Some Hilbert Spaces related with the Dirichlet space

We do a preliminary study of the reproducing kernel Hilbert space having as kernel $k^d$, where $d$ is a positive integer and $k$ is the reproducing kernel of the analytic Dirichlet space.


Introduction
Consider the Dirichlet space D on the unit disc {z ∈ C : |z| < 1} of the complex plane. It can be defined as the Reproducing Kernel Hilbert Space (RKHS) having kernel (zw) n n + 1 .
We are interested in the spaces D d having kernel k d , with d ∈ N. D d can be thought of in terms of function spaces on polydiscs, following ideas of Aronszajn [Aro]. To explain this point of view, note that the tensor d-power D ⊗d of the Dirichlet space has reproducing kernel k d (z 1 , · · · , z d ; w 1 , . . . , w d ) = Π d j=1 k(z j , w j ). Hence, the space of restrictions of functions in D ⊗d to the diagonal z 1 = · · · = z d has the reproducing kernel k d , and therefore coincides with D d .
We will provide several equivalent norms for the spaces D d and their dual spaces in Theorem 1. Then we will discuss the properties of these spaces. More precisely, we will investigate: -D d and its dual space HS d in connection with Hankel operators of Hilbert-Schmidt class on the Dirichlet space D; -the complete Nevanlinna-Pick property for D d ; -the Carleson measures for these spaces. Concerning the first item, the connection with Hilbert-Schmidt Hankel operators served as our original motivation for studying the spaces D d .
Note that the spaces D d live infinitely close to D in the scale of weighted Dirichlet spaces D s , defined by the norms where dA(z) π is normalized area measure on the unit disc. Notation: We use multiindex notation. If n = (n 1 , . . . , n d ) belongs to N d , then |n| = n 1 + · · · + n d . We write A ≈ B if A and B are quantities that depend on a certain family of variables, and there exist independent constants 0 < c < C such that cA ≤ B ≤ CA. .
Define now the holomorphic space HS d by the norm: Furthermore, the norm can be written as where {e n } ∞ n=0 is the canonical orthonormal basis of D, e n (z) = z n √ n+1 . The remainder of this section is devoted to the proof of Theorem 1. The expression for ϕ D d in (1) follows by expanding (k z ) d as a power series. The equivalence ϕ D d ≈ [ϕ] d , as well as ϕ HS d ≈ [ϕ] HS d , are consequences of the following lemma. We denote by c, C positive constants which are allowed to depend on d only, whose precise value can change from line to line.
Lemma 1. For each d ∈ N there are constants c, C > 0 such that for all k ≥ 0 we have Consequently, if t ∈ (0, 1), then Proof of Lemma 1. We will prove the Lemma by induction on d ∈ N. It is obvious for d = 1. Thus let d ≥ 2 and suppose the lemma is true for d − 1. Also we observe that there is a constant c > 0 such that for all k ≥ 0 and 0 ≤ n ≤ k we have Next, we prove the equivalence [ϕ] HS d ≈ |[ϕ]| HS d which appears in (5).
Given the Lemma, we expand obtaining the desired conclusion.
Proof of Lemma 2. The case d = 1 is obvious, leaving us to consider d ≥ 2. We will also assume that k ≥ 2. Then by Lemma 1 we have Now, the duality between D d and HS d under the duality pairing given by the inner product of D is easily seen by considering [·] d and [·] HS d . They are weighted ℓ 2 norms and duality is established by means of the Cauchy-Schwarz inequality.
Next we will prove that [ϕ] d ≈ |[ϕ]| d . This is equivalent to proving that the dual space of HS d , with respect to the Dirichlet inner product · , · D , is the Hilbert space with the norm |[·]| d .
A positive function W on the unit disc is said to satisfy the Bekollé-Bonami condition (B2) if there exists c > 0 such that for every Carleson square S h (e is ). If d ∈ N and if W d (z) is defined as before, then by Lemma 3, at least if 0 < h < 1/2. Observe that both W d and 1/W d are positive and integrable in the unit disc, hence it follows that the estimate holds for all 0 < h ≤ 1. Thus W d satisfies the condition (B2). Furthermore, note that if f (z) = ∞ k=0f (k)z k is analytic in the open unit disc, then .

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A special case of Theorem 2.1 of Luecking's paper [L] says that if W satisfies the condition (B2) by Bekollé and Bonami [BB], then one has a duality between the spaces L 2 a (W dA) and L 2 a ( 1 W dA) with respect to the pairing given by |z|<1 f gdA. Thus, we have This finishes the proof of (5). It remains to demonstrate (6). We defer its proof to the next section.
By Theorem 1 we have the following chain of inclusions: with duality w.r.t. D linking spaces with the same index. It might be interesting to compare this sequence with the sequence of Banach spaces related to the Dirichlet spaces studied in [ARS2]. Note that for d ≥ 3 the reproducing kernel of HS d is continuous up to the boundary. Hence functions in HS d extend continuously to the closure of the unit disc, for d ≥ 3.

Hilbert-Schmidt norms of Hankel-type operators
Let {e n } be the canonical orthonormal basis of D, e n (z) = z n √ n+1 . Equation (6) follows from the computation Polarizing this expression for · HS d , the inner product of HS d can be written ψ 1 , ψ 2 HS d = (n 1 ,...,n d ) ψ 1 , e n 1 . . . e n d D e n 1 . . . e n d , ψ 2 D .

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Hence, for any λ, ζ ∈ D, . . e n d D e n 1 . . . e n d , k ζ D = n∈N d e n 1 (λ) . . . e n d (λ)e n 1 (ζ) . . . e n d (ζ) That is, Analogous interpretations can be given for d ≥ 3, but then function spaces on polydiscs are involved. We consider the case d = 3, which is representative. Consider first the operator The formula uniquely defines an operator, whose action is Similarly, we can consider U b : D ⊗ D → D defined by The action of this operator is given by The Hilbert-Schmidt norm of U b is still b HS 3 . holds with a constant C(µ) which is independent of f . The characterization [ARS1] shows that, since the (B2) condition holds, then Theorem 2. For d ∈ N, a measure µ ≥ 0 on {|z| < 1} is Carleson for D d if and only if for |a| < 1 we have: The characterization extends to HS 2 , with the weight log −1 1 1−|z| 2 . Since functions in HS d are continuous for d ≥ 3, all finite measures are Carleson measures for these spaces. Once we know the Carleson measures, we can characterize the multipliers for D d in a standard way.
The complete Nevanlinna-Pick property for D d Next, we prove that the spaces D d have the Complete Nevanlinna-Pick (CNP) Property. Much research has been done on kernels with the CNP property in the past twenty years, following seminal work of Sarason and Agler. See the monograph [AMcC] for a comprehensive and very readable introduction to this topic. We give here a definition which is simple to state, although perhaps not the most conceptual. An irreducible kernel k : X × X → C has the CNP property if there is a positive definite function F : X → D and a nowhere vanishing function δ : X → C such that: k(x, y) = δ(x)δ(y) 1 − F (x, y) whenever x, y lie in X. The CNP property is a property of the kernel, not of the Hilbert space itself.
Theorem 3. There are norms on D d such that the CNP property holds.
Proof. A kernel k : D × D → C of the form k(w, z) = ∞ k=0 a k (zw) k has the CNP property if a 0 = 1 and the sequence {a n } ∞ n=0 is positive and log-convex: a n−1 a n ≤ a n a n+1 .
The CNP property has a number of consequences. For instance, we have that the space D d and its multiplier algebra M(D d ) have the same interpolating sequences. Recall that a sequence Z = {z n } ∞ n=0 is interpolating for a RKHS H with reproducing kernel k H if the 8 weighted restriction map R : ϕ → ϕ(zn) k H (zn,zn) 1/2 ∞ n=0 maps H boundedly onto ℓ 2 ; while Z is interpolating for the multiplier algebra M(H) if Q : ψ → {ψ(z n )} ∞ n=0 maps M(H) boundedly onto ℓ ∞ . The reader is referred to [AMcC] and to the second chapter of [S] for more on this topic.
It is a reasonable guess that the universal interpolating sequences for D d and for its multiplier space M(D d ) are characterized by a Carleson condition and a separation condition, as described in [S] (see the Conjecture at p. 33). See also [B], which contains the best known result on interpolation in general RKHS spaces with the CNP property. Unfortunately we do not have an answer for the spaces D d .