A remark on the multipliers on spaces of weak products of functions

If $\mathcal{H}$ denotes a Hilbert space of analytic functions on a region $\Omega \subseteq \mathbb{C}^d$, then the weak product is defined by $$\mathcal{H}\odot\mathcal{H}=\left\{h=\sum_{n=1}^\infty f_n g_n : \sum_{n=1}^\infty \|f_n\|_{\mathcal{H}}\|g_n\|_{\mathcal{H}}<\infty\right\}.$$ We prove that if $\mathcal{H}$ is a first order holomorphic Besov Hilbert space on the unit ball of $\mathbb{C}^d$, then the multiplier algebras of $\mathcal{H}$ and of $\mathcal{H}\odot\mathcal{H}$ coincide.


Introduction
Let d be a positive integer and let R = d i=1 z i ∂ ∂z i denote the radial derivative operator. For s ∈ R the holomorphic Besov space B s is defined to be the space of holomorphic functions f on the unit ball B d of C d such that for some nonnegative integer k > s Here dV denotes Lebesgue measure on B d . It is well-known that for any f ∈ Hol(B d ) and any s ∈ R the quantity f k,s is finite for some nonnegative integer k > s if and only if it is finite for all nonnegative integers k > s, and that for each k > s · k,s defines a norm on B s , and that all these norms are equivalent to one another, see [2]. For s < 0 one can take k = 0 and these spaces are weighted Bergman spaces. In particular, B −1/2 = L 2 a (B d ) is the unweighted Bergman space. For s = 0 one obtains the Hardy space of B d and one has that for each k ≥ 1 f 2 k,0 is equivalent to ∂B d |f | 2 dσ, where σ is the rotationally invariant probability measure on ∂B d . We also note that for s = (d − 1)/2 we have B s = H 2 d , the Drury-Arveson space. If d = 1 and s = 1/2, then B s = D, the classical Dirichlet space of the unit disc.
Let H ⊆ Hol(B d ) be a reproducing kernel Hilbert space such that 1 ∈ H. The weak product of H is denoted by H ⊙ H and it is defined to be the collection of all functions h ∈ Hol(B d ) such that there are . We define a norm on H ⊙ H by In what appears below we will frequently take H = B s , and will use the same notation for this weak product.
Weak products have their origin in the work of Coifman, Rochberg, and Weiss [5]. In the frame work of the Hilbert space H one may consider the weak product to be an analogue of the Hardy , see [5]. For the Dirichlet space D the weak product D ⊙ D has recently been considered in [1], [4], [9], [6], and [7]. The space H 2 d ⊙ H 2 d was used in [10]. For further motivation and general background on weak products we refer the reader to [1] and [9].
Let B be a Banach space of analytic functions on B d such that point evaluations are continuous and such that 1 ∈ B. We use M(B) to denote the multiplier algebra of B, The multiplier norm ϕ M is defined to be the norm of the associated multiplication operator M ϕ : B → B. It is easy to check and is well- For those cases M(B s ) has been described by a certain Carleson measure condition, see [3,8].
It is easy to see that M Note that when d ≤ 2, then B s is an algebra for all s > 1. Thus for each d ∈ N the nontrivial range of the Theorem is 0 < s ≤ 1. If d = 1 then the theorem applies to the classical Dirichlet space of the unit disc and for d ≤ 3 it applies to the Drury-Arveson space.

Preliminaries
For z = (z 1 , ..., z d ) ∈ C d and t ∈ R we write e it z = (e it z 1 , ..., e it z d ) and we write z, w for the inner product in C d . Furthermore, if h is a function on B d , then we define T t f by (T t f )(z) = f (e it z). We say that a space H ⊆ Hol(B d ) is radially symmetric, if each T t acts isometrically on H and if for all t 0 ∈ R, T t → T t 0 in the strong operator topology as For example, for each s ∈ R the holomorphic Besov space B s is radially symmetric when equipped with any of the norms · k,s , k > s.
It is elementary to verify the following lemma.

Lemma 2.1. If H ⊆ Hol(B d ) is radially symmetric, then so is H ⊙ H.
Note that if h and ϕ are functions on B d , then for every t ∈ R we have (T t ϕ)h = T t (ϕT −t h), hence if a space is radially symmetric, then T t acts isometrically on the multiplier algebra. For 0 < r < 1 we write f r (z) = f (rz). Proof. Let ϕ ∈ M(H ⊙ H) and h ∈ H ⊙ H, then for 0 < r < 1 we have This implies Thus, ϕ r M (H⊙H) ≤ ϕ M (H⊙H) .

Multipliers
The following Proposition is elementary. As explained in the Introduction, the following will establish Theorem 1.1. Here for each s we have the norm on B s to be · k,s , where k is the smallest natural number > s.
Proof. We first do the case 0 < s < 1. Then k = 1, and f 2 . For later reference we note that a short calculation shows that B d |Rf | 2 dV s ≤ f 2 Bs . We write Rϕ Ca(Bs) for the Carleson measure norm of |Rϕ| 2 , i.e.
First we note that if b is holomorphic in a neighborhood of B d and where we have continued to write · * for · Bs⊙Bs .
Let ϕ ∈ M(B s ⊙ B s ) and let 0 < r < 1. Then for all f ∈ B s we have Bs Rϕ r Ca(Bs) . Next we take the sup of the left hand side of this expression over all f with f Bs = 1 and we obtain Rϕ r 2 Ca(Bs) ≤ 4 ϕ M (Bs⊙Bs) Rϕ r Ca(Bs) which implies that Rϕ r Ca(Bs) ≤ 4 ϕ M (Bs⊙Bs) holds for all 0 < r < 1. Thus, for 0 < s < 1 the result follows from Fatou's lemma as r → 1.