J. London Math. Soc. (2) 75 (2007) 369–390

e 2007 London Mathematical Society

C

doi:10.1112/jlms/jdm002

MULTIPLIERS ON BANACH SPACES OF FUNCTIONS ON A LOCALLY COMPACT ABELIAN GROUP VIOLETA PETKOVA Abstract Let E be a Banach space of functions on a locally compact Abelian group G satisfying certain conditions. It has  been proved that for every bounded operator M on E commuting with translations there exists hM ∈ L∞ (G E) f = hM f˜ ∀f ∈ Cc (G), where G  is a suitable subset of the group of the continuous morphisms such that M E  from G into C∗ and g is a generalized Fourier transform of g defined on G E.

1. Introduction  the Let G be a locally compact Abelian (LCA) group with unit element 0 = 0G . Denote by G dual group of G, that is, the set of continuous morphisms χ : G −→ T, where T denotes the unit  respectively, circle. Throughout the paper, dx and dχ denote the Haar measures on G and G, normalized so that  F : L2 (G) −→ L2 (G), the Fourier transform, is an isometry. Recall that for f ∈ L1 (G), F(f ) is defined by   F(f )(χ) = f (x)χ(−x) dx ∀χ ∈ G G

 we have a.e. and if g = F(f ) ∈ L (G) 1

f (x) = F −1 (g)(x) =

  G

g(χ)χ(x) dχ.

Sometimes we will write fˆ instead of Ff . The reader may find the basic properties of the LCA groups in [18]. Denote by Cc (G) the space of continuous complex-valued functions on G with compact support and denote by CK (G) the space of functions in Cc (G) with support included in a compact K ⊂ G. For f ∈ Cc (G), set f ∞ = supx∈G |f (x)|, so that the spaces CK (G) are Banach algebras with respect to the norm  · ∞ . We have Cc (G) = lim CK (G), and we equip → Cc (G) with the locally convex topology τ associated to this inductive limit. It is well known that a sequence (fn )n1 of elements of Cc (G) converges to zero with respect to τ if and only if the following two conditions hold: (1)  limn→+∞ fn ∞ = 0; (2) n∈N supp(fn ) is relatively compact in G. It is also well known that a linear map φ from Cc (G) into a locally convex space E is continuous if and only if limn→+∞ φ(fn ) = 0 for every sequence (fn )n1 which converges to 0 in Cc (G) with respect to τ . In other terms, every sequentially continuous linear map from Cc (G) into a locally convex space is continuous [1, Chapter 7; 4, 5]. Let L1loc (G) be the space of all measurable complex-valued functions f on G such that f |K is integrable with respect to the Haar measure on every compact set K ⊂ G. For x ∈ G, let Sx

Received 6 January 2006; published online 18 April 2007. 2000 Mathematics Subject Classification 43A22, 43A25.

370

V. PETKOVA

be the operator defined on

L1loc (G)

by

Sx f (y) = f (y − x)

a.e.

 denote by Γχ the operator f −→ χf on L1 (G). Let E ⊂ L1 (G) be a Banach For χ ∈ G, loc loc space, and assume that the identity map i : E −→ L1loc (G) is continuous. It follows from the closed graph theorem that if Sx (E) ⊂ E for x ∈ G, the operator Sx is bounded from E into  the operator Γχ is bounded from E into E. We will be E. Likewise, if Γχ (E) ⊂ E for χ ∈ G, interested in Banach spaces E satisfying the following conditions: (H1) Cc (G) ⊂ E ⊂ L1loc (G), with continuous inclusions, and Cc (G) is dense in E; (H2) for every x ∈ G, Sx (E) ⊂ E and supx∈K Sx  < +∞ for every compact set K ⊂ G;  Γχ (E) ⊂ E and sup  Γχ  < +∞. (H3) for every χ ∈ G, χ∈G Set |||f ||| = supχ∈G Γχ f  for f ∈ E. The norm ||| · ||| is equivalent to the norm of E and, without loss of generality, we can consider below that Γχ is an isometry on E for every  From now, Sx for x ∈ G (respectively, Γχ for χ ∈ G)  will denote the restriction of χ ∈ G. Sx (respectively, Γχ ) on E. Definition 1.

A multiplier on E is a bounded linear operator M : E −→ E

such that Sx M = M S x

∀x ∈ G.

The algebra of the multipliers on E is denoted by M(E). Let B(E) be the closed algebra generated by {Sx }x∈G . Given a commutative Banach algebra A, we will denote by A the set of characters of A. Denote by σ({Sx }x∈G ) the joint spectrum of {Sx }x∈G defined by  σ({Sx }x∈G ) = {(γ(Sx ))x∈G , γ ∈ B(E)}. In the particular case E = Lp (G) for 1  p < ∞, it is well known [15] (see [12] for G = R)  such that that for M ∈ M(E), there exists h ∈ L∞ (G) f = hf M

∀f ∈ Cc (G).

(1.1)

The function h is called the symbol of M . The aim of this paper is to obtain a representation theorem analogous to (1.1) for multipliers on a general Banach space E satisfying only hypotheses (H1), (H2) and (H3). We are motivated by a result concerning the multipliers on E = L2ω (R) (see [16]), where ω is a measurable non-negative function on R. Let I = [− ln ρ(S−1 ), ln ρ(S1 )]. For a ∈ I and for f ∈ E, set (f )a (t) = eat f (t) a.e. for f ∈ E, and set Ω = {z ∈ C | Im z ∈ I}. We have the following result. Theorem 1 [16]. Let E = L2ω (R) and suppose that Sx is bounded on E for every x ∈ R. Let M ∈ M(E). (i) We have (M f )a ∈ L2 (R), for f ∈ Cc∞ (R) and a ∈ I. (ii) For a ∈ I, there exists a function ha ∈ L∞ (R) such that   (M f )a (t) = ha (t)(f )a (t)

∀f ∈ Cc∞ (R) a.e.

Moreover, there exists C > 0 such that ha ∞  CM  for every a ∈ I. ◦

(iii) If ρ(S1 ) > 1/ρ(S−1 ), there exists a function h ∈ H∞ (Ω) such that f = hf˜ M

∀f ∈ Cc∞ (R), ◦

  f (x + ia) = (M where M f )a (x), and f˜(x + ia) = (f )a (x), for x + ia ∈ Ω.

MULTIPLIERS ON BANACH SPACES

371

Here, the Fourier transform of M f and the symbol of M are defined on a strip Ω ⊂ C isomorphic to R × [− ln ρ(S−1 ), ln ρ(S1 )]. The situation when G = Z and E is a Banach space satisfying (H1), (H2) and (H3) has been considered by  the author in [17]. Denote by F (Z) the set of the finite sequences on Z. Set formally u ˜(z) = n∈Z un z n for u = (un )n∈Z ∈ CZ , and it ˜(reit ) for r  0 and t ∈ R. Denote by S the shift operator S1 and denote by set u (r) (e ) = u spec(S) the spectrum of S considered as an operator on E. Theorem 2 [17]. Let M ∈ M(E). The following properties hold: (i) spec(S) = {z ∈ C, 1/ρ(S −1 )  |z|  ρ(S)}; ∞ −1   := M  (ii) set M (e0 ) for M ∈ M(E). Then M ), ρ(S)] and (r) ∈ L (T) for r ∈ [1/ρ(S  M(r) ∞  M ; ◦  ∈ H∞ (spec(S)) and (iii) if 1/ρ(S −1 ) < ρ(S), then M u = M u M ˜

∀u ∈ F (Z).

In this case the Fourier transform of M f and the symbol of M are defined on the annulus spec(S) ≈ T × [1/ρ(S −1 ), ρ(S)].  be the set of the continuous morphisms from G Now let G be a general LCA group. Let G ∗  + into C and let G be the set of the continuous morphisms from G into R+ = (0, +∞). We  with the topology characterized by uniform convergence on every compact subset. equip G  the integral Notice that for f ∈ Cc (G) and θ ∈ G,  f (x)θ(x)−1 dx G

  We observe that f˜|  = fˆ is well defined. For f ∈ Cc (G), set f˜(θ) = G f (x)θ(x)−1 dx for θ ∈ G. G ˜  and f is a ‘generalized Fourier transform’ of f defined on G.   The first problem is to find a subset G E of G such that for every M ∈ M(E) and for −1 2  every f ∈ Cc (G) the function (M f )θ is in L (G) for θ ∈ G E . Then we will be able to define −1 naturally the Fourier transform of (M f )θ . The difficulty here is that we have very little information about M f . Taking into account the arguments from [16, 17], it is natural to consider  ˜  ∀f ∈ Cc (G)}, G E = {θ ∈ G | |f (θ)|  Mf  where Mf ∈ M(E) is the convolution operator g −→ f ∗ g on E (see the beginning of Section 2). Notice that if G is a compact group, then the range of |θ| is the trivial compact subgroup of    R+ for every θ ∈ G E , and so GE = G.    Recall that a domain X of Cn is a Reinhardt domain It is obvious that χGE = GE for χ ∈ G. −1 if Y (Y(X)) = X, where Y : Cn (z1 , . . . , zn ) −→ (|z1 |, . . . , |zn |) ∈ (R+ )n . We observe that if (x1 , . . . , xn ) ∈ Gn are ‘independent’, in the sense that the system χ(xi ) = i ,  for every ( 1 , . . . , n ) ∈ Tn , then the set 1  i  n, has a solution χ in G  {(θ(x1 ), . . . , θ(xn )), θ ∈ G E}  + +    is a Reinhardt domain. Set G E = GE ∩ G . It is clear that if θ ∈ GE , then |θ| : G x −→ |θ(x)|  + +   belongs to G E . Trivially we have GE = GE G. We prove in Section 2 the following crucial facts.

372

V. PETKOVA

Theorem 3. Let E be a Banach space satisfying conditions (H1), (H2) and (H3).  + (i) The set G E is non-empty, compact and logarithmically convex. (ii) We have

 G E =

 | |θ(x)−1 |  sup |η(x)−1 | θ∈G

∀x ∈ G .

 η∈G E

(1.2)

We recall that a set X is said to be logarithmically convex if xt y 1−t ∈ X for every x, y ∈ X ∀t ∈ [0, 1]. Sometimes we will write log-convex instead of logarithmically convex. If G is a  discrete or a compact group, we will see that G E satisfies the following property: −1   G (1.3) |  ρ(Sx ) ∀x ∈ G}. E = {θ ∈ G | |θ(x) We conjecture that property (1.3) holds for a general LCA group. We will establish a   homeomorphism between G E and A(G), where A(G) is the closed algebra generated by  is non-empty. Let U be an open subset of C. A function {Mφ }φ∈Cc (G) . We will see that A(E)  Π : U λ −→ Π(λ) ∈ G will be said to be analytic on U if, for every x ∈ G, the function U λ −→ Π(λ)(x) ∈ C  +  is analytic on U . Denote by d the discrete measure on G E and by m the Haar measure on G. We obtain the following general result. Theorem 4. Let E be a Banach space satisfying conditions (H1), (H2) and (H3).  + −1  ∈ L2 (G). Set for δ ∈ G (i) Let M ∈ M(E) and θ ∈ G E . For every f ∈ Cc (G), (M f )θ E,  for almost every χ ∈ G,  f (δχ) = (M M f )δ −1 (χ).  Then there exists a function hM ∈ L∞ (G E , d ⊗ m) such that we have  (M f ) = hM f˜

∀f ∈ Cc (G)

and hM ∞  CM , where C is a constant independent of M.  (ii) Let U be an open subset of Cp . Let Π : U −→ G E be an analytic function. There exists ∞  ∞  a function HM,Π ∈ L (G, H (U )) such that, for λ ∈ U, for almost every χ ∈ G, f (Π(λ)χ) = HM,Π (χ)(λ)f˜(Π(λ)χ) M

∀f ∈ Cc (G).

Now we turn to some interpretations of this theorem. Set L = {z ∈ C | Re z ∈ [0, 1]}. Fix φ  + and ψ ∈ G E . Suppose that φ = ψ. The function Π : L λ −→ φλ ψ 1−λ ◦

is analytic on L. Set Ωφ,ψ = Π(L).   Since G E is log-convex, Π(L) ⊂ GE . It is clear that Ωφ,ψ is isomorphic to L. We see that for θ = φλ ψ 1−λ ∈ Ωφ,ψ , we get the representation  φ(x) Re λ 1−Re λ ∀x ∈ G. θ(x) = φ(x) ψ(x) exp iIm λ ln ψ(x) Introduce the morphism  φ(x) ∈T γ : G x −→ exp i ln ψ(x)

373

MULTIPLIERS ON BANACH SPACES

and set  | χ(x) = (γ(x))t γ R = {χ ∈ G

∀x ∈ G for some t ∈ R}.

Denote by Sφ,ψ the convex set defined by

 + t 1−t Sφ,ψ = η ∈ G E | η(x) = φ(x) ψ(x)

 ∀x ∈ G for some t ∈ [0, 1] .

Notice that Ωφ,ψ is isomorphic to Sφ,ψ × γ R . Observe that Theorem 4 (ii) gives some analytic  +   symbol for M ∈ M(E) as soon as G E is not a singleton even if GE has an empty interior in G. Now we discuss some examples. Example 1.

 G = Z. Then every φ ∈ G E is of the form φ : Z n −→ z n ∈ C,

where z ∈ C and 1/ρ(S −1 )  |z|  ρ(S). If ρ(S) > 1/ρ(S −1 ), we can choose φ and ψ so that ∀n ∈ Z,

φ(n) = ρ(S)n ψ(n) = ρ(S

−1 −n

)

∀n ∈ Z.

Observe that the set Sφ,ψ is isomorphic to the segment [1/ρ(S −1 ), ρ(S)] and  = T. γR ≈ Z So we obtain

 Ωφ,ψ ≈

z∈C|

1  |z|  ρ(S) ρ(S −1 )

and Theorem 4 gives exactly the result of [17]. Example 2. G = Zk . Set ej = (ej,1 , . . . , ej,k ), with ej,i = 0 for i = j, and ej,j = 1. Define Sj = Sej . It follows from (1.2) that k ≈ σ(S , . . . ., S ). Z 1 k E k has the form φ = φ , where z = (z , . . . , z ) ∈ C∗k and φ is defined on Zk Each φ ∈ Z z 1 k z E by the formula φz (n1 , . . . , nk ) = z1n1 . . . zknk . In other terms if we set z n = z1n1 . . . zknk for k has the form φ = φ , where z = (z1 , . . . , zk ) ∈ Ck and n = (n1 , . . . , nk ) ∈ Zk , each φ ∈ Z z E n k ∗k φz (n) = z for n ∈ Z for some z ∈ C . Set k }. FE = {z ∈ C∗k |φz ∈ Z E

Notice that FE satisfies the following two properties. (1) We have (z1 eiθ1 , . . . , zk eiθk ) ∈ FE for every (z1 , . . . , zk ) ∈ FE and every (θ1 , . . . , θk ) ∈ Rk. (2) The set {(log |z1 |, . . . , log |zk |)}(z1 ,...,zk )∈FE is convex. ◦

◦

◦

In particular FE is a log-convex Reinhardt domain if FE = ∅. Assume that FE = ∅ and set ◦ ◦ k is analytic. It follows from Theorem 4 that there Π(z) = φz for z ∈ FE . Then Π : FE −→ Z E ◦ k , H∞ (FE )) satisfying, for each u ∈ F (Zk ) and exists for M ∈ M(E) a function HM,Π ∈ L∞ (Z k , for almost every χ ∈ Z u(φz χ) = HM,Π (χ)(z)˜ u(φz χ) M

◦

∀z ∈ FE .

k for which the above formula holds. Let δ ∈ Tk be such that Now choose χ = (χ1 , . . . , χk ) ∈ Z χ = φδ . Define θM : FE z −→ HM,Π (χ)(zδ −1 ) ∈ C,

374

V. PETKOVA ◦

where zδ −1 = (z1 δ1−1 , . . . , zk δk−1 ). Then θM ∈ H∞ (FE ). Obviously, we have φzδ = φz φδ for every z ∈ C∗k . We obtain for u ∈ F (Zk ) for z ∈ FE u(φzδ−1 φδ ) = θM (z)˜ u(φz ) = M u(φzδ−1 φδ ) = θM (z)˜ u(φz ). M Thus we have the following result. Let E be a Banach space of sequences on Zk satisfying (H1), (H2) and ◦ ◦ ∞ k = ∅. Then, for M ∈ M(E), there exists θ (H3). Suppose that Z M ∈ H (FE ) such that for E f ∈ F (Zk ) Corollary 1.

◦

f (φz ) = θM (z)f˜ (φz ) ∀z ∈ FE . M Example 3.

Define E : C∗k

k is of the form G = Rk . Every element of R

ψa : x −→ e−ia,x for some a ∈ C∗k . k and note that E −1 (ψ) ∈ C∗k ∀ψ ∈ R k .

a −→ ψa ∈ R E E ◦

k = ∅. Set Suppose that R E

 UE = E

−1

◦



◦

 k+ ≈ Rk + iR E .

k R E

◦

 k+ We recall that the set R E is log-convex. Notice that when k = 1, UE is a strip of C. Let k . Π : UE a −→ ψa ∈ R E For every x ∈ Rk , the function a −→ Π(a)(x) = e−ia,x is obviously analytic on UE . Fix M ∈ M(E). Applying Theorem 4 we get, for almost every k , χ∈R f (ψa χ) = HM,Π (χ)(a)f˜(ψa ) ∀a ∈ UE , M k , H∞ (UE )). Fix χ ∈ R k such that the above formula holds. Let δ ∈ Rk where HM,Π ∈ L∞ (R be such that χ = ψδ . We have f (ψaδ−1 ψδ ) = HM,Π (ψδ )(a)f˜(ψaδ−1 ψδ ) ∀a ∈ UE ∀f ∈ Cc (Rk ). M We obtain f (ψa ) = HM,Π (ψδ )(a)f˜(ψa ) M

∀a ∈ UE ∀f ∈ Cc (Rk ).

Set JM (a) = HM,Π (ψδ )(a)

∀a ∈ UE .

∞

Hence we get JM ∈ H (UE ) and f (ψa ) = JM (a)f˜(ψa ) M

∀a ∈ UE ∀f ∈ Cc (Rk ).

This proves the following corollary. Corollary ◦2. Let E be a Banach space of functions on Rk satisfying (H1), (H2) and (H3). k = ∅. Then, for M ∈ M(E), there exists J ∈ H∞ (U ) such that Suppose that R M E E f (ψa ) = JM (a)f˜(ψa ) M

∀a ∈ UE ∀f ∈ Cc (Rk ).

375

MULTIPLIERS ON BANACH SPACES

Now we will give some examples of Banach spaces satisfying our conditions. Example 4. Let ω be a continuous non-negative real function on G. For 1  p < +∞ set Lpω (G) the space of the measurable functions on G such that  |f (x)|p ω(x)p dx < +∞ G

and set



(1/p)

f ω,p =

|f (x)|p ω(x)p dx

for f ∈ Lpω (G).

G

Obviously, the Banach space Lpω (G) satisfies our hypotheses (H1) and (H3). Condition (H2) holds if and only if we have ω(x + y) < +∞ ω(y) y∈G

∀x ∈ G.

sup

(1.4)

For a complete study of the representation of the multipliers on L2ω (R) see [16]. For illustration, we will give one concrete example. Set ∀(n, k) ∈ Z2

ω(n, k) = emax(n,k)

and E = lω2 (Z2 ). In this case we have Sn,k  = max(en , ek ) and ρ(Sn,k ) = max(en , ek ). Here, 2 ≈ σ(S the set Z 1,0 , S0,1 ) and using Theorem 3 we get E 2 ≈ {(z , z ) ∈ C2 |1  |z |  e, Z 1 2 i E

i ∈ {1, 2}, |z1 ||z2 | = e}.

Notice, that σ(S1,0 , S0,1 ) = spec(S1,0 ) × spec(S0,1 ) and that the interior of σ(S1,0 , S0,1 ) is ◦ ◦ +  empty although spec(S1,0 ) = ∅ and spec(S0,1 ) = ∅. However, it is clear that Z2E has at least two elements. Example 5.

Let ω be a continuous weight on G. Set C0,ω (G) = {f ∈ C(G)|f ω ∈ C0 (G)}.

We equip C0,ω (G) with the norm f  = f ω∞ . It is clear that C0,ω (G) satisfies conditions (H1) and (H3) and if ω has the property 0 < sup x∈G

ω(x + y) < +∞ ω(x)

∀y ∈ G,

hypotheses (H2) holds in C0,ω (G). Example 6. Let A be a real-valued continuous function on [0, +∞[, such that A(0) = 0 and let (A(y)/y) be non-decreasing for y > 0. Let LA (G) be the set of all complex-valued, measurable functions on G such that    |f (x)| dx < +∞ A t G for some positive number t and let







f A = inf t > 0|

A G

|f (x)| t



dx  1

for f ∈ LA (G). Then LA (G) is a Banach space called a Birnbaum–Orlicz space [3]. It is easy to check that LA (G) satisfies (H1), (H2) and (H3).

376

V. PETKOVA

 2. The set G E Notice that for every φ ∈ CK (G), g ∈ E, the function G x −→ φ(x)Sx g ∈ E is uniformly continuous on G and  φ(x)Sx g dx  φ∞ g sup Sx m(K) < +∞. 

x∈K

K

We conclude that K φ(x)Sx g dx is well defined as a Bochner integral with respect to the strong operator topology [11, Chapter 3]. We have  Mφ = φ(x)Sx dx. (2.1) G

Indeed let K be a compact subset of G. We have Mφ (CK (G)) ⊂ CK+supp(φ) (G) and the restriction of G φ(x)Sx dx to CK (G) can be considered as a Bochner integral on CK (G) with values in CK+supp(φ) (G). It is clear that for x ∈ G, the map f −→ f (x) is a continuous linear form on CK0 , for every compact K0 . Since Bochner integrals commute with continuous linear forms we obtain, for g ∈ Cc (G),   Mφ g(x) = (φ ∗ g)(x) = φ(y)g(x − y) dy = φ(y)(Sy g)(x) dy G supp(φ)   =

φ(y)Sy g (x)

∀x ∈ G

supp(φ)

and formula (2.1) follows from the density of Cc (G) in E. Let A(E) be the closure in M(E) of the algebra {Mφ }φ∈Cc (G) . We have the following proposition. Proposition 1. If f ∈ Cc (G), f  0, f = 0, then we have ρ(Mf ) > 0. Proof. Fix f ∈ Cc (G) such that f  0 and f = 0. Let V be a compact neighbourhood of 0 such that supp(f ) ⊂ V and let F be a finite subset of G such that V + V ⊂ F + V . We set nV := {s1 + · · · + sn , s1 , . . . , sn ∈ V } and we notice that Denote f 1 =

 G

nV ⊂ (n − 1)F + V f (x) dx. We get 

∀n  1.

f ∗n (x) dx = f n1

(2.2)

∀n  1,

nV

where f ∗n is the product of n times f in the convolution algebra L1 (G). From (2.2) it follows that   ∗n f (x) dx = f ∗n (x) dx nV (n−1)F +V   = S−s f ∗n (x) dx s∈(n−1)F

 CV



V

s∈(n−1)F

S−s f ∗n E ,

377

MULTIPLIERS ON BANACH SPACES

where CV is a constant such that  |g(x)| dx  CV gE

∀g ∈ E.

V

Let k be the number of the elements of F and let D = maxs∈F S−s . Then we have  S−s   k n−1 Dn−1 s∈(n−1)F

and consequently  n+1 f ∗(n+1) (x) dx  CV k n Dn f ∗(n+1)   CV k n Dn (Mf )n f  f 1 =

∀n  1.

G

We deduce that (Mf )n  

f n+1 1 CV f (kD)n

and therefore we get ρ(Mf ) 

f 1 > 0. kD

Observe that the proposition above implies that the algebra A(E) is not radical. Next notice that Sx ◦ Mφ = MSx (φ) , hence R ◦ T ∈ A(E) for each R ∈ B(E) and for each T ∈ A(E). Let  We have γ ∈ A(E). γ(R ◦ T2 ) γ(R ◦ T1 ) = γ(T1 ) γ(T2 ) / Ker(γ), and for R ∈ B(E), T1 , T2 ∈ A(E) \ Ker(γ). Now choose φ ∈ Cc (G) such that Mφ ∈ define Δγ : B(E) −→ C by the formula Δγ (R) =

γ(R ◦ Mφ ) γ(Mφ )

∀R ∈ B(E).

(2.3)

It is clear that Δγ (R1 ◦ R2 ) =

γ(R1 ◦ R2 ◦ Mφ2 ) γ(R1 ◦ R2 ◦ Mφ ) = = Δγ (R1 )Δγ (R2 ) γ(Mφ ) γ(Mφ2 )

 Notice that we have for R1 , R2 ∈ B(E). Since Δγ (I) = 1, we obtain Δγ ∈ B(E). 1  |Δγ (Sx )|  ρ(Sx ) ρ(S−x )

∀x ∈ G.

(2.4)

We need the following lemma. Lemma 1. Let E be a Banach space satisfying (H1) and (H2).   (i) For θ ∈ G E there exists γθ ∈ A(E) such that θ(x) =

γθ (Mφ ) γθ (Sx ◦ Mφ )

for φ ∈ Cc (G) such that Mφ ∈ / Ker(γθ ) and we have 1  |θ(x)−1 |  ρ(Sx ) ρ(S−x )

∀x ∈ G.

(2.5)

378

V. PETKOVA

  (ii) The map T : θ −→ γθ is a homeomorphism from G E onto A(E), with respect to the Gelfand topology.  Proof. Fix θ ∈ G E and define



φ(x)θ(x)−1 dx

γθ (Mφ ) =

∀φ ∈ Cc (G).

(2.6)

G

 We will prove that Taking into account the Fubini theorem, it is easy to see that γθ ∈ A(E). θ(x) = We have, for y ∈ G, γθ (MSy φ ) =

 G

=

γθ (Mφ ) . γθ (Sx ◦ Mφ )

(2.7)



φ(x − y)θ(x)−1 dx  φ(x)θ(x + y)−1 dx = θ(y)−1 φ(x)θ(x)−1 dx = θ(y)−1 γθ (Mφ ) Sy φ(x)θ(x)

−1

dx =

G

G

G

and (2.7) holds. Next observe that (2.4) implies (2.5). This completes the proof of assertion (i).  and fix ψ ∈ Cc (G) such that Mψ ∈ Fix γ ∈ A(E) / Ker(γ). Set θγ (x) =

γ(Mψ ) γ(Sx ◦ Mψ )

∀x ∈ G.

Suppose that (φn )n0 ⊂ CK (G) is a sequence converging to φ ∈ CK (G) uniformly on K. For every g ∈ E we get Mφn g − Mφ g  φn − φ∞ sup Sy gm(K) y∈K

and this implies that limn→+∞ Mφn − Mφ  = 0. This shows that the linear map φ −→ Mφ is sequentially continuous and hence continuous from Cc (G) into A(E). Since the map x −→ Sx (φ) is continuous from G into Cc (G), we conclude that the map x −→ Sx ◦ Mφ = MSx (φ)  is continuous from G into A(E). Thus θγ is continuous on G. Now we will prove that θγ ∈ G E. Set η : Cc (G) φ −→ γ(Mφ ). The map η is a continuous linear form on Cc (G) and hence there exists some measure μ [10, Chapter 3] such that  η(φ) = φ(x) dμ(x) ∀φ ∈ Cc (G). G

This implies that for every f , φ ∈ Cc (G) we have  γ(Mφ ◦ Mf ) = (φ ∗ f )(t) dμ(t)  G  = φ(x)f (t − x) dx dμ(t). G

G

Using the Fubini theorem, we obtain     γ(Mφ ◦ Mf ) = φ(x) f (t − x) dμ(t) dx = φ(x)γ(Sx ◦ Mf ) dx G

G

G

MULTIPLIERS ON BANACH SPACES

and



φ(x)θ(x)−1 dx

γ(Mφ ) =

∀φ ∈ Cc (G).

379

(2.8)

G

   Consequently, we conclude that  G φ(x)θ(x)−1 dx = |γ(Mφ )|  Mφ  and θγ ∈ G E . Define  −→ G,  R : A(E)  by the formula for γ ∈ A(E), R(γ)(x) =

γ(Mφ ) γ(Sx ◦ Mφ )

∀x ∈ G

 / Ker(γ). We have T R = RT = I. Let θ0 ∈ G for φ ∈ Cc (G) such that Mφ ∈ E . For fixed φ1 , . . . , φk ∈ Cc (G) and > 0 we have      −1 −1  θ(x) φi (x) dx <

sup  θ0 (x) φi (x) dx − i=1,...,k

G

G

 for every θ ∈ G E such that sup

|θ0 (x)−1 − θ(x)−1 |
0 such that |γα (Mf )|  δ, ∀α  β. We have R(γα )(x)−1 = ((γα (Sx ◦ Mf ))/(γα (Mf ))) and we obtain |(R(γα )(x))−1 − (R(γα )(y))−1 |  δ −1 γα Sx ◦ Mf − Sy ◦ Mf   δ −1 Sx ◦ Mf − Sy ◦ Mf  ∀x, y ∈ G, ∀α  β. Thus we deduce that the family (R(γα ))αβ is equicontinuous on G. Since (R(γα )) converges to R(γ) simply, (R(γα )) converges uniformly on every compact of G to R(γ). Consequently, R is continuous and the proof is complete. Proposition 2. Let G be a topological group and let φ : G → R be a unital morphism. Then φ is continuous if and only if φ is locally bounded. Proof. Every continuous morphism is clearly locally bounded. Now assume that φ is locally bounded and let U be a neighbourhood of the unit element 0G , and M > 0 such that φ(U ) ⊂ [−M, M ]. Let > 0, and let n  1 be such that M/n < . There exists a neighbourhood V of 0 such that nx ∈ U, so that n|φ(x)|  M for every x ∈ V. Hence |φ(x)| < for every x ∈ V, which shows that φ is continuous at 0 and hence continuous on G. Corollary 3. Let E ⊂ L1loc (G) be a Banach space satisfying hypotheses (H1) and (H2).  Then the map x −→ |χ(Sx )| is continuous on G. Let χ ∈ B(E). Proof. Let K = K −1 be a compact neighbourhood of 0. It follows from hypotheses (H2) that 1  M := supx∈K Sx  < +∞. For x ∈ K we have |χ(Sx )|  M,

|χ(Sx )|−1  M.

We obtain − log M  log |χ(Sx )|  log M

∀x ∈ K,

380

V. PETKOVA

and it follows from Proposition 2 that the map x −→ log |χ(Sx )| is continuous on G. This completes the proof.  Next we will use (H3). Recall that this condition implies that G E has the following property:   χG E = GE

 ∀χ ∈ G.

It is well known that a Reinhardt domain X such that 0 ∈ X is monomially convex if and only if it is a logarithmically convex domain and contains the polydisc of radius Y(z), for every z ∈ X \ {0}. Recall that the definition of a Reinhardt domain and the definition of the map Y are given in Section 1. Moreover, a Reinhardt domain X such that 0 ∈ X is monomially convex if and only if X is the domain of convergence of a power series [14]. The following proof of Theorem 3 is related to analogous results for Reinhardt domains contained in C∗k .  Proof of Theorem 3. Taking into account Proposition 1 and Lemma 1, it is clear that G E   + + is not empty. First, we prove that GE is equicontinuous. Let γ ∈ GE . We have 1  γ(x)−1  ρ(Sx ) ρ(S−x )

∀x ∈ G.

Hence for a compact neighbourhood V0 of 0G , we have 1 C−V0

 γ(x)−1  CV0

∀x ∈ V0 ,

where CV0 = supx∈V0 ρ(Sx ) < +∞ and C−V0 = supx∈V0 ρ(S−x ) < +∞. Fix δ > 0. There exists n > 0 such that (CV0 )1/n − 1 < δ and 1 − (1/(C−V0 )1/n ) < δ. The map x −→ nx is continuous on G and there exists a neighbourhood Wδ of 0 such that nx ∈ V0 , for x ∈ Wδ . Then 1  γ(nx)−1  CV0 C−V0

∀x ∈ Wδ

and 1  γ(x)−1  (CV0 )1/n (C−V0 )1/n

∀x ∈ Wδ .

It follows that 1 − δ  γ(x)−1  1 + δ and hence γ(0)−1 − δ  γ(x)−1  γ(0)−1 + δ, for   + + is equicontinuous at 0 and so G is equicontinuous on G. x ∈ W . Now it is clear that G δ

E

E

is bounded for every x ∈ G and it follows from the We have seen that the set {θ(x)−1 }θ∈G  E  +   definition of GE that GE is closed in G with respect to the uniform convergence on every  + compact subset. A standard version of the Ascoli theorem [19] implies that G E is compact.   + + Next we will prove that GE is log-convex. Suppose that GE has at least two elements. Let η1  and η2 be in G E and assume that |η1 | = |η2 |. Set L = {z ∈ C | Im z ∈ [0, 1]}. For λ ∈ L, define θλ (x) = |η1 (x)|λ |η2 (x)|1−λ ,

x ∈ G.

For f ∈ Cc (G) and for every x ∈ supp(f ) we have  −1

sup |f (x)θλ (x) λ∈L

|  f ∞ sup l∈[0,1]

sup x∈supp(f )

 |η1 (x)|

l

sup

|η2 (x)|

1−l

< +∞.

x∈supp(f )

The function G × L : (x, λ) −→ f (x)θλ (x)−1 ∈ C is separately continuous and uniformly bounded and so this function is measurable on G × L [13]. Then using the Morera theorem and the Fubini theorem, we obtain that for f ∈ Cc (G)

381

MULTIPLIERS ON BANACH SPACES

the function F defined by



f (x)θλ (x)−1 dx

F : λ −→ G ◦

is analytic on the strip L. By the Phragmen–Lindel¨ of theorem, we obtain     |F (λ)|  max  f (x)θλ (x)−1 dx  Mf . Re λ∈{0,1}

Then we have

G

      |η1 (x)|ia  |η2 (x)|ia    dx , sup f (x)|η1 (x)| dx . |F (λ)|  max sup  f (x)|η2 (x)| |η2 (x)|ia  a∈R  |η1 (x)|ia  a∈R 

G

G

 + ia 1−ia 1−ia   Since |η2 | ∈ G |η1 |ia ∈ G |η2 |ia ∈ E , ∀a ∈ R. Similarly |η1 | E and |η1 /η2 | ∈ G, we have |η2 |  G E ∀a ∈ R and we obtain |F (λ)|  Mf 

∀λ ∈ L.

 This implies that θλ ∈ G E ∀λ ∈ L.  +  + and suppose that θ ∈ Now we prove (ii). Let θ ∈ G /G E . Then there exists φ ∈ Cc (G) and

> 0 such that      φ(x)θ(x)−1 dx > Mφ  + . (2.9)   G

 + Let K = supp(φ). Since the family {φη −1 , η ∈ G E ∪ {θ}} is equicontinuous on K, for every x ∈ K there exists a neighbourhood Vx of x in K such that

 + ∀η ∈ G sup |φ(y)η(y)−1 − φ(x)η(x)−1 | < E ∪ {θ}. 3m(K) y∈Vx The family {Vx }x∈K is an open covering of K, hence there exists a1 , . . . , ap ∈K such that p K ⊂ i=1 Vai . Set Vaci = {x ∈ K | x ∈ / Vai }. Define K1 = Va1 and Ki = Vai ∩ ( j =i Vaj )c for 1 < i  p. We see that   p      −1 −1 φ(ai )η(ai ) m(Ki )  φ(x)η(x) dx −  K  i=1   p      (φ(x)η(x)−1 − φ(ai )η(ai )−1 ) dx =   i=1 Ki  p p  



m(Ki ) = .  dx = 3m(K) 3m(K) i=1 3 i=1 Ki  + Let η ∈ G E . Since we have   p      φ(ai )η(ai )−1 m(Ki )   φ(x)η(x)−1 dx −   K i=1   p      φ(ai )θ(ai )−1 m(Ki )   φ(x)θ(x)−1 dx −   K i=1

and

     −1  φ(x)η(x)−1 dx −  > , φ(x)θ(x) dx   K

K

, 3

3

382

V. PETKOVA

we obtain

 p  p   

  −1 −1 φ(ai )η(ai ) m(Ki ) − φ(ai )θ(ai ) m(Ki ) > .    3 i=1 i=1

Hence we get  + ∀η ∈ G E.

(θ(a1 )−1 , . . . , θ(ap )−1 ) = (η(a1 )−1 , . . . , η(ap )−1 ) Set

C = (log |η(a1 )|, . . . , log |η(ap )|),

  η∈G E .

Since the set C is closed and convex, we have (log θ(a1 ), . . . , log θ(ap )) ∈ / C. Hence, there exists a linear form L on Rp such that L ((log θ(a1 ), . . . , log θ(ap ))) > sup L ((log |η(a1 )|, . . . , log |η(ap )|)) .  η∈G E

Set

⎧ ⎨ Δ = (α1 , . . . , αp ) ∈ Rp |α1 log θ(a1 ) + · · · + αp log θ(ap ) ⎩ ⎫ ⎬ > sup (α1 log η(a1 ) + · · · + αp log η(ap )) . ⎭  + η∈GE

Since sup

 η∈G+ E

| log η(ai )| < +∞ for 1  i  p, Δ is open, and since λΔ ⊂ Δ for λ > 0 we have

Δ ∩ Z = ∅. Now let (n1 , . . . , np ) ∈ Δ ∩ Zp . We have p

1 1 p −1 p −1 θ(a−n . . . a−n ) > sup |η(a−n . . . a−n ) | p p 1 1

 η∈G E

and this shows that





 | |θ(x)−1 |  sup |η(x)−1 | θ∈G  η∈G E

∀x ∈ G

 ⊂G E,

which proves (ii). Corollary 4.

We have −1   |  ρA(E) (Sx ) G E = {θ ∈ G | |θ(x)

∀x ∈ G},

where ρA(E) (Sx ) = supγ∈A(E)  |Δγ (Sx )|. We recall that the definition of Δγ is given by formula (2.3). Corollary 5.

  G E is connected if and only if G is connected.

383

MULTIPLIERS ON BANACH SPACES

3. A representation theorem 3.1. Quasimeasures In this section we give some useful results about the representation of a multiplier as a convolution operator. Let BG be the family of Borel of G. A measure on G is a map μ : BG,μ −→ C of the form μ = μ1 − μ2 + iμ3 − iμ4 , where μi is a positive measure on G for i = 1, . . . , 4 such that μi (K) < +∞, for every K compact subset of G and where BG,μ = {U ∈ BG | sup μi (U ) < +∞, i = 1, . . . , 4}. Denote by M (G) the set of the measures on G. The Riesz representation theorem shows that every continuous linear form on Cc (G) can be uniquely represented in the form  L(f ) = G f (x) dμL (x), where μL ∈ M (G). We will denote by Mb (G) the set of all measures on G of bounded variations, equipped with the norm μMb (G) = |μ|(G), which can be identified as above with the dual of C0 (G). Denote by Mc (G) the set of measures on G with compact support and equip M (G) with the vague topology. Recall that a net (μα ) ⊂ M (G) converges vaguely to μ ∈ M (G) if and only if   f (x) dμ(x) = lim f (x) dμα (x) ∀f ∈ Cc (G). α

G

G

We need the following definitions in order to introduce the result of Gaudy proved in [9]. Definition 2. DK (G) =

Let K be a compact subset of G. Define

u ∈ Cc (G)|u =

∞ 

fi ∗ gi , fi , gi ∈ CK (G)

and

i=1

∞ 

fi ∞ gi ∞ < +∞ .

i=1

We equip DK (G) with the norm ∞ ∞   uDK (G) = inf fi ∞ gi ∞ | u = fi ∗ gi , fi , gi ∈ CK (G) i=1

and

i=1 ∞ 



fi ∞ gi ∞ < +∞ .

i=1

Definition 3. Consider in the category of locally convex spaces the inductive limit   D(G) = limDK (G). Denote by D(G) the dual space of D(G). The elements of D(G) are → called quasimeasures. We have the following important result. Theorem 5 [9]. For every continuous linear operator M from Cc (G) into the space of measures M (G), which commutes with convolutions by functions in Cc (G), there exists a quasimeasure μ such that M f = μ ∗ f ∀f ∈ Cc (G). Notice that for all f ∈ Cc (G) the application g −→ f ∗ g is continuous from D(G) into D(G) and for f ∈ Cc (G), μ ∗ f is a quasimeasure defined by μ ∗ f, g = μ, f ∗ g

∀g ∈ D(G).

384

V. PETKOVA

If f ∈ D(G), then μ ∗ f is a continuous function on G defined by: y −→ μx , f (x − y). Applying Theorem 5, the restriction to Cc (G) of a multiplier on Lp (G), for 1  p < +∞, is characterized as a convolution with a quasimeasure [9, 2]. Otherwise, Edwards gave a representation theorem for multipliers from Cc (G) into Mc (G). Set  B(G) = {fˆ, f ∈ L1 (G)} and fˆB(G) = f L1 (G)  . We will denote by P (G) the dual of B(G). The elements of P (G) are called pseudomeasures. Denote by P 1 (G) the set of all pseudomeasures s such that s ∗ f ∈ Mb (G) for all f ∈ Cc (G). We have the following theorem. Theorem 6 [7]. (1) The continuous linear operators T from Cc (G) into Mb (G) which commute with translations are precisely the operators having the form Tf = s ∗ f

∀f ∈ Cc (G),

where s ∈ P (G). (2) The continuous linear operators T from Cc (G) into Mc (G) which commute with translations are precisely the operators having the form 1

Tf = s ∗ f

∀f ∈ Cc (G),

where s is a pseudomeasure with compact support. Furthermore, this theorem implies that a quasimeasure with compact support is a pseudomeasure [9]. The Fourier transform sˆ of a pseudomeasure s is defined as follows: sˆ is the continuous linear form on L1 (G) given by sˆ : L1 (G) f −→ s(fˆ) ∈ C.  Following Theorem 6, for a linear Notice that sˆ can be identified with an element of L∞ (G). operator T from Cc (G) into Mb (G) continuous with respect to the inductive limit topology on Cc (G) and the vague topology on M (G) and commuting with convolutions by functions in Cc (G) we have (3.1) Tf = hfˆ ∀f ∈ Cc (G),  Such a representation holds also for multipliers on Lp (G), 1  p < +∞ where h ∈ L∞ (G). (see [2]). 3.2. Approximation of a multiplier We need several lemmas. If E ⊂ L1loc (G) is a Banach space satisfying (H1), (H2) and (H3) we will denote as before by Mφ : f −→ f ∗ φ the convolution operator associated with a function φ ∈ Cc (G). Following some arguments of Gaudry and Figa-Talamanca [8, 9], we obtain the following. Lemma 2. Let E ⊂ L1loc (G) be a Banach space satisfying (H1), (H2) and (H3). For every M ∈ M(E) there exists a net (φα ) ⊂ Cc (G) such that (i) M = limα Mφα with respect to the strong operator topology; (ii) we have Mφα   CM , where C is a constant independent of M.

385

MULTIPLIERS ON BANACH SPACES

Proof. Fix M ∈ M(E). Let μ be the quasimeasure such that Mf = μ ∗ f

∀f ∈ Cc (G).

 −→ E by Fix f ∈ E and define Tf : G Tf (χ) = χM (χf ),

 χ ∈ G.

 into E. Denote by 1  the unit element of G.  It is easy to see that Tf is continuous from G G 1  Let (kα ) ⊂ L (G) be a net such that k α ∈ Cc (G), kα L1 (G)  = 1, kα (χ)  0, for every α and  limα χ∈V kα (χ) dχ = 0, for every neighbourhood V of 1G . We have / lim(kα ∗ Tf )(1G ) = Tf (1G ) = M f, α

Observe that



−1

 G

kα (χ)χ

and it shows that

−1

M (χ

 f )dχ 

  G

 G

∀f ∈ E.

kα (χ)M χ−1 f dχ  M f 

(3.2)

kα (χ)χ−1 M (χ−1 f ) dχ

 with values in E. Define Yα : E −→ E by the formula is well defined as a Bochner integral on G  Yα (f ) = (kα ∗ Tf )(1G ) = kα (χ)χ−1 M (χ−1 f ) dχ.  G

We claim that for f ∈ DK (G) and x ∈ G, we have  Yα f (x) = kα (χ)χ(x)−1 M (χ−1 f )(x) dχ.  G

(3.3)

We will justify formula (3.3). First, we will show that if f ∈ DK (G) then M (f χ−1 ) is continuous on G. Indeed, we have M (χ−1 f )(x) = μy , χ(y − x)−1 f (y − x)

∀x ∈ G.

Notice that for g ∈ DK (G), for x0 ∈ G and a compact K0 which contains both K and K + Vx0 , where Vx0 is a neighbourhood of x0 , we get Sx g − Sx0 gDK0 (G)

    inf Sx fi − Sx0 fi ∞ gi ∞ , s.t. g = fi ∗ gi , fi ∞ gi ∞ < +∞ . i

i

i

It is clear that lim Sx fi − Sx0 fi ∞ = 0

x→x0

for every i and since



fi ∞ gi ∞ < +∞

i

we obtain lim Sx g − Sx0 gDK0 (G) = 0.

x→x0

It follows that the function M (χ−1 f ) is continuous. Secondly we need the bound sup sup |M (f χ−1 )(x)| < +∞.

 x∈G χ∈G

(3.4)

386

V. PETKOVA

Let Sx DK (G)→DK0 (G) be the norm of the operator Sx considered as an operator on DK (G)  we obtain into DK0 (G). Then, for χ ∈ G, (M (χ−1 f ))(x0 ) = μy , f (y − x0 )χ(y − x0 )−1   μSx0 DK (G)→DK0 (G) f χ−1 DK (G) . It is easy to see that f χ−1 DK (G) = f DK (G) and hence Sx0 DK (G)→DK0 (G)  1. This implies sup |M (f χ−1 )(x0 )| < μf 

 χ∈G

and (3.4). Let x0 ∈ G and let K0 be a compact neighbourhood of x0 . Let ψ ∈ CK0 (G) be such that ψ(x0 ) = 1. Clearly,   −1 −1 kα (χ)ψχ M (χ f )∞ dχ  A kα (χ) dχ,  G

 G

where A is a constant. Consequently,  kα (χ)ψχ−1 M (χ−1 f ) dχ  G

is well defined as a Bochner integral with values in CK0 (G). Therefore, since ψχ−1 M (χ−1 f ) ∈ CK0 (G), repeating the argument of the beginning of Section 2, we obtain    kα (χ)ψχ−1 M (χ−1 f ) dχ (x) = kα (χ)ψ(x)χ(x)−1 M (χ−1 f )(x) dχ ∀x ∈ K0 ,  G

 G

which implies that    −1 −1 kα (χ)χ M (χ f ) dχ (x0 ) = kα (χ)χ(x0 )−1 M (χ−1 f )(x0 ) dχ.  G

 G

 of kα (χ)ψχ−1 M (χ−1 f ) Taking into account condition (H1), the Bochner integral on G considered with respect to the norm of E defines the same function as the Bochner integral on  of kα (χ)ψχ−1 M (χ−1 f ) considered with respect to the norm of CK (G) and this completes G 0 the proof of (3.3). Now, for f ∈ D(G) and x ∈ G, we have  kα (χ)χ(x)−1 M (χ−1 f )(x) dχ Yα f (x) =  G  = kα (χ)χ(x)−1 μy , (χ−1 f )(y − x)dχ  G = kα (χ)μy , χ(−y)f (y − x)dχ  G    = μy , kα (χ)χ(−y) dχ f (y − x)  G

 = μy , k α (y)f (y − x) = (kα μ ∗ f )(x). Moreover, taking into account (3.2), we have a control of the norm of the operator Yα . Indeed, we have Yα   M . To approximate M with respect to the strong operator topology, it is sufficient to approximate the operators Yα , which are the convolutions with the quasimeasures  with compact support k α μ. In order to do this we fix α and we set ν = kα μ. Recall that ν

387

MULTIPLIERS ON BANACH SPACES

is also a pseudomeasure. Denote by V the operator of convolution with ν. Consider a net (hβ ) ⊂ Cc (G) ∗ Cc (G) satisfying the following conditions: (i) hβ L1 (G) = 1 ∀β; (ii) hβ vanishes outside some fixed compact K; (iii) hβ (x)  0 ∀x ∈ G;  hβ (x)dx = 0. (iv) For every neighbourhood O of 0G , limβ x∈O / Set Vβ (f ) = V (hβ ∗ f ), for f ∈ E. Taking into account our hypotheses on E and the properties of (hβ ), it is easy to see that limβ hβ ∗ f − f  = 0 ∀f ∈ E. It implies that limβ Vβ f − V f  = 0 for f ∈ E. Moreover, it is easy to control the norm of Vβ . In fact, for f ∈ Cc (G), we have Vβ f = V f ∗ hβ and      Vβ f  = hβ ∗ V f  =  hβ (y)Sy (V f )dy   G   |hβ (y)|Sy (V f ) dy  (sup Sy )V f . y∈K

G

Hence we obtain Vβ   CV , where C is a constant. On the other hand, Vβ f = (ν ∗ hβ ) ∗ f for f ∈ Cc (G). For g ∈ D(G) we have    ν ∗ hβ , g = ν, g ∗ hβ  = νx , (Sy hβ )(x)g(y) dy . G

We recall that ν ∈ P (G) and Sy hβ ∈ Cc (G) ∗ Cc (G). It is easy to see that Cc (G) ∗ Cc (G) ⊂  we have B(G). Indeed, for f1 , f2 ∈ Cc (G), f1 ∗ f2 = F −1 (Ff1 Ff2 ) and since Ff1 , Ff2 ∈ L2 (G), 1  Ff1 Ff2 ∈ L (G). Observe that   Sy hβ B(G) |g(y)| dy = hβ B(G) |g(y)| dy < +∞ G

G

 and hence G g(y)Sy hβ dy is a Bochner integral with values in B(G). Since Bochner integrals commute with continuous linear forms, we have  ν ∗ hβ , g = νx , (Sy hβ )(x)g(y)dy ∀g ∈ D(G). G

We deduce that the quasimeasure ν ∗ hβ is the function defined by (ν ∗ hβ )(y) = νx , hβ (x − y) = ν, Sy hβ  We will check that the function G : G y −→



 χ −→ G

∀y ∈ G. 

 (Sy hβ )(x)χ(x) dx

 ∈ L1 (G)

G

 We have is continuous from G into L (G). 1

β (χ−1 ) G(y) : χ −→ χ(y)h  the function for all y ∈ G. It is easy to see that for φ ∈ Cc (G)    χ −→ χ(y)φ(χ) ∈ L1 (G) G y −→ G  is dense in L1 (G),  we obtain that G is the uniform limit of is continuous. Indeed, since Cc (G) a sequence of continuous functions and hence G is continuous. We conclude that the function G y −→ Sy hβ ∈ B(G) is continuous from G into B(G). We deduce that ν ∗ hβ ∈ Cc (G) and the proof is complete. Now we are ready to prove our main theorem.

388

V. PETKOVA

 Proof of Theorem 4. Fix M ∈ M(E) and θ ∈ G E . Let (φα ) ⊂ Cc (G) be a net such that (Mφα ) converges to M with respect to the strong operator topology and such that Mφα    CM . Fix θ ∈ G E . We have         φα θ−1 (χ) =  φα (x)θ(x)−1 χ(x)−1 dx  Mφα   CM  ∀χ ∈ G. G

−1 ) converges to a function h If we replace (φα ) by a suitable subnet, we obtain that (φ αθ M,θ ∈ ∞  ∗ ∞  1  L (G) for the weak topology σ(L (G), L (G)). Moreover, we have hM,θ ∞  CM . Taking into account that   −1 (χ)g(χ) dχ =  lim φ θ hM,θ (χ)g(χ) dχ ∀g ∈ L1 (G), α α

 G

 G

we obtain   −1 (χ)f −1 (χ)g(χ) dχ =  lim φ θ θ hM,θ (χ)f θ−1 (χ)g(χ) dχ α α

 G

 G

 ∀f ∈ Cc (G), ∀g ∈ L2 (G).

This implies that for each f ∈ Cc (G), the net (F((Mφα f )θ−1 )) = (F((φα ∗ f )θ−1 )) = −1 f   Consequently, (φ θ−1 ) converges to hM,θ f θ−1 with respect to the weak topology of L2 (G). αθ θ−1 ) lim(Mφα f )θ−1 = F −1 (hM,θ f α

∀f ∈ Cc (G)

with respect to the weak topology of L2 (G). On the other hand, lim Mφα f − M f  = 0 α

∀f ∈ E

and by (H1), for g ∈ Cc (G), we get     −1  lim  g(y)θ(y) (Mφα f (y) − M f (y))dy  = 0. α G

θ−1 ) define the same linear functional on Cc (G) We see that functions (M f )θ−1 and F −1 (hM,θ f for every f ∈ Cc (G). We obtain for all f ∈ Cc (G), θ−1 )(x), a.e. (M f )(x)θ(x)−1 = F −1 (hM,θ f  we have θ−1 ) ∈ L2 (G). Then, for almost every χ ∈ G, Notice that (M f )θ−1 = F −1 (hM,θ f F((M f )θ−1 )(χ) = hM,θ (χ)F(f θ−1 )(χ)

∀f ∈ Cc (G).

 +  We set for M ∈ M(E), for δ ∈ G E and for almost every χ ∈ G hM (δχ) = hM,δ (χ).  +  Then, for every δ ∈ G E and for almost every χ ∈ G we have f (δχ) = hM (δχ)f˜(δχ) M

∀f ∈ Cc (G),

 f (δχ) = (M where M f )δ −1 (χ) a.e. This completes the proof of (i).  Now we will prove assertion (ii). Let U be an open subset of Cp and let Π : U −→ G E be an  , we have analytic function. Since for every λ ∈ U , Π(λ) ∈ G E   −1   sup (Π(λ)(x))  sup ρ(Sx )  sup Sx  < +∞ x∈K

x∈K

x∈K

 the function for every compact K ⊂ G. For χ ∈ G, −1

G × U (x, λ) −→ φα (x) (Π(λ)(x))

χ(x)−1

is separately continuous and uniformly bounded and so is a measurable function on G × U (see [13]). Let D1 , . . . , Dp be open discs of C such that D1 × · · · × Dp ⊂ U . For fixed λj , for

389

MULTIPLIERS ON BANACH SPACES

j = i and for every triangle T ⊂ Di , using the Fubini theorem, we have   φα (x)(Π(λ1 , . . . , λp )(x))−1 χ(x)−1 dλi dx T G    = φα (x)χ(x)−1 (Π(λ1 , . . . , λp )(x))−1 dλi dx. G

T −1

Since λi −→ (Π(λ1 , . . . , λp )(x)) function

is analytic on Ui , we obtain by the Morera theorem that the 

φα (x)(Π(λ1 , . . . , λp )(x))−1 χ(−x) dx

Di λi −→ G

is analytic. We obtain that −1

U λ −→ F(φα Π(λ)



φα (x)(Π(λ)(x))−1 χ(−x) dx

)(χ) = G

is separately analytic, hence analytic on U . Set α (Π(·)χ) ∈ H∞ (U ).  χ −→ φ Δα : G For every α we have Δα L∞ (G,H  ∞ (U ))  CMφα .  H∞ (U )) is uniformly bounded. Equip U with the Lebesgue measure. The net (Δα ) ⊂ L∞ (G, We can identify the dual of L1 (U ) with L∞ (U ), the duality being implemented by the formula  f, g = f (x)g(x) dx ∀f ∈ L1 (U ) ∀g ∈ L∞ (U ). U ∞

The space H (U ) is closed with respect to the topology σ(L∞ (U ), L1 (U )). Set ∞ H⊥ (U ) = {f ∈ L1 (U )|f, g = 0

∀g ∈ H∞ (U )}.

∞ We can identify the dual of H∗∞ (U ) := L1 (U )/H⊥ (U ) with H∞ (U ). We set

P(f ), g = f, g

∀f ∈ L1 (U ) ∀g ∈ H∗∞ (U ),

∞  with the where P : L1 (U ) −→ L1 (U )/H⊥ (U ) denotes the canonical surjection. Now, equip G  H∞ (U )) with L∞ (G,  H∞ (U )), the duality being Haar measure and identify the dual of L1 (G, ∗ implemented by the formula   H∞ (U )) ∀g ∈ L∞ (G,  H∞ (U )). f, g = f (χ), g(χ)dχ ∀f ∈ L1 (G, ∗  G

 H∞ (U )) with We can extract from (Δα ) a subnet convergent to a function HM,Π ∈ L∞ (G, ∗ ∞  ∞ 1  ∞ respect to the weak topology σ(L (G, H (U )), L (G, H∗ (U ))). We will denote also by (Δα ) this convergent subnet. We have    H∞ (U )). lim g(χ)(·), Δα (χ)(·)dχ = g(χ)(·), HM,Π (χ)(·)dχ ∀g ∈ L1 (G, ∗ α

 G

 G

 Set Lλ : H (U ) F −→ F (λ), for λ ∈ U . Notice that Lλ ∈ H∗∞ (U ), for λ ∈ U . Fix g ∈ L1 (G)  H∞ (U )) by the formula G(χ)(λ) = g(χ)Lλ for every λ ∈ U , for almost and define G ∈ L1 (G, ∗  For almost every χ ∈ G  we have every χ ∈ G. ∞

G(χ)(·), Δα (χ)(·) = g(χ)Δα (χ)(λ)

∀λ ∈ U

and G(χ)(·), HM,Π (χ)(·) = g(χ)HM,Π (χ)(λ)

∀λ ∈ U.

390

MULTIPLIERS ON BANACH SPACES

It follows that

 lim α

 G

α (Π(λ)χ) dχ = g(χ)φ

  G

g(χ)HM,Π (χ)(λ) dχ

∀λ ∈ U.

 Using the definition of hM given in the proof of (i), we conclude that for almost every χ ∈ G HM,Π (χ)(λ) = hM (Π(λ)χ)

∀λ ∈ U.

 we obtain Finally, for all λ ∈ U and for almost all χ ∈ G, f (Π(λ)χ) = HM (χ)(λ)f˜(Π(λ)χ) M Acknowledgement. ment.

∀f ∈ Cc (G).

The author thanks Jean Esterle for his useful advice and encourage-

References 1. N. Bourbaki, General topology (Springer, Berlin, 1998). 2. B. Brainerd and R. E. Edwards, ‘Linear operators which commute with translations. I. Representation theorems’, J. Austral. Math. Soc. 6 (1966) 289–327. 3. I. M. Bund, ‘Birnbaum–Orlicz spaces of functions on groups’, Pacific J. Math. 58 (1975) 351–359. 4. R. M. Dudley, ‘On sequential convergence’, Trans. Amer. Math. Soc. 112 (1964) 483–507. 5. R. E. Edwards, Functional analysis: theory and applications (Holt, Rinehart & Winston, New York, 1965). 6. R. E. Edwards, ‘Operators commuting with translations’, Pacific J. Math. 16 (1966) 259–265. 7. R. E. Edwards and G. I. Gaudry, Littlewood–Paley and multiplier theory (Springer, Berlin, 1977). 8. A. Figa-Talamanca and G. I. Gaudry, ‘Density and representation theorems for multipliers of type (p, q)’, J. Austral. Math. Soc. 7 (1967) 1–6. 9. G. I. Gaudry, ‘Quasimeasures and operators commuting with convolution’, Pacific J. Math. 18 (1966) 461–476. 10. E. Hewitt and K. A. Ross, Abstract harmonic analysis, vol. 1 (Springer, Berlin, 1970). 11. E. Hille and R. S. Phillips, Functional analysis and semi-groups (American Mathematical Society, Providence, 1957). ¨ rmander, ‘Estimates for translation invariant operators in Lp spaces’, Acta Math. 104 (1960) 93–140. 12. L. Ho 13. B. E. Johnson, ‘Separate continuity and measurability’, Proc. Amer. Math. Soc. 20 (1969) 420–422. 14. L. Kaup and B. Kaup, Holomorphic functions of several variables (de Gruyter, Berlin, 1983). 15. R. Larsen, The multiplier problem (Springer, Berlin, 1969). 16. V. Petkova, ‘Symbole d’un multiplicateur sur L2ω (R)’, Bull. Sci. Math. 128 (2004) 391–415. 17. V. Petkova, ‘Multipliers and Toeplitz operators on Banach spaces of sequences’, 2006, https:// hal.archives-ouvertes.fr/hal-00121335. 18. A. Weil, L’int´ egration dans les groupes topologiques (Hermann, Paris, 1965). 19. J. D. Weston, ‘A generalization of Ascoli’s theorem’, Mathematika 6 (1959) 19–24.

Violeta Petkova Laboratoire Bordelais d’Analyse et G´eom´etrie UMR 5467 Universit´e Bordeaux 1 351, cours de la Lib´eration 33405 Talence France [email protected]