (ψ 0 ) is linear, norm-decreasing, injective, and has dense range. 3) For all h L (G) and f L p (G), we have 4) For all f L p (G), we have F p (h f) = [λ(h)f p (f)]. F p (J p f) = F p (f). Proof. ) First suppose that f K(G). Then, using Proposition 2.2, we conclude from Lemma 4.4 that F p (f) L q (ψ 0 ) with F p (f) q f p. Thus we have defined a norm-decreasing mapping F p K(G) : L p (G) L q (ψ 0 ). Furthermore F p K(G) is linear: for all f, f 2 K(G) and all ξ K(G) we have (f + f 2 ) q ξ = f q ξ + f2 q ξ so that F p (f + f 2 ) = [F p (f ) + F p (f 2 )] by Proposition 2.5. Now F p K(G) extends by continuity to a norm-decreasing linear mapping F p : L p (G) L q (ψ 0 ). We claim that for all f L p (G), we have F p(f) = F p (f). this will prove ). Let f L p (G). Then F p(f) L q (ψ 0 ) and K(G) D(F p(f)) by Lemma 2.6. On the other hand, by the remarks at the beginning of this section, F p (f) is closed, densely defined, and ( q )-homogeneous, an K(G) D(F p(f)). Thus ny Lemma 2.7, to conclude that F p(f) = F p (f) we just have to show that ξ K(G) : F p(f)ξ = F p (f)ξ. Now, take f n K(G) such that f n f in L p (G). Then for all ξ K(G), we have F p (f n )ξ = f n q ξ On the other hand, since F p is continuous, f q ξ = Fp (f)ξ in L p (G). F p (f n )ξ = F p(f n )ξ F p(f)ξ in L 2 (G) by Lemma 2.6. We conclude that F p (f)ξ = F p(f)ξ as desired. Thus ) is proved.

26 572 M. TERP 2) By the proof of ), we just have to show that F p is injective and has dense range. The injectivity is evident: if F p (f) = 0 for some f L p (G), then f q ξ = 0 for all ξ K(G), and thus f = 0. That F p (L p (G)) is dense will be proved later. 3) For all h L (G), f L p (G), and ξ K(G) we have (in L p (G)). Thus by Proposition 2.5, h (f q ξ) = (h f) q ξ λ(h)f p (f)] = F p (h f). 4)Let f K(G). Then for ξ, η K(G) we have (F p (J p f)ξ η) = (J p f q ξ η) = ( q ξ (J p f) η) = (ξ q ( p f η)) = (ξ f q η) = (ξ F p (f)η), so that F p (J p f) K(G) (F p (f) K(G) ). By Proposition 2.5, we conclude that F p (J p f) = F p (f). By the continuity of J p, F p, and, this holds for all f L p (G). Finally, let us show that F p (L p (G)) is dense in L q (ψ 0 ). By the duality between L q (ψ 0 ) and L p (ψ 0 ), this is equivalent to proving that if T L p (ψ 0 ) satisfies [Fp (f)t ] = 0 for all f L p (G) for all f L p (G), then T = 0. Suppose that T L p (ψ 0 ) is such that f L p (G) : [F p (f)t ] = 0. Let f L p (G). Then for all h L (G) we have [F p (h f)t ] = 0. Alternatively stated, since [F p (h f)t ] = [[λ(h)f p (f)]t ] = [λ(h)[f p (f)t ]], we have h L (G) : [λ(h)[f p (f)t ]] = 0. We conclude that the normal functional on M defined by [F p (f)t ] L (ψ 0 ) is 0, so that [F p (f)t ] = 0. Changing f into J p f and using 4) this gives f L p (G) : [F p (f) T ] = 0.

27 L p FOURIER TRANSFORMATION 573 Now let ξ D(T ). Then using [2, II, Proposition 5],[] we find that f, η K(G) : (T ξ f q η) = (T ξ F p (f)η) = [F p (f) T ], λ(ξ)λ(η) = 0. Thus T ξ = 0. This proves that T = 0 as wanted. Proposition 4.6. Let p [, 2]. Let f L p (G) and F p (f) 0 if and only if ξ K(G) : f(x)(ξ J p ξ)(x)dx 0. Proof. We have (F p (f)ξ ξ) = (f p ξ)(x)ξ(x)dx = f(x)(ξ p ˇξ)(x)dx for all ξ K(G). The result follows by changing ξ into ξ and recalling that F p (f) = [F p (f) K(G) ]. The L p Fourier transformations are well-behaved with respect to convolution as the following proposition shows. The result generalizes 3) of theorem. Proposition 4.7. Let p, p 2, p [, 2] such that p + p 2 =. Define q p [2, ] by p + q =. Let f L p (G) and f 2 L p 2 (G). Then F p (f q f 2 ) = [F p (f )F p2 (f 2 )]. Proof. By Lemma., we have f q f 2 L p (G), and (f, f 2 ) F p (f q f 2 ) maps L p (G) L p 2 (G) continuously into L q (ψ 0 ) (where + = ). Also p q [F p (f )F p2 (f 2 )] is continuous as a function of (f, f 2 ) L p (G) L p 2 (G) with values in L q (ψ 0 ). Thus we need only prove the statement for f, f 2 K(G). Since (f q f 2 ) q ξ = f q (f 2 q 2 ξ) (where p 2 + q 2 = ) for all f, f 2, ξ K(G), the result follows by Proposition 2.5 as usual. We conclude this section by the following characterization of the image of L p (G) under F p : Proposition 4.8. Let p ], 2] and p + q =. Let T Lq (ψ 0 ). ) If T = F p (f) for some f L p (G), then for any approximate identity (ξ i ) i I in K(G) + we have T ξ i f in L p (G). In particular, lim i I T ξ i p = f p <. 2) Conversely, suppose that for some approximate identity (ξ i ) i I in K(G) + we have T ξ i L p (G) for all i I and Then T F p (L p (G)). lim inf i I T ξ i p <.

28 574 M. TERP Proof. The first part is obvious since T ξ i = f q ξi f in L p (G) and therefore T ξ i p f p. Now suppose that the hypothesis of 2) holds for some (ξ i ) i I. We the proceed as in the proof of the surjectivity of P (Theorem 3.2). for all η, ζ K(G) we have (η q ζ T ξi ) =(η (T ξ i ) q ζ) =(η T (ξ i ζ)) =(T η ξ i ζ) (T η ζ) = (η T ζ). Thus we can define a linear functional F on K(G) K(G) by F (ξ) = lim ξ(x)(t ξ i )(x)dx. i I Since ξ(x)(t ξ i )(x)dx ξ q T ξ i p we have F (ξ) (lim inf T ξ i p ) ξ q. i I Now since K(G) K(G) is dense in L q (G), F extends to a bounded functional on L q (G) and therefore is given by some f L p (G): F (ξ) = ξ(x)f(x)dx. In particular, (η T ζ) = F (η q ζ) = (η q ζ)(x)f(x)dx for all η, ζ K(G). Since (η q ζ)(x)f(x)dx = η(x)(f q ζ)(x)dx = (η Fp (f)ζ), (for counter- this implies that ζ K(G) : T ζ = F p (f)ζ, and we conclude by Proposition 2.5 that T = F p (f). Remark 4.9. For p =, part 2) of the above proposition fails. example, take T = λ(x), x G.) 5. The L p Fourier contransformation Definition 5.. Let p [, 2] and + =. For each T p q Lp (ψ 0 ), denote by F p (T ) the unique function in L q (G) such that h(x)f p (T )(x)dx = [F p (h)t ]

29 L p FOURIER TRANSFORMATION 575 for all h L p (G) (or just h K(G), or h K(G), or h K(G) K(G)). The mapping F p : L p (ψ 0 ) L q (G) thus define will be called the L p fourier transformation. For p =, we write F = F. Note that if < p 2, then F p is simply the transpose of F p : L p (G) L q (ψ 0 ) when we identify the dual spaces of L p (G) and L q (ψ 0 ) with L q (G) and L p (ψ 0 ), respectively. The mapping F takes an element T L (ψ 0 ) into the unique function ϕ A(G) that defines the same element of M as T does; in particular, ( ) dϕ F = ϕ for all ϕ (M ) + A(G) +. In view of these remarks, we obviously have Theorem 5.2. ) Let p ], 2] and p + q F p : L p (ψ 0 ) L q (G) =. Then is linear, norm-decreasing, injective, and has dense range. 2) The mapping F : L (ψ 0 ) A(G) is an isometry of L (ψ 0 ) onto A(G). Remark 5.3. With our definition of the contransformations, F 2 is not exactly the inverse of P; they are related by the formula T L 2 (ψ 0 ) : F 2 (T ) = P (T ) (since for all h L 2 (G) we have h(x)f 2 (T )(x)dx = [F 2 (h)t ] = (F 2 (h) T ) L 2 (ψ 0 ) = ( h P (T ) ) = h(x)p (T )(x)dx. L 2 (G) It follows that F 2 : L 2 (ψ 0 ) L 2 (G) is unitary. The classical Hausdorff Young theorem [24, p.0] has a second part, stating that with each c l p (Z), p 2, we can associate a function f L q (T) with f q c p, such that c is the sequence of Fourier coefficients of f. Theorem 5.2 is a generalization of this result. Indeed, let T L p (ψ 0 ) and put g = q F p (T )ˇ. Then g L q (G) and g q = F p (T ) q T p, and we shall see that T is close to being the L q Fourier transform of g in the sense that T ξ = g p ξ for certain ξ (note that we do not in general define L q Fourier transforms for q 2). Proposition 5.4. Let p [, 2] and p + q =. Then for all T Lp (ψ 0 ), we have F p (T ) = J q (F p (T )).