C. R. Acad. Sci. Paris, t. 33?, Série I, p. 1–??, 2002 - PXMA????.TEX - Analyse mathématique/Mathematical Analysis

Besov spaces and Bergman projections on the ball

H. Turgay Kaptanoğlu^{a}

a Mathematics Department, Middle East Technical University, Ankara 06531, Turkey E-mail: kaptan@math.metu.edu.tr

(Reçu le jour mois année, accepté après révision le jour mois année)

Abstract. A class of radial differential operators are investigated yielding a natural classification of diagonal Besov spaces on the unit ball of ℂ^{N}. Precise conditions are given for the boundedness of Bergman projections from certain L^{p} spaces onto Besov spaces. Right inverses for these projections are also provided. Applications to complex interpolation are presented. \begin{matrix}c \\ ◯ \end{matrix} 2002 Académie des sciences/Éditions scientifiques et médicales Elsevier SAS

Les espaces de Besov et les projections de Bergman dans la boule

Résumé. Nous étudions une class d’opérateurs différentiels radiaux conduisant à une classification naturelle des espaces de Besov diagonaux dans la boule unité de ℂ^{N}. Nous donnons les conditions précises pour la bornitude des projections de Bergman de certains espaces L^{p} sur des espaces de Besov. Nous déterminons aussi des inverses à droite pour ces projections. Nous présentons des applications à l’interpolation complexe. \begin{matrix}c \\ ◯ \end{matrix} 2002 Académie des sciences/Éditions scientifiques et médicales Elsevier SAS

1. Introduction

Let 𝔹 denote the unit ball of ℂ^{N} and H(𝔹) the space of holomorphic functions on 𝔹. Let ν be the normalized volume measure on 𝔹 and let dν_{q}(z) = (1 − |z|^{2})^{q} dν(z). Diagonal Besov spaces BSV_{p}^{v} are defined in one of two ways. One definiton requires an f ∈ H(𝔹) to satisfy (1 − |z|^{2})^{m−v} R^{m}f(z) ∈ L^{p}(ν_{−1}) for some integer m > v, where v is real (see [6]). Here R is the radial derivative; see below. Another requires an f ∈ H(𝔹) to satisfy R^{1+v}f(z) ∈ L^{p}(ν_{p−1}), where v > 0, R^{v} is a vth-order radial derivative, and p > 1 (see [12]).

The purpose of this note is to give a new definition that characterizes these spaces using exactly the same parameters as those of the weighted Bergman spaces A_{q}^{p} = L^{p}(ν_{q})∩H(𝔹): p > 0 and q ∈ ℝ. The function whose pth power is considered is not f any more, but a product of a tth order radial derivative of f and the tth power of 1 − |z|^{2}. We call the spaces thus defined B_{q}^{p}, and it turns out that they naturally extend the A_{q}^{p} spaces to all q ∈ ℝ while satisfying A_{q}^{p} = B_{q}^{p} for q > −1. The value of t turns out to be irrelevant as long as (Re t)p + q > −1. Equation (2) summarizes the new classification.

Detailed proofs and further results will be presented elsewhere.

Note présentée par First name NAME

S0764-4442(00)0????-?/FLA \begin{matrix}c \\ ◯ \end{matrix} 2002 Académie des sciences/Éditions scientifiques et médicales Elsevier SAS. Tous droits réservés. 1

H. Turgay Kaptanoğlu

2. Radial Differential Operators

The radial derivative at z of a holomorphic function f is Rf(z) = ∑_{k=1}^{∞} k f_{k}(z), where f_{k} is the kth term in the homogeneous expansion of f. Following [7], we define some more general linear operators.

Definition . – Let f ∈ H(𝔹). We define

D_{s}^{t}f = ∑_{k=0}^{∞} \frac{(s + t)_{k}}{(s)_{k}} f_{k} (s ∈ ℂ \ −ℕ, s + t ∈ ℂ \ −ℕ),

where (a)_{0} = 1 and (a)_{m} = a(a + 1) ⋯ (a + m − 1) for positive integer m. If, say, s = −m with m ∈ ℕ, we write the corresponding factor as (λ)_{k}, divide it by (−1)^{m}(λ + m), and then let λ → −m. So

D_{s}^{t}f = f_{0}+⋯+f_{m}+ ∑_{k=m+1}^{∞} \frac{(t − m)_{k}}{m! (1)_{k−m−1}} f_{k}, (s = −m ∈ −ℕ, s + t ∈ ℂ \ −ℕ).

The definition of D_{s}^{t} when s + t or both s and s + t are nonpositive integers is similar.

It is shown in [5] that D_{s}^{t} is a continuous operator on H(𝔹). If d_{s,t,k} is the coefficient of f_{k} in D_{s}^{t}f, Stirling’s formula gives d_{s,t,k} ∼ k^{t} as k → ∞. Thus D_{s}^{t} is a radial differential operator of order t ∈ ℂ. Every D_{s}^{t} is a bijection on H(𝔹) and thus invertible. A case by case checking reveals that D_{s+t}^{r}D_{s}^{t} = D_{s}^{t+r}. Thus (D_{s}^{t})^{−1} = D_{s+t}^{−t} for all r, s, t ∈ ℂ.

3. B_{q}^{p} Spaces

Definition . – Let q ∈ ℝ and 0 < p ≤ ∞ with −qp + q > −1 (read −q > 0 when p = ∞). We define B_{q}^{p} = B_{q}^{p}(𝔹) as the space of f ∈ H(𝔹) for which the function (1 − |z|^{2})^{−q} D_{−q}^{−q}f(z) belongs to L^{p}(ν_{q}). The B_{q}^{p} norm of f is defined as the L^{p}(ν_{q}) norm of (1 − |z|^{2})^{−q} D_{−q}^{−q}f(z).

We use the term norm even when 0 < p < 1. More explicitly,

B_{q}^{p}(𝔹) = { f ∈ H(𝔹) : ∥f∥_{B_{q}^{p}}^{p} = ∫_{𝔹} |D_{−q}^{−q}f(z)|^{p} dν_{−qp+q}(z) < ∞ }  (0 < p < ∞),
B_{q}^{∞}(𝔹) = { f ∈ H(𝔹) : ∥f∥_{B_{q}^{∞}} = sup_{z∈𝔹}(1 − |z|^{2})^{−q} |D_{−q}^{−q}f(z)| < ∞ }.

Let’s also define B_{q}^{0}(𝔹) as that subspace of B_{q}^{∞}(𝔹) with the same norm consisting of functions f for which (1 − |z|^{2})^{−q} |D_{−q}^{−q}f(z)| belongs to C_{0}(𝔹), continuous functions on 𝔹 with 0 boundary values. Since every D_{s}^{t} is invertible, there is no nonzero f ∈ B_{q}^{p} for which ∥f∥_{B_{q}^{p}} is zero.

Theorem 1. – An f ∈ H(𝔹) belongs to B_{q}^{p} if and only if for some s and t satisfying (Re t)p + q > −1 (read Re t > 0 when p = ∞) the function (1 − |z|^{2})^{t} D_{s}^{t}f(z) belongs to L^{p}(ν_{q}). The L^{p}(ν_{q}) norm of (1 − |z|^{2})^{t} D_{s}^{t}f(z) is equivalent to the B_{q}^{p} norm of f.

The case q = −(N + 1) is handled in [15]. Also, the B_{q}^{p} spaces are the spaces A_{−qp+q+1,−q}^{p} of [9], which considers only −qp + q + 1 > 0.

Definition . – Let q ∈ ℝ and 0 < p ≤ ∞. We define B_{q}^{p} = B_{q}^{p}(𝔹) as the space of f ∈ H(𝔹) for which the function (1 − |z|^{2})^{t} D_{s}^{t}f(z) for some s and t satisfying (Re t)p + q > −1 (read Re t > 0 for p = ∞) belongs to L^{p}(ν_{q}).

Corollary . – The spaces B_{q}^{p} for q > −1 and the Bergman spaces A_{q}^{p} coincide.

By Theorem 1, B_{q}^{∞} spaces are the same for all q ∈ ℝ. This space is the Bloch space ℬ. Similarly, the spaces B_{q}^{0} are all the same for q ∈ ℝ and are the little Bloch space ℬ_{0}. The next result is in [15] for s > N.

Corollary . – A function f ∈ H(𝔹) belongs to ℬ (resp. ℬ_{0}) if and only if for some s and t with Re t > 0 the function (1 − |z|^{2})^{t} D_{s}^{t}f(z) is uniformly bounded on 𝔹 (resp. belongs to C_{0}(𝔹)).

Besov spaces and Bergman projections on the ball

Corollary . – Our space B_{q}^{p} and the diagonal Besov space BSV_{p}^{−(q+1)/p} coincide.

Thus B_{q}^{p} spaces for p ≥ 1 are Banach spaces. Each B_{q}^{2} space has several equivalent inner products

(f, g)_{s,t} = ∫_{𝔹}(1 − |z|^{2})^{t} D_{s}^{t}f(z) \overline{(1 − |z|^{2})^{t} D_{s}^{t}g(z)} dν_{q}(z),  (1)

one for each s and each t satisfying 2 Re t + q > −1. The monomials {z^{α}} form an orthogonal set with respect to each of these inner products.

Theorem 2. – Each B_{q}^{2} is a Hilbert space with reproducing kernel K_{q}(z, w) = (1 − ⟨z, w⟩)^{−(N+1+q)} for q > −(N + 1) and

K_{q}(z, w) = 1 + ∑_{k=1}^{∞} \frac{(k − 1)!}{(−N − q)_{k}} ⟨z, w⟩^{k}

for q ≤ −(N + 1); in particular, K_{−(N+1)}(z, w) = 1 − log(1 − ⟨z, w⟩).

Similar descriptions in [13] and [2] led the author to this research. The spaces in Theorem 2 are known as Dirichlet-type spaces, B_{−(N+1)}^{2} being the Dirichlet space 𝒟 and B_{−1}^{2} being the Hardy space H^{2}. The space B_{−N}^{2} attracts a lot of attention in operator theory (see [3] and [8]) due to the universal property of its kernel in Nevanlinna-Pick interpolation (see [1]) and is denoted 𝒫 here. Another description of B_{q}^{2} spaces for q ≥ −(N + 1) without any derivatives is given in [4]. The same reference contains an example that shows that the inclusion B_{q_{1}}^{p} ⊂ B_{q_{2}}^{p} for q_{1} < q_{2} is proper for q ≥ −(N + 1), which works for all real q.

Let 0 < p_{1} < 1 < p_{2} < 2 < p_{3} and q_{1} < −(N + 1) < −N < q_{2} < −1 < 0 < q_{3}. Then

\begin{aligned}& ⋮ & & ⋮ & & ⋮ & & ⋮ & & ⋮ & & ⋮ \\ ⋯ & A_{q_{3}}^{p_{1}} & ⊃ & A_{q_{3}}^{1} & ⊃ & A_{q_{3}}^{p_{2}} & ⊃ & A_{q_{3}}^{2} & ⊃ & A_{q_{3}}^{p_{3}} & ⊃ & ℬ & q_{3} = q \\ & ∪ & & ∪ & & ∪ & & ∪ & & ∪ & & ∥ \\ ⋯ & A^{p_{1}} & ⊃ & A^{1} & ⊃ & A^{p_{2}} & ⊃ & A^{2} & ⊃ & A^{p_{3}} & ⊃ & ℬ & 0 = q \\ & ∪ & & ∪ & & ∪ & & ∪ & & ∪ & & ∥ \\ ⋯ & B_{−1}^{p_{1}} & & B_{−1}^{1} & & B_{−1}^{p_{2}} & & H^{2} & & B_{−1}^{p_{3}} & & ℬ & −1 = q \\ & ∪ & & ∪ & & ∪ & & ∪ & & ∪ & & ∥ \\ ⋯ & B_{q_{2}}^{p_{1}} & & B_{q_{2}}^{1} & & B_{q_{2}}^{p_{2}} & & B_{q_{2}}^{2} & & B_{q_{2}}^{p_{3}} & & ℬ & q_{2} = q \\ & ∪ & & ∪ & & ∪ & & ∪ & & ∪ & & ∥ \\ ⋯ & B_{−N}^{p_{1}} & & B_{−N}^{1} & & B_{−N}^{p_{2}} & & 𝒫 & & B_{−N}^{p_{3}} & & ℬ & −N = q \\ & ∪ & & ∪ & & ∪ & & ∪ & & ∪ & & ∥ \\ ⋯ & B_{−(N+1)}^{p_{1}} & & ℳ & ⊂ & B_{−(N+1)}^{p_{2}} & ⊂ & 𝒟 & ⊂ & B_{−(N+1)}^{p_{3}} & ⊂ & ℬ & −(N +1)=q \\ & ∪ & & ∪ & & ∪ & & ∪ & & ∪ & & ∥ \\ ⋯ & B_{q_{1}}^{p_{1}} & & B_{q_{1}}^{1} & & B_{q_{1}}^{p_{2}} & & B_{q_{1}}^{2} & & B_{q_{1}}^{p_{3}} & & ℬ & q_{1} = q \\ & ⋮ & & ⋮ & & ⋮ & & ⋮ & & ⋮ & & ⋮ \end{aligned} (2)

The inclusions of the level q = −(N + 1) follow from their Möbius invariance and are in [10] for p > 1. Further, B_{−(N+1)}^{1} is the minimal Möbius-invariant space ℳ; see [11]. Thus also ℳ ⊂ B_{−(N+1)}^{p_{2}}.

4. Bergman Projections

We restrict ourselves to 1 ≤ p ≤ ∞ from now on. Bergman projections are the linear operators P_{s} with s ∈ ℂ defined by

P_{s}f(z) = C ∫_{𝔹} \frac{(1 − |w|^{2})^{s}}{(1 − ⟨z, w⟩)^{N+1+s}} f(w) dν(w) (z ∈ 𝔹)

for f ∈ L^{1}(ν_{s}). It is clear that P_{s}f ∈ H(𝔹). The coefficient C is a normalization constant. The following theorem appears in [11] for real s > −1.

Theorem 3. – Let 1 ≤ p < ∞ and q ≤ −1. The Bergman projection P_{s} maps L^{p}(ν_{q}) boundedly onto B_{q}^{p} if and only if q + 1 < p(Re s + 1) and N + 1 + s is not a nonpositive integer. Given such an s, if t satisfies (Re t)p + q > −1, then P_{s}((1 − |z|^{2})^{t} D_{N+1+s}^{t}f(z)) = C′f(z) for all f ∈ B_{q}^{p}.

Corollary . – The Bergman projection P_{s} maps L^{∞}(ν_{q}) = L^{∞}(ν) boundedly onto ℬ and each of C(\overline{𝔹}) and C_{0}(𝔹) onto ℬ_{0} if and only if Re s > −1. Given such an s, if t satisfies Re t > 0, then P_{s} [(1 − |z|^{2})^{t} D_{N+1+s}^{t}f(z)] = C′′f(z) for all f ∈ ℬ and hence for all f ∈ ℬ_{0} too.

5. Duality and Interpolation

Theorem 4. – Let 1 ≤ p < ∞ and q ≤ −1. The dual space (B_{q}^{p})^{∗} can be identified with B_{q}^{p′} under the pairing (·, ·)_{N+1,−q} of (1), where 1/p + 1/p′ = 1. In particular, the Bloch space ℬ is the dual space of all B_{q}^{1}. The dual space ℬ_{0}^{∗} can be identified with each of B_{q}^{1} under the pairings (·, ·)_{N+1,−q} of (1).

Theorem 5. – Suppose 1 ≤ p_{0} < p < p_{1} ≤ ∞ with 1/p = (1 − θ)/p_{0} + θ/p_{1} for some θ ∈ (0, 1). Then the complex interpolation space [B_{q}^{p_{0}}, B_{q}^{p_{1}}]_{θ} is B_{q}^{p}.

The case q = −(N + 1) of Theorem 4 is in [15]. For the definitions on complex interpolation, see [14]. Let ψ be a holomorphic automorphism of 𝔹. It is shown in [9] using (1) that the Bergman spaces A_{q}^{p} for

0 < p ≤ ∞ and q > −1 are invariant under each of the isometries

U_{ψ}^{p,q}f(z) = f(ψ(z)) (Jψ(z))^{\frac{2}{p} (1+ \frac{q}{N+1} )}.

Theorem 6. – Suppose 2 ≤ p ≤ ∞, −(N + 1) < q ≤ −1, and ψ as above. Then U_{ψ}^{p,q} is a bounded linear transformation on B_{q}^{p}.

Acknowledgements. The author expresses his gratitude to Daniel Alpay of Ben-Gurion University of the Negev for his extensive support and getting him involved in the space 𝒫.

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