Multipliers on homogeneous Banach spaces with respect to Jacobi polynomials

Rend. Circ. Mat. Palermo (2013) 62:409–425DOI 10.1007/s12215-013-0133-7

Multipliers on homogeneous Banach spaces with respectto Jacobi polynomials

Sina Degenfeld-Schonburg

Received: 29 March 2012 / Accepted: 28 August 2013 / Published online: 10 September 2013© Springer-Verlag Italia 2013

Abstract A bounded linear operator is called multiplier with respect to Jacobi polynomialsif and only if it commutes with all Jacobi translation operators on [−1, 1]. Multipliers onhomogeneous Banach spaces on [−1, 1] determined by the Jacobi translation operator areintroduced and studied. First we prove four equivalent characterizations of a multiplier for anarbitrary homogeneous Banach spaces B on [−1, 1]. One of them implies the existence of analgebra isomorphism from the set of all multipliers on B into the set of all pseudomeasures.Further, we study multipliers on specific examples of homogeneous Banach spaces on [−1, 1].Amongst others, multipliers on the Wiener algebra, on the Beurling space and on Sobolevspaces are analyzed. We obtain that the multiplier spaces of the Wiener algebra, the Beurlingspace and of all Sobolev spaces are isometric isomorphic to each other. Furthermore, thesemultiplier spaces are all isometric isomorphic to the set of all pseudomeasures.

Keywords Hypergroup · Jacobi Polynomials · Homogeneous Banach Spaces ·Multipliers

Mathematics Subject Classification (2000) 43A62 · 33C45 · 43A22

1 Introduction

In the context of Fourier analysis homogeneous Banach spaces on the unit circle are ofgreat interest, see [14,17,23]. For instance, in homogeneous Banach spaces we can applyall the classical approximation procedures on functions on the unit circle and their Fourierexpansions.

Dales and Pandey [3] have studied the class Sp of Segal algebras and proved weaklyamenability. Using this results, Ghahramani and Lau [9,10] characterized for various classesof Segal algebras derivations and multipliers from a Segal algebra into itself and into its

S. Degenfeld-Schonburg (B)Centre of Mathematics, Munich University of Technology, 85748 Garching , Germanye-mail: [email protected]

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410 S. Degenfeld-Schonburg

dual module. Furthermore, multipliers of Segal algebras on locally compact groups are alsoinvestigated in [2,5,11,12,15,20–22]. Multipliers into general homogeneous Banach spaceson groups are investigated by Feichtinger [6].

Segal algebras are a specific type of homogeneous Banach spaces. They are often evendefined equivalent to homogeneous Banach spaces. Homogeneous Banach spaces determinedby the Jacobi translation operator are introduces by Fischer and Lasser [7]. They give a lotof examples of homogeneous Banach spaces and Banach algebras consisting of functionson S = [−1, 1]. These spaces are determined by the Jacobi translation operator, which isgenerated by the Jacobi polynomials R(α,β)n (x).

Our intention here is to characterize multipliers on homogeneous Banach spaces deter-mined by the Jacobi translation operator and to specify these characterizations on certainexamples of such homogeneous Banach spaces, e.g. the Wiener algebra, the Beurling spaceor the Sobolev space.

Before we do so, we give for the convenience of the reader a short introduction of the basictools we use and some facts of the Jacobi polynomials and the continuous Jacobi translation.

1.1 Homogeneous Banach spaces determined by the Jacobi translation operator

Using the Jacobi polynomials (R(α,β)n (x))n∈N, α, β > −1, we can define a polynomialhypergroup on N0. We denote the Haar measure on N0 by h. It is well known that

h(n) = (2n + α + β + 1)(α + β + 1)n(α + 1)n(α + β + 1)n!(β + 1)n

, n ∈ N0,

see [18]. Corresponding to this polynomial hypergroup, we can find a dual hypergroup onS := [−1, 1] as follows.

The Jacobi polynomials (R(α,β)n (x))n∈N, α, β > −1, are orthogonal with respect toπ(α,β),where suppπ(α,β) = [−1, 1] = S. We choose the normalization R(α,β)n (1) = 1.

If

(α, β) ∈ J :={(α, β) : α ≥ β > −1 and

(β ≥ −1

2or α + β ≥ 0

)},

then for any x, y ∈ [−1, 1] exists a probability Borel measure μ(α,β)x,y ∈ M(S) such that

R(α,β)n (x)R(α,β)n (y) =∫S

R(α,β)n (z)dμ(α,β)x,y (z)

for all n ∈ N0, see [7,8] or [18]. Here, M(S) denotes the set of all bounded complex measureson S.

For the sake of simplicity we fix the parameters (α, β) ∈ J and omit those from now onat all the notations of this paper.

Let L p(S, π), 1 ≤ p ≤ ∞, be the Banach space endowed with the norm

‖ f ‖p =⎛⎝∫

S| f (x)|pdπ(x)

⎞⎠

1/p

,

for p < ∞ and ‖ f ‖∞ = supx∈S | f (x)| for p = ∞. C(S) denotes the Banach space of allcontinuous function on S with norm ‖ f ‖∞ = supx∈S | f (x)|. Using the probability Borelmeasure μx,y ∈ M(S), we are in the position to define a Jacobi translation operator by

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Multipliers on homogeneous Banach 411

L y f (x) :=∫S

f (z)dμx,y(z)

for all f ∈ L1(S, π) and y ∈ S. This operator is not the usual translation operator. Since S isendowed with a hypergroup structure corresponding to μx,y , see [18], all the known resultsfor translation operators on hypergroups are valid in this context and we have in particularL y f ∈ L p(S, π) for all f ∈ L p(S, π) and

∥∥L y f∥∥

p ≤ ‖ f ‖p for 1 ≤ p < ∞. Furthermore,

L y f ∈ C(S) for all f ∈ C(S) and∥∥L y f

∥∥∞ ≤ ‖ f ‖∞.Moreover, we can define a Jacobi transform

f (n) :=∫S

f (x)Rn(x)dπ(x)

for f ∈ L1(S, π) and n ∈ N0, and an inverse Jacobi transform

d(x) :=∞∑

k=0

d(k)Rk(x)h(k)

for d ∈ l1(N0, h) and x ∈ S. By l p(N0, h), 1 ≤ p < ∞, we denote the spaces of allsequences d = (d(n))n∈N0 such that

∞∑k=0

|d(k)|ph(k) < ∞.

We endow those spaces with the usual norm ‖d‖p := (∑∞k=0 |d(k)|ph(k)

)1/p .Just like the Fourier transform, the Jacobi transform also admits the uniqueness theorem

stating that f = 0 implies f = 0 in L1(S, π), see [1].Moreover, the Jacobi transform is also an isometric and isomorphic mapping from

L2(S, π) onto l2(N0, h) and conversely from (l1(N0, h), ‖.‖2) into L2(S, π). Hence, wecan also extend the inverse Jacobi transform to a Plancherel transformation ℘ such that℘ : l2(N0, h) → L2(S, π), see Definition 2.2.21 in [1].

Using the Jacobi translation operator we define a commutative convolution on L1(S, π) by

f ∗ g :=∫S

f (x)L y g(x)dπ(x)

for f, g ∈ L1(S, π), y ∈ S. f ∗ g is again an element in L1(S, π) and we have ‖ f ∗ g‖1 ≤‖ f ‖1 ‖g‖1. Similarly L1(S, π) acts on L p(S, π), 1 ≤ p < ∞, see [7] or for details [1]. Iff ∈ C(S) ⊂ L1(S, π) the convolution of f with g ∈ L p(S, π), 1 ≤ p < ∞, is nothingelse than a L p(S, π)-valued integral,

f ∗ g =∫S

f (x)Lx gdπ(x),

since Lx g(y) = L y g(x) for all x, y ∈ S, see [7].A simple consequence is

(L y f ) = Rn(y) f (n)

for all f ∈ L1(S, π), y ∈ S.Next we introduce the concept of a homogeneous Banach space determined by the Jacobi

translation operator.

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Definition 1 We call a linear subspace B of L1(S, π) a homogeneous Banach space on S(with respect to (α, β)), if it is endowed with a norm ‖.‖B such that

(B1) Rn ∈ B for all n ∈ N0.(B2) B is complete with respect to ‖.‖B and ‖.‖1 ≤ ‖.‖B .(B3) For every f ∈ B, x ∈ S we have Lx f ∈ B and ‖Lx f ‖B ≤ ‖ f ‖B .(B4) For every f ∈ B the map x → Lx f , S → B is continuous.

A homogeneous Banach space is called character-invariant, if

(B5) For every f ∈ B, n ∈ N0 we have Rn · f ∈ B and ‖Rn · f ‖B ≤ ‖ f ‖B .

Every homogeneous Banach space B on S with respect to (α, β) is a L1(S, π)-module,since for each g ∈ B and f ∈ L1(S, π) we have f ∗ g ∈ B and ‖ f ∗ g‖B ≤ ‖ f ‖1 ‖g‖B .Furthermore, B is a Banach algebra with convolution as multiplication, see [7].

Some obvious examples for a homogeneous Banach space B on S with respect to (α, β)are B = L p(S, π), 1 ≤ p < ∞, with norm ‖.‖p and B = C(S) with norm ‖.‖∞.

1.2 Pseudomeasures

Before we are in the position to study multipliers on certain homogeneous Banach spaces onS,we need to introduce the concept of pseudomeasures. We already introduced pseudomeasuresin [4]. Pseudomeasures enable us to give a useful characterization of multipliers on all kindsof Banach spaces.

Denote by W (S) := {d : d ∈ l1(N0, h)}. By the uniqueness theorem of the inverseJacobi transformation is ‖d‖W := ‖d‖1 a norm on W (S), see [18] and Theorem 2.2.35 of[1]. With this norm W (S) is a Banach space. The space of all continuous linear functionalson W (S) is denoted by P(S), the elements σ of P(S) are called pseudomeasures on Sand ‖σ‖P = sup{ |σ(d)| : ‖d‖W ≤ 1} is the norm on the dual space P(S). P(S)is isometrically isomorphic to l∞(N0), the space of all bounded sequences (ϕ(n))n∈N0 onN0. (We recall that the dual space of l1(N0, h) is isometrically isomorphic to l∞(N), see[13, Theorem 12.18].) The mapping � : l∞(N0) → P(S) where for each ϕ ∈ l∞(N0) theelement �(ϕ) = σ ∈ P(S) is uniquely determined by

∞∑k=0

d(k)ϕ(k)h(k) = �(ϕ)(d) = σ(d) for d ∈ l1(N0, h),

defines an isometric isomorphism� from the Banach space l∞(N0)onto P(S). A convolutionof σ1, σ2 ∈ P(S) is determined by

σ1 ∗ σ2 = �(�−1(σ1)�

−1(σ2)),

so that � is also an algebra isomorphism from l∞(N0) onto P(S). These facts are proven in[4] and the proof follows the lines of [16, Theorem 4.2.2]. We shall call �−1(σ ) the Jacobitransform of σ ∈ P(S). If μ ∈ M(S) then

μ(d) =∫S

d(x)dμ(x) =∫S

∞∑k=0

d(k)Rk(x)h(k)dμ(x) =∞∑

k=0

d(k)μ(k)h(k)

for all d ∈ l1(N0, h). Hence each measure μ ∈ M(S) is a pseudomeasure and μ =�−1(μ). Moreover, we have ‖μ‖P = ‖μ‖∞ ≤ ‖μ‖. Furthermore,

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∞∑k=0

d(k)�−1(μ1 ∗ μ2)(k)h(k) =∫S

d(x)dμ1 ∗ μ2(x) for all d ∈ l1(N0, h).

Hence the convolution μ1 ∗μ2 of two measures μ1, μ2 ∈ M(S) agrees with the convolutionof μ1 and μ2 seen as pseudomeasures. Obviously, we have

�−1(σ1 ∗ σ2) = �−1(σ1) �−1(σ2) for σ1, σ2 ∈ P(S).

The above conclusions are summarized as follows.

Proposition 1 The Jacobi transform �−1 : P(S) → l∞(N0) determined by∞∑

k=0

d(k) �−1(σ )(k) h(k) = σ(d) for all d ∈ l1(N0, h)

is an isometric algebra isomorphism of P(S) onto l∞(N0).

We shall say that a pseudomeasure σ ∈ P(S) belongs to L2(S, π) if there is a g ∈L2(S, π) such that

σ(d) =∫S

d(x) g(x) dπ(x) for all d ∈ l1(N0, h).

Since {d : d ∈ l1(N0, h)} is dense in L2(S, π), g is uniquely determined. If σ ∈ P(S)belongs to L2(S, π) and g is the corresponding element from L2(S, π) then Parseval’sformula yields for d ∈ l1(N0, h)

∞∑k=0

d(k)�−1(σ )(k)h(k) = σ(d) =∫S

d(x)g(x)dπ(x) =∞∑

k=0

d(k)g(k)h(k)

Hence we conclude�−1(σ ) = g ∈ l2(N0, h), that is the Jacobi transform of the pseudomea-sure σ agrees with the Jacobi transform of g.

Conversely, let σ ∈ P(S) such that �−1(σ ) ∈ l2(N0, h) then σ belongs to L2(S, π).Indeed, putting g = ℘(�−1(σ )) ∈ L2(S, π) we obtain, using Parseval’s formula

σ(d) =∞∑

k=0

d(k)�−1(σ )(k)h(k) =∫S

d(x) g(x) dπ(x) for all d ∈ l1(N0, h).

We summarize the latter discussion in the following proposition, see also [4].

Proposition 2 A pseudomeasure σ ∈ P(S) belongs to L2(S, π) if and only if �−1(σ ) ∈l2(N0, h). Moreover, the Jacobi transform of σ as a pseudomeasure coincides with the Jacobitransform of the corresponding g ∈ L2(S, π).

It should be noted that every g ∈ L2(S, π) determines a pseudomeasure σ ∈ P(S) suchthat σ(d) = ∫

S d(x) g(x) dπ(x) is true for all d ∈ l1(N0, h). Indeed, put σ = �(g). Then

�(g) =∞∑

k=0

d(k)�−1(�(g))(k)h(k) =∞∑

k=0

d(k)g(k)h(k) =∫S

d(x)g(x)dπ(x)

In particular, the convolution σ ∗ g = �(�−1(σ )g) of σ ∈ P(S) and g ∈ L2(S, π) iswell-defined as a convolution of pseudomeasures.

If σ ∈ P(S) belongs to L2(S, π) we will further on designate the corresponding elementof L2(S, π) also by σ.

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2 Multipliers on homogeneous Banach spaces

Let B be a homogeneous Banach space on S with respect to (α, β).

Definition 2 We call a bounded linear operator T on B multiplier, if and only if T commuteswith the Jacobi translation operator L y for all y ∈ S, i.e. T ◦ L y = L y ◦ T . We denote byM(B) the set of all multipliers on B.

Before we take a look on multipliers on some specific homogeneous Banach spaces onS, we will first characterize multipliers on a general homogeneous Banach space B on S.Later on, we will investigate some examples for homogeneous Banach spaces on S and theircorresponding multiplier spaces.

Theorem 1 A bounded linear operator T on B is a multiplier for B, T ∈ M(B), if andonly if

T ( f ∗ g) = f ∗ T g

for all f ∈ L1(S, π) and g ∈ B. Moreover, we have T f ∗ g = T ( f ∗ g) = f ∗ T g for allf, g ∈ B.

Proof Let T ∈ M(B). That means T ◦ Lx = Lx ◦T for all x ∈ S. Since, B is a homogeneousBanach space, we have Lx f ∈ B and ‖Lx f ‖B ≤ ‖ f ‖B for all x ∈ S, f ∈ B. Moreover, themapping x → Lx f , S → B is continuous. For f ∈ C(S) and g ∈ B the convolution f ∗ gis a B-valued integral, such that

f ∗ T g =∫S

f (x)Lx (T g)dπ(x) =∫S

f (x)T (Lx g)dπ(x).

The linearity and the continuity of T implies

f ∗ T g = T

⎛⎝∫

Sf (x)(Lx g)dπ(x)

⎞⎠ = T ( f ∗ g)

for all f ∈ C(S) and g ∈ B.For f ∈ L1(S, π) we can choose a sequence ( fn)n∈N in C(S) such that ‖ fn − f ‖1 → 0

as n tends to infinity. Hence,

‖ f ∗ T g − T ( f ∗ g)‖B

≤ ‖ f ∗ T g − fn ∗ T g‖B + ‖ fn ∗ T g − T ( fn ∗ g)‖B + ‖T ( fn ∗ g)− T ( f ∗ g)‖B

≤ ‖ f − fn‖1 ‖T g‖B + ‖ fn ∗ T g − T ( fn ∗ g)‖B + ‖T ‖ ‖ f − fn‖1 ‖g‖B → 0

as n tends to infinity. Thus, we have f ∗ T g = T ( f ∗ g) for all f ∈ L1(S, π) and g ∈ B.Choosing f ∈ B ⊂ L1(S, π) and interchanging the roles of f and g proves the secondstatement.

Conversely let T be a bounded linear operator T on B such that T ( f ∗ g) = f ∗ T g forall f ∈ L1(S, π) and g ∈ B. Fix x ∈ S. For f ∈ L1(S, π) and g ∈ B follows

f ∗ (Lx ◦ T )g = Lx f ∗ T g = T (Lx f ∗ g) = T ( f ∗ Lx g) = f ∗ (T ◦ Lx )g.

Since L1(S, π) has an approximate identity, we obtain (T ◦ Lx )g = (Lx ◦ T )g in L1(S, π)and further T ◦ Lx = Lx ◦ T as operators on B. �

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The next Theorem shows that each multiplier T on B is uniquely determined by its valueson Rn for n ∈ N0.

Theorem 2 A bounded linear operator T on B is a multiplier for B, i.e. T ∈ M(B), if andonly if there exists a unique function ϕ ∈ l∞(N0) such that

(T f ) = ϕ f

for all f ∈ B. Moreover, we have ϕ(n) = (T Rn )(n)h(n) for all n ∈ N0 and ‖ϕ‖∞ ≤ ‖T ‖.

Proof By (B1) in Definition 1 is Rn ∈ B for all n ∈ N0 and we have Rn(m) = δm,nh(n)−1,where δm,n denotes the Kronecker symbol. Thus, for every n ∈ N0 exists Rn ∈ B withRn(n) = h(n)−1 �= 0. Further, for n ∈ N0 and f, g ∈ B such that f (n) �= 0 and g(n) �= 0we have by Theorem 1

(T f )(n)/ f (n) = (T g)(n)/g(n).

This equation shows that the definition

ϕ(n) := (T f )(n)/ f (n) = (T Rn )(n)h(n)

is independent on the choice of f ∈ B. Hence, ϕ(n) is well-defined on N0. If f (n) �= 0 andg(n) = 0 then (T g)(n) f (n) = (T f )(n)g(n) = 0. Hence the equation

(T g)(n) = ϕ(n)g(n)

is valid for all g ∈ B and all n ∈ N0.If ψ ∈ l∞(N0) is a second function with (T g)(n) = ψ(n)g(n) for all n ∈ N0 and g ∈ B,

then we obtain (ψ(n)− ϕ(n))g(n) = 0 for all n ∈ N0 and all g ∈ B. This implies ϕ = ψ .To prove that ϕ is bounded, we define

Kn := supf ∈B

{∣∣∣ f (n)∣∣∣ : ‖ f ‖B = 1

}.

By∥∥∥ f

∥∥∥∞ ≤ ‖ f ‖1 ≤ ‖ f ‖B = 1 for all f ∈ B ⊂ L1(S, π), we have 0 < Kn ≤ 1. Further

holds∣∣g(n)∣∣ ≤ Kn ‖g‖B and we obtain

∣∣ϕ(n)g(n)∣∣ = ∣∣(T g)(n)∣∣ ≤ Kn ‖T g‖B ≤ Kn ‖T ‖ ‖g‖B

for all g ∈ B. By choosing only those g ∈ B with ‖g‖B = 1 and g(n) �= 0 we have

|ϕ(n)| ≤ Kn ‖T ‖ inf

{1∣∣g(n)∣∣ : ‖g‖B = 1 and g(n) �= 0

}= ‖T ‖.

The last equation is valid, since those g ∈ B with ‖g‖B = 1 and g(n) = 0 do not contributeto the value of Kn . Hence, ϕ is bounded by ‖ϕ‖∞ ≤ ‖T ‖.

Conversely, let T be a bounded linear operator on B such that T is for a given ϕ ∈ l∞(N0)

defined by (T f ) = ϕ f for all f ∈ B. Then

(T ◦ Lx f )(n) = ϕ(n)(Lx f )(n) = ϕ(n)Rn(x) f (n) = Rn(x)(T f )(n) = (Lx ◦ T f )(n)

for all n ∈ N0 and f ∈ B. By the uniqueness theorem of the Jacobi transform followsT ◦ Lx = Lx ◦ T . �

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The next Theorem depends on whether the homogeneous Banach space B is a subset inL2(S, π) or not. If B ⊂ L2(S, π), we have f ∈ l2(N0, h) for all f ∈ B and we can showthe existence of a pseudomeasure σ ∈ P(S), such that T f = σ ∗ f and σ ∗ f belongs toL2(S, π) for all f ∈ B.

Theorem 3 Let B ⊂ L2(S, π). For a bounded linear operator T on B the following condi-tions are equivalent:

(i) T ∈ M(B), i.e. T ◦ Lx = Lx ◦ T for all x ∈ S.(ii) There exists a unique pseudomeasure σ ∈ P(S), such that T f = σ ∗ f for all f ∈ B.

Moreover, there exists an continuous algebra isomorphism form M(B) into P(S) such that‖σ‖P ≤ ‖T ‖.

Proof Let T ∈ M(B). By Theorem 2 exists a unique ϕ ∈ l∞(N0) such that T f = ℘(ϕ f ) forall f ∈ B. Furthermore, by Proposition 1 exists an isometric isomorphism �−1 : P(S) →l∞(N0). Set σ := �(ϕ). Moreover, for f ∈ B ⊂ L2(S, π) exists σ ∗ f as a pseudomeasureand we obtain by definition

�−1(σ ∗ f ) = �−1(σ ) f = ϕ f ∈ l2(N0, h).

Hence, �(ϕ f ) = σ ∗ f belongs to L2(S, π) and

σ ∗ f = �(ϕ f ) = ℘(ϕ f ) = T f.

Since � is isometric, we have by Theorem 2 ‖σ‖P = ‖ϕ‖∞ ≤ ‖T ‖. Conversely is everybounded linear operator T on B, which is defined by T f = σ ∗ f for σ ∈ P(S), a multiplierfor B. Indeed, T f = σ ∗ f = �(�−1(σ ) f ) = ℘(ϕ f ) for ϕ = �−1(σ ) ∈ l∞(N0). The restfollows by Theorem 2. �

If B �⊂ L2(S, π) we have to make a compromise. We are only able to prove a muchweaker result for T ∈ M(B).

Theorem 4 Let B �⊂ L2(S, π). If a bounded linear operator T on B is a multiplier for B,i.e. T ◦ Lx = Lx ◦ T for all x ∈ S, then there exists a unique pseudomeasure σ ∈ P(S),such that T f = σ ∗ f for all f ∈ B ∩ L2(S, π).

Moreover, there exists an continuous algebra isomorphism on M(B) into P(S) such that‖σ‖P ≤ ‖T ‖.

Proof The proof follows the lines of the first part of proof of Theorem 3. �

3 Multipliers on the Wiener algebra W(S)

An interesting example of homogeneous Banach spaces determined by the Jacobi translationoperator is the Wiener algebra. Denote by

W (S) :={

f ∈ C(S) : f ∈ l1(N0, h)}.

We call W (S)with norm ‖ f ‖W :=∥∥∥ f

∥∥∥1

the Wiener algebra (We introduced W (S) already

in section 1.2).(W (S), ‖ ‖W ) is a homogeneous Banach space on S, see [7]. Furthermore, W (S) is a

Banach algebra with respect to the convolution and with respect to the point wise multipli-cation of functions.

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Theorem 5 For a bounded linear operator T on W (S), the following conditions are equiv-alent:

(i) T ∈ M(W (S)), i.e. T ◦ Lx = Lx ◦ T for all x ∈ S.(ii) For all f, g ∈ W (S) we have T f ∗ g = T ( f ∗ g) = f ∗ T g.

(iii) There exists a unique bounded function ϕ on N0, such that(T f ) = ϕ f for all f ∈ W (S).

Moreover, ‖ϕ‖∞ = ‖T ‖.

We will present an alternative proof and show the implication i) to i i) again, even thoughwe have already proven this equivalency in Theorem 1.

Proof Let T ∈ M(W (S)). Since W (S) is a homogeneous Banach space, we have Lx f ∈W (S) for all x ∈ S and f ∈ W (S).

By

‖ f ‖W =∥∥∥ f

∥∥∥1

≤ ‖ f ‖1 +∥∥∥ f

∥∥∥1

≤ ‖ f ‖∞ +∥∥∥ f

∥∥∥1

≤ 2∥∥∥ f

∥∥∥1

= 2 ‖ f ‖W

the two norms ‖.‖W and ‖.‖1 := ‖.‖1 + ∥∥.∥∥1 are equivalent on W (S). Thus each continuouslinear functional F on W (S) with respect to ‖ ‖W is also continuous with respect to ‖.‖1.

Further, it is evident that the mapping � : W (S) → L1(S, π) × l1(N0, h) defined by�( f ) := ( f, f ) for each f ∈ W (S) is a linear isometry of (W (S), ‖.‖1) into the Banachspace L1(S, π)× l1(N0, h) equipped with the sum of the corresponding norms as product-norm. Thus we may consider W (S) as a closed linear subspace of L1(S, π) × l1(N0, h).Since the dual space of L1(S, π)× l1(N0, h) is isomorphic to L∞(S, π)× l∞(N0), by thetheorem of Hahn-Banach we can consider every continuous linear functional F on W (S)with respect to ‖.‖1 to be of the following form:

F( f ) =∫S

f (x)a(x)dπ(x)+∞∑

k=0

f (k)b(k)h(k),

for (a, b) ∈ L∞(S, π)× l∞(N0). (Of course the pair (a, b) corresponding to a given func-tional may not be unique.)Now let F be such a continuous functional on W (S) with respect to ‖.‖1. Then F ◦ T is alsoa continuous linear functional on W (S) with respect to ‖.‖1. Hence, there exist (a, b) and(α, β) in L∞(S, π)× l∞(N0) such that for each f ∈ W (S) we have

F( f ) =∫S

f (x)a(x)dπ(x)+∞∑

k=0

f (k)b(k)h(k)

F ◦ T ( f ) =∫S

f (x)α(x)dπ(x)+∞∑

k=0

f (k)β(k)h(k).

Consequently, for f, g ∈ W (S) is

F(T f ∗ g) =∫S(T f ∗ g)(x)a(x)dπ(x)+

∞∑k=0

(T f ∗ g)(k)b(k)h(k)

=∫S

∫S

L y T f (x)g(y)dπ(y)a(x)dπ(x)+∞∑

k=0

(T f )∫S

g(y)Rk(y)dπ(y)b(k)h(k)

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=∫S

g(y)∫S

L y T f (x)a(x)dπ(x)dπ(y)+∫S

g(y)∞∑

k=0

(T f )Rk(y)b(k)h(k)dπ(y)

=∫S

g(y)

⎡⎣∫

ST L y f (x)a(x)dπ(x)+

∞∑k=0

(T L y f )b(k)h(k)

⎤⎦ dπ(y)

=∫S

g(y)F ◦ T (L y f )dπ(y)

=∫S

g(y)

⎡⎣∫

SL y f (x)α(x)dπ(x)+

∞∑k=0

(L y f )β(k)h(k)

⎤⎦ dπ(y)

=∫S

∫S

L y f (x)g(y)dπ(y)α(x)dπ(x)+∞∑

k=0

f∫S

g(y)Rk(y)dπ(y)β(k)h(k)

=∫S( f ∗ g)(x)α(x)dπ(x)+

∞∑k=0

( f ∗ g)(k)β(k)h(k)

= F ◦ T ( f ∗ g) = F(T ( f ∗ g))

Since this holds for every continuous linear functional F on W (S), we obtainT f ∗ g = T ( f ∗ g). Changing the roles of f and g proves i i). Moreover, we know byTheorem 2 the equivalency of i) and i i i). Hence there exists a unique ϕ ∈ l∞(N0) such that(T f ) = ϕ f for all f ∈ W (S) and ‖T ‖ ≥ ‖ϕ‖∞. We obtain further

‖T f ‖W = ∥∥(T f )∥∥

1 =∥∥∥ϕ f

∥∥∥1

≤ ‖ϕ‖∞ ‖ f ‖W .

Hence ‖T ‖ = ‖ϕ‖∞. � Theorem 6 For T ∈ B(W (S)) the following conditions are equivalent:

(i) T ∈ M(W (S)), i.e. T ◦ Lx = Lx ◦ T for all x ∈ S.(ii) There exists a unique pseudomeasureσ ∈ P(S), such that T f = σ ∗ f for all f ∈ W (S).

Moreover, there exists an isometric algebra isomorphism form M(W (S)) onto P(S).

Proof The assertion follows by Theorem 3, since W (S) ⊂ L2(S, π). Further is the mapping� : l∞(N0) → P(S) isometric and we have by Theorem 5

‖σ‖P = ‖ϕ‖∞ = ‖T ‖.Moreover, each ϕ ∈ l∞(N0) defines a multiplier for W (S) by T f := (ϕ f ). Hence

M(W (S)) � l∞(N0) � P(S),where � denotes an isometric isomorphism. �

4 Multipliers on A p(S)

Motivated by the Wiener algebra, W (S), we introduce further homogeneous Banach spaces

Ap(S) :={

f ∈ L1(S, π) : f ∈ l p(N0, h)},

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1 ≤ p < ∞, with norm ‖ f ‖p := ‖ f ‖1 +∥∥∥ f

∥∥∥p. Notice that (A1(S), ‖.‖1) is the Wiener

algebra W (S) with a different but equivalent norm. The equivalence of the two norms ‖.‖1

and ‖.‖W was shown in the proof of Theorem 5.Larsen introduced in [16], chapter 6, these spaces on groups. He presents various char-

acterizations for multipliers for functions with Fourier transform in L p(G). Many of theseresults are transferable to commutative hypergroups. Here we focus on Ap(S).

We want to remark, that the following relations between these spaces holds obviously for1 ≤ p ≤ q ≤ 2 ≤ r ≤ s < ∞,

W (S) ⊂ Ap(S) ⊂ Aq(S) ⊂ A2(S) = L2(S, π) ⊂ Ar (S) ⊂ As(S) ⊂ L1(S, π)Before we can characterize the multipliers on Ap(S), we need to check, if these spaces areindeed homogeneous Banach spaces on S. The proof equals partial the proof of Proposition3.6 in [7].

Proposition 3 Ap(S) :={

f ∈ L1(S, π) : f ∈ l p(N0, h)}

, 1 ≤ p < ∞, is with norm

‖ f ‖p := ‖ f ‖1 +∥∥∥ f

∥∥∥p

a character-invariant homogeneous Banach space on S.

Proof Ap(S) is obviously a linear subspace of L1(S, π) and ‖·‖p is a norm by the uniquenesstheorem of the Jacobi transform. Since Rm(n) = h(n)−1δm,n ∈ l p(N0, h), where δm,n

denotes the Kronecker symbol, we have Rn ∈ Ap(S) for all n ∈ N0. Hence (B1) holds.

For f ∈ Ap(S)we have ‖ f ‖1 ≤ ‖ f ‖1+∥∥∥ f

∥∥∥p. Further, Ap(S) is complete with respect to

‖·‖p , because each Cauchy sequence ( fn)n∈N in Ap(S) is a Cauchy sequence in L1(S, π) and( fn)n∈N is a Cauchy sequence in l p(N0, h). Since L1(S, π) and l p(N0, h) are complete withrespect to ‖.‖1 and ‖.‖p respectively, there exists f ∈ L1(S, π)with limn→∞ ‖ fn − f ‖1 = 0

and d ∈ l p(N0, h) such that limn→∞∥∥∥ fn − d

∥∥∥p. Furthermore, f = d , since

∥∥∥ f − d∥∥∥∞ ≤

∥∥∥ f − fn

∥∥∥∞ +∥∥∥ fn − d

∥∥∥p

≤ ‖ f − fn‖1 +∥∥∥ fn − d

∥∥∥p

→ 0

as n tends to infinity. Thus, we have proven (B2).To show (B3) notice that supx∈S |Rn(x)| = 1 for all n ∈ N0. Hence, for each f ∈ Ap(S)

and x ∈ S we have Lx f ∈ L1(S, π) and (Lx f )(n) = Rn(x) f (n) is an element in l p(N0, h).Further follows

‖Lx f ‖p = ‖Lx f ‖1+( ∞∑

k=0

∣∣∣Rk(x) f (k)∣∣∣p

h(k)

)1/p

≤‖ f ‖1 + supk∈N0

|Rk(x)|∥∥∥ f

∥∥∥p≤‖ f ‖p.

Now we want to show the continuity of x → Lx f , S → Ap(S) for all f ∈ Ap(S). Fix

f ∈ Ap(S) and let x0 ∈ S, ε > 0. There exists N ∈ N and g ∈ Ap(S) such that∥∥∥g − f

∥∥∥p<

ε4 , g(n) = f (n) for all n ≤ N and g(n) = 0 for all n > N . Indeed, since f ∈ l p(N0, h) there

exists a N ∈ N0 such that∑∞

k=N+1 | f (k)|ph(k) < ε4 . We choose g = ∑N

k=0 f (k)Rkh(k).Furthermore, since g(n) = 0 for all n > N exists δ > 0 with

∥∥(Lx g)− (Lx0 g)∥∥

p <ε2 for

all x ∈ S such that |x − x0| < δ. Thus we obtain∥∥(Lx f )− (Lx0 f )

∥∥p < ε for all x ∈ S

with |x − x0| < δ and finally∥∥Lx f − Lx0 f

∥∥p< 2ε for all x ∈ S with |x − x0| < δ′, since

L1(S) is a homogeneous Banach space. Thus, the map x → Lx f , S → B is continuous forall f ∈ Ap(S) and (B4) is proven.

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We obtain Rn · f ∈ Ap(S), since Rn = 1h(n) δn and so

(Rn · f ) = Rn ∗ f ∈ l1(N0, h) ∗ l p(N0, h) ∈ l p(N0, h).

Moreover,

‖Rn · f ‖p = ‖Rn · f ‖1 + ∥∥(Rn · f )∥∥

p

≤ ‖Rn‖∞ ‖ f ‖1 +∥∥∥Rn ∗ f

∥∥∥p

≤ ‖ f ‖1 +∥∥∥Rn

∥∥∥1

∥∥∥ f∥∥∥

p= ‖ f ‖1 +

∥∥∥∥ 1

h(n)δn

∥∥∥∥1

∥∥∥ f∥∥∥

p= ‖ f ‖p.

for all n ∈ N0. Hence, Ap(S) is a character-invariant homogeneous Banach space on S. �

Remark 1 A2(S) is with norm ‖ f ‖ :=∥∥∥ f

∥∥∥2

also a character-invariant homogeneous Banachspace.

Now we continue characterizing multipliers on Ap(S).

Theorem 7 For a bounded linear operator T on Ap(S), 1 ≤ p < ∞, the following condi-tions are equivalent:

(i) T ∈ M(Ap(S)), i.e. T ◦ Lx = Lx ◦ T for all x ∈ S.(ii) For all f, g ∈ Ap(S) we have T f ∗ g = T ( f ∗ g) = f ∗ T g.

(iii) There exists a unique bounded function ϕ on N0, such that(T f ) = ϕ f for all f ∈ Ap(S).

Moreover, ‖T ‖ ≥ ‖ϕ‖∞. For 1 ≤ p ≤ 2 is further ‖T ‖ ≤ 2 ‖ϕ‖∞.

Proof The proof follows directly by Theorem 1 and Theorem 2. (One could also follow thelines of the proof of Theorem 5.)

For the last statement we have for 1 ≤ p ≤ 2

‖T f ‖p ≤ ‖T f ‖2 + ‖ϕ‖∞∥∥∥ f

∥∥∥p

= ∥∥(T f )∥∥

2 + ‖ϕ‖∞∥∥∥ f

∥∥∥p

≤ ‖ϕ‖∞∥∥∥ f

∥∥∥2+ ‖ϕ‖∞

∥∥∥ f∥∥∥

p≤ 2 ‖ϕ‖∞

∥∥∥ f∥∥∥

p≤ 2 ‖ϕ‖∞ ‖ f ‖p.

� The next characterization of multipliers in Ap(S) depends on whether p ≤ 2 or p > 2. Thisis do to the fact that for 1 ≤ p ≤ 2, we have Ap(S) ⊂ L2(S, π).

Conversely is L2(S, π) � Ap(S)whenever 2 < p. Indeed, for p > 2 and 1/p +1/q = 1suppose L2(S, π) = A2(S, π) = Ap(S, π).

Theorem 8 For a bounded linear operator T on Ap(S) and 1 ≤ p ≤ 2, the followingconditions are equivalent:

(i) T ∈ M(Ap(S)), i.e. T ◦ Lx = Lx ◦ T for all x ∈ S.(ii) There exists a unique pseudomeasure σ ∈ P(S), such that T f = σ ∗ f for all f ∈

Ap(S).Moreover, there exists an continuous algebra isomorphism from M(Ap(S)) onto P(S) suchthat ‖σ‖P ≤ ‖T ‖ ≤ 2 ‖σ‖P .

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Proof The proof follows by Theorem 3 and Theorem 7,since P(S) and l∞(N0) are isometricisomorphic. Moreover, each ϕ ∈ l∞(N0) defines by T f := ℘(ϕ f ) for all f ∈ Ap(S) a mul-tiplier for Ap(S). Hence, the algebra isomorphism from M(Ap(S)) into P(S) is surjective.

�

Proposition 4 Let 1 ≤ p ≤ 2. Then there exists an norm-increasing algebra isomorphismfrom M(W (S)) into M(Ap(S)).

Proof We have W (S) ⊂ Ap(S). Hence M(Ap(S)) ⊂ M(W (S)), since for T ∈ M(Ap(S))and f ∈ W (S) we obtain (T f ) = ϕ f ∈ l1(N0, h), for ϕ ∈ l∞(N0) as in Theorem 7, andT f = (ϕ f ) ∈ C(S). Hence T f ∈ W (S). Furthermore,

‖T f ‖W = ∥∥(T f )∥∥

1 ≤ ‖ϕ‖∞∥∥∥ f

∥∥∥1

= ‖ϕ‖∞ ‖ f ‖W .

Thus, T is a bounded linear operator on W (S), which commutes with translation, i.e. T ∈M(W (S)). Further, by Theorem 6 and Theorem 8 exists an isomorphism between M(W (S))and P(S) and between P(S) and M(Ap(S)). Hence M(W (S)) and M(Ap(S)) are algebraicisomorphic. Moreover, we have

‖T ‖W = ‖ϕ‖∞ ≤ ‖T ‖p,

where ‖T ‖W denotes the operator norm of T defined on W (S) and ‖T ‖p denotes the operatornorm of T defined on Ap(S). Hence, the algebra isomorphism is norm-increasing. �

Remark 2 We want to point out, that by Proposition 4 the multiplier spaces of W (S) andAp(S) coincide, despite that fact that W (S) is a proper linear subset in Ap(S) for p > 1and the norms of these spaces are not equivalent. This leads to the observation that theset of multipliers for a homogeneous Banach space contribute little information about thehomogeneous Banach space itself.

Moreover, there exists also an norm-increasing algebra isomorphism from M(Ap(S)) intoM(Aq(S)) for 1 ≤ p < q ≤ 2.

Remark 3 In contrast to 1 ≤ p < q ≤ 2 where M(Ap(S)) = M(Aq(S)), for 2 ≤ p < q <∞ we can only show that

M(Aq(S)) ⊂ M(Ap(S)).

Since for T ∈ M(Aq(S)) and f ∈ Ap(S) ⊆ Aq(S) we have (T f ) = ϕ f ∈ l p(N0) asf ∈ Ap(S) and ϕ is bounded. Further is

‖T f ‖p ≤ ‖T f ‖1 + ‖ϕ‖∞∥∥∥ f

∥∥∥p

≤ ‖T f ‖q + ‖ϕ‖∞∥∥∥ f

∥∥∥p

≤ ‖T ‖q ‖ f ‖q + ‖ϕ‖∞∥∥∥ f

∥∥∥p

≤ (‖T ‖q + ‖ϕ‖∞) ‖ f ‖p,

where ‖T ‖q denotes the operator norm of T defined on Aq(S). Hence T is a bounded linearoperator on Ap(S) and T commutes with all Jacobi translation operators.

Remark 4 Using Theorem 4 in [4, pp. 17], we have for all 1 ≤ p < ∞l1(N0, h) = M(Ds) � M(L1(S, π)) ⊂ M(Ap(S)) ⊂ M(W (S)) � P(S).

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5 Multipliers on the Beurling space W∗(S)

The space

W∗(S) :={

f ∈ W (S) :∞∑

k=0

supl≥k

| f (l)|h(k) < ∞}

is called Beurling space. The Beurling space W∗(S) is with norm

‖ f ‖W∗ :=∞∑

k=0

supl≥k

| f (l)|h(k)

a homogeneous Banach space, see [7]. Furthermore, W∗(S) is a Banach algebra with respectto the convolution and with respect to the point wise multiplication of functions.

Fischer and Lasser proved in [7] that W (S)\W∗(S) �= ∅ by giving an example of a functionin W (S)\W∗(S). Hence it makes indeed sense to examine the multipliers for W∗(S), eventhough we already know the multipliers for W (S). Despite the fact that the two homogeneousBanach spaces are different and their norms are not equivalent, we will prove that theirmultiplier spaces coincide.

It is easy to see, that the multiplier space M(W (S)) is included in M(W∗(S)). Indeed,choose a multiplier T ∈ M(W (S)) with the corresponding function ϕ ∈ l∞(N0) such that(T g) = ϕg for all g ∈ W (S) and let f ∈ W∗(S) ⊂ W (S). We obtain

‖T f ‖W∗ =∞∑

k=0

supl≥k

|(T f )(l)|h(k) =∞∑

k=0

supl≥k

|ϕ(l)|| f (l)|h(k) ≤ ‖ϕ‖∞ ‖ f ‖W∗ .

Thus, T is a bounded linear operator on W∗(S), which commutes with Jacobi translationoperators. Hence T ∈ M(W∗(S)). In particular each ϕ ∈ l∞(N0) defines a multiplier forW∗(S), see Theorem 6

There is even more we can say about the multipliers on W∗(S):

Theorem 9 For a bounded linear operator T on W∗(S) the following conditions are equiv-alent:

(i) T ∈ M(W∗(S))(ii) For all f, g ∈ W∗(S) we have T f ∗ g = T ( f ∗ g) = f ∗ T g.

(iii) There exists an unique function ϕ ∈ l∞(N0) such that (T f ) = ϕ f for all f ∈ W∗(S).Moreover is ‖T ‖ = ‖ϕ‖∞.

Proof The equivalencies of i), i i) and i i i) follow by Theorem 1 and 2.Furthermore, we have ‖T ‖ ≥ ‖ϕ‖∞ by Theorem 2. By

‖T f ‖W∗ =∞∑

k=0

supl≥k

|(T f )(l)|h(k) =∞∑

k=0

supl≥k

|ϕ(l)|| f (l)|h(k) ≤ ‖ϕ‖∞ ‖ f ‖W∗ .

we obtain ‖ϕ‖∞ = ‖T ‖. � As a consequence of Theorem 9 we obtain

Theorem 10 For a bounded linear operator T on W∗(S) the following conditions are equiv-alent:

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(i) T ∈ M(W∗(S))(ii) There exists a unique pseudomeasure σ ∈ P(S) such that T f = σ ∗ f , for all f ∈

W∗(S).Moreover, there exists an isometric algebra isomorphism from M(W∗(S)) onto P(S).Proof The proof follows by Theorem 3 and by Theorem 9.

Moreover, the isometric algebra isomorphism from M(W∗(S)) into P(S) is surjective,since each ϕ ∈ l∞(N0) defines a multiplier for W∗(S) by T f := (ϕ f ). Hence, we have anisometric algebra isomorphism from M(W∗(S)) onto l∞(N0) � P(S). � Corollary 1 There exists an isometric algebra isomorphism between M(W (S)) andM(W∗(S)).

6 Multipliers on the Sobolev space H(1)2 (S)

We denote by sn := R′n(1) for n ∈ N.

Define a subspace

D(B) :={

f ∈ L2(S, π) :∞∑

k=0

s2k | f (k)|2h(k) < ∞

}

and an operator acting on D(B)

B : D(B) → L2(S, π), B f :=∞∑

k=0

sk f (k)Rkh(k)

for all f ∈ D(B). Furthermore, we put

H (1)2 (S) :=

{f ∈ L2(S, π) : lim

x→1−f − Lx f

1 − xexists in L2(S, π)

}

and call

D : H (1)2 (S) → L2(S, π), D f := lim

x→1−f − Lx f

1 − x

the Jacobi differential operator with respect to (α, β) ∈ J .Fischer and Lasser showed in [7] that the Jacobi differential operator D fullfils

(i) H (1)2 (S) = D(B)

(ii) D f = limx→1− f −Lx f1−x = ∑∞

k=0 sk f (k)Rkh(k) = B f for all f ∈ H (1)2 (S)

We call the space H (1)2 (S) Sobolev space induced by D and choose

‖ f ‖2,1 := ‖ f ‖2 + ‖D f ‖2

as norm on H (1)2 (S). With this norm H (1)

2 (S) becomes a homogeneous Banach space on Swith respect to (α, β) ∈ J , see [7].

Sobolev spaces are very important in the theory of partial differential equations. Sobolevspaces defined on the torus T are investigated in [19].

Theorem 11 For a bounded linear operator T on H (1)2 (S) the following conditions are

equivalent:

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(i) T is a multiplier for H (1)2 (S), i.e. T ∈ M(H (1)

2 (S)).(ii) For all f, g ∈ H (1)

2 (S) we have T f ∗ g = T ( f ∗ g) = f ∗ T g.

(iii) There exists a unique function ϕ ∈ l∞(N0) such that (T f ) = ϕ f for all f ∈ H (1)2 (S).

Moreover, ‖ϕ‖∞ = ‖T ‖.

Proof The equivalencies i), i i) and i i i) follow by Theorem 1 and 2. Theorem 2yields‖ϕ‖∞ ≤ ‖T ‖. Further, we obtain ‖T ‖ ≤ ‖ϕ‖∞ by

‖T f ‖2,1 = ‖T f ‖2 + ‖D(T f )‖2 = ∥∥(T f )∥∥

2 +∥∥∥∥∥

∞∑k=0

sk(T f )(k)Rkh(k)

∥∥∥∥∥2

=∥∥∥ϕ f

∥∥∥2+

∥∥∥∥∥∞∑

k=0

skϕ(k) f (k)Rkh(k)

∥∥∥∥∥2

≤ ‖ϕ‖∞∥∥∥ f

∥∥∥2

+( ∞∑

k=0

s2k (ϕ(k) f (k))2h(k)

)1/2

≤ ‖ϕ‖∞ ‖ f ‖2 + ‖ϕ‖∞

( ∞∑k=0

s2k ( f (k))2h(k)

)1/2

= ‖ϕ‖∞ (‖ f ‖2 + ‖D f ‖2) = ‖ϕ‖∞ ‖ f ‖2,1.

� Theorem 12 Let T be a bounded linear operator on H (1)

2 (S). The following conditions areequivalent:

(i) T ∈ M(H (1)2 (S)).

(ii) There exists a unique pseudomeasure σ ∈ P(S) such that T f = σ ∗ f for all f ∈H (1)

2 (S).

Moreover, there exists an isometric algebra isomorphism from M(H (1)2 (S)) onto P(S).

Proof The proof follows by Theorem 3 and Theorem 11.Furthermore, each ϕ ∈ l∞(N0) defines by T f := ℘(ϕ f ) a multiplier for H (1)

2 (S). Indeed,we have

‖T f ‖2,1 = ‖T f ‖2 +( ∞∑

k=0

s2k (T f )(k)2h(k)

)1/2

=∥∥∥ϕ f

∥∥∥2+

( ∞∑k=0

s2kϕ(k)

2 f (k)2h(k)

)1/2

≤ ‖ϕ‖∞∥∥∥ f

∥∥∥2+ ‖ϕ‖∞

( ∞∑k=0

s2k f (k)2h(k)

)1/2

= ‖ϕ‖∞ ‖ f ‖2,1 < ∞

for all f ∈ H (1)2 (S). Hence T f is a bounded linear operator on H (1)

2 (S), which commutes

with Jacobi translation operators. Thus T ∈ M(H (1)2 (S)) and we obtain an isometric algebra

isomorphism from M(H (1)2 (S)) onto P(S). �

Corollary 2 There exists an isometric algebra isomorphism between the spaces M(W (S)),M(W∗(S)) and M(H (1)

2 (S)).

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Remark 5 We want to point out, that the homogeneous Banach spaces W (S), W∗(S) andH (1)

2 (S) are all very different in their structure and the spaces of bounded operators B(W (S)),B(W∗(S)) and B(H (1)

2 (S)) on W (S), W∗(S) and H (1)2 (S), respectively, differ. However, their

multiplier spaces coincide. The basic tool to prove this quite remarkable fact is the theory ofpseudomeasures.

Acknowledgments The authors gratefully acknowledges the support of the TUM Graduate School’s FacultyGraduate Center International School of Applied Mathematics (ISAM) at Technische Universität München,Germany.

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Multipliers on homogeneous Banach spaces with respect to Jacobi polynomials