Besov spaces and Bergman projections on the ball

C. R. Acad. Sci. Paris, Ser. I 335 (2002) 729–732 Analyse mathématique/Mathematical Analysis

Besov spaces and Bergman projections on the ball

H. Turgay Kaptano˘glu

Mathematics Department, Middle East Technical University, Ankara 06531, Turkey Received 2 September 2002; accepted 13 September 2002

Note presented by Jean-Pierre Kahane.

Abstract A class of radial differential operators are investigated yielding a natural classification of diagonal Besov spaces on the unit ball ofC^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. To cite this article: H.T. Kaptano˘glu, C. R. Acad. Sci. Paris, Ser. I 335 (2002) 729–732.

2002 Académie des sciences/Éditions scientifiques et médicales Elsevier SAS

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

naturelle des espaces de Besov diagonaux dans la boule unité deC^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. Pour citer cet article : H.T.

Kaptano˘glu, C. R. Acad. Sci. Paris, Ser. I 335 (2002) 729–732.

2002 Académie des sciences/Éditions scientifiques et médicales Elsevier SAS

1. Introduction

LetB denote the unit ball of C^N and H (B) the space of holomorphic functions on B. Let ν be the normalized volume measure onB and let dνq(z)= (1 − |z|²)^qdν(z). Diagonal Besov spaces BSV^v_p are defined in one of two ways. One definiton requires an f∈ H (B) to satisfy (1 − |z|²)^m−vR^mf (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+vf (z)∈ L^p(νp−1), where v > 0, R^vis a v-th-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^p_q= L^p(ν_q)∩ H (B): p > 0 and q ∈ R. The function whose p-th power is considered is not f any more, but a product of a t -th order radial derivative of f and the t -th power of 1− |z|². We call the spaces thus defined B_q^p, and it turns out that they naturally extend the A^p_q spaces to all q∈ R while satisfying A^pq = Bq^pfor q >−1. The value of t turns out to be irrelevant as long as (Re t)p+ q > −1. Eq. (2) summarizes the new classification.

Detailed proofs and further results will be presented elsewhere.

2. Radial differential operators

The radial derivative at z of a holomorphic function f is Rf (z)=
_∞

k=1kfk(z), where fk is the k-th term in the homogeneous expansion of f . Following [7], we define some more general linear operators.

E-mail address: kaptan@math.metu.edu.tr (H.T. Kaptano˘glu).

2002 Académie des sciences/Éditions scientifiques et médicales Elsevier SAS. Tous droits réservés

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H.T. Kaptano˘glu / C. R. Acad. Sci. Paris, Ser. I 335 (2002) 729–732

DEFINITION. – Let f ∈ H (B). We define D_s^tf =^∞

k=0

(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^tf = f0+ · · · + fm+ ^∞

k=m+1

(t− m)k

m!(1)k−m−1f_k (s= −m ∈ −N, s + t ∈ C \ −N).

The definition of D^t_s when s+ t or both s and s + t are nonpositive integers is similar.

It is shown in [5] that D^t_s is a continuous operator on H (B). If ds,t,k is the coefficient of fk in D^t_sf , Stirling’s formula gives ds,t,k∼ k^t as k→ ∞. Thus D^ts is a radial differential operator of order t ∈ C.

Every D^t_sis a bijection on H (B) and thus invertible. A case by case checking reveals that D^rs+tD^t_s= D^ts^+r. Thus (D^t_s)⁻¹= Ds^−t+t for all r, s, t∈ C.

3. Bq^pspaces

DEFINITION. – Let q∈ R and 0 < p
∞ with −qp + q > −1 (read −q > 0 when p = ∞). We define B_q^p= Bq^p(B) as the space of f ∈ H (B) for which the function (1 − |z|²)^−qD_−q^−qf (z) belongs to L^p(ν_q).

The B_q^pnorm of f is defined as the L^p(ν_q) norm of (1− |z|²)^−qD^−q_−qf (z).

We use the term norm even when 0 < p < 1. More explicitly, B_q^p(B) =



f∈ H (B) : f ^p_B^p

q =



B

D_−q^−qf (z)^pdν_−qp+q(z) <∞



(0 < p <∞), B_q^∞(B) =

f ∈ H (B) : f B^∞_q = sup

z∈B

1− |z|² _−qD_−q^−qf (z)<∞ .

Let us also define B_q⁰(B) as that subspace of B_q^∞(B) with the same norm consisting of functions f for which (1− |z|²)^−q|D_−q^−qf (z)| belongs to C0(B), continuous functions on B with 0 boundary values. Since every D^t_s is invertible, there is no nonzero f∈ Bq^pfor whichf _Bq^pis zero.

THEOREM 1. – An f∈ H (B) belongs to Bq^pif 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)^tD_s^tf (z) belongs to L^p(ν_q). The L^p(ν_q) norm of (1− |z|²)^tD_s^tf (z) is equivalent to the Bq^pnorm of f .

The case q= −(N + 1) is handled in [15,16]. Also, the Bq^pspaces are the spaces A^p_{−qp+q+1,−q}of [9], which considers only−qp + q + 1 > 0.

DEFINITION. – Let q∈ R and 0 < p
∞. We define Bq^p= Bq^p(B) as the space of f ∈ H (B) for which the function (1− |z|²)^tD_s^tf (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 Bq^pfor q >−1 and the Bergman spaces A^pq 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_q⁰are all the same for q∈ R and are the little Bloch space B0. The next result is in [16] for s > N . COROLLARY. – A function f ∈ H (B) belongs to B (resp. B0) if and only if for some s and t with Re t > 0 the function (1− |z|²)^tD_s^tf (z) is uniformly bounded onB (resp. belongs to C0(B)).

COROLLARY. – Our space B_q^pand the diagonal Besov space BSV^−(q+1)/p_p coincide.

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To cite this article: H.T. Kaptano˘glu, C. R. Acad. Sci. Paris, Ser. I 335 (2002) 729–732

Thus B_q^pspaces for p 1 are Banach spaces. Each B_q²space has several equivalent inner products (f, g)_s,t=



B

1− |z|² t

D^t_sf (z)

1− |z|² t

D_s^tg(z) dνq(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_q²is 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

(k− 1)!

(−N − q)kz, 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. In fact, the spaces in Theorem 2 are known as Dirichlet-type spaces, B_−(N+1)² being the Dirichlet spaceD and B₋₁² being the Hardy space H². The space B_−N² 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 denotedP here. Another description of Bq²

spaces for q −(N + 1) without any derivatives is given in [4]. The same reference contains an example that shows that the inclusion Bq^p₁⊂ Bq^p₂ for q1< q2is proper for q −(N + 1), which actually works for all real q .

Let 0 < p1< 1 < p2< 2 < p3and q1<−(N + 1) < −N < q2<−1 < 0 < q3. Then

... ... ... ... ... ...

· · · A_q^p₃¹ ⊃ A¹_q3 ⊃ A_q^p₃² ⊃ A²_q3⊃ A^p_q3³ ⊃ B q₃= q

∪ ∪ ∪ ∪ ∪ 

· · · A^p1 ⊃ A¹ ⊃ A^p2 ⊃ A² ⊃ A^p3 ⊃ B 0= q

∪ ∪ ∪ ∪ ∪ 

· · · B₋₁^p1 B₋₁¹ B₋₁^p2 H² B₋₁^p3 B −1 = q

∪ ∪ ∪ ∪ ∪ 

· · · B_q^p₂¹ B_q¹

2 B_q^p2

2 B_q²

2 B_q^p3

2 B q₂= q

∪ ∪ ∪ ∪ ∪ 

· · · B_−N^p1 B_−N¹ B_−N^p2 P B_−N^p3 B −N = q

∪ ∪ ∪ ∪ ∪ 

· · · B_−(N+1)^p1 M ⊂ B_−(N+1)^p2 ⊂ D ⊂ B_−(N+1)^p3 ⊂ B −(N +1)=q

∪ ∪ ∪ ∪ ∪ 

· · · Bq^p₁¹ B_q¹

1 B_q^p2

1 B_q²

1 B_q^p3

1 B q₁= q

... ... ... ... ... ...

(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 spaceM; see [11]. Thus also M ⊂ B_−(N+1)^p2 . 4. Bergman projections

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

P_sf (z)= C



B

(1− |w|²)^s

(1− z, w)^N+1+sf (w) dν(w) (z∈ B)

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H.T. Kaptano˘glu / C. R. Acad. Sci. Paris, Ser. I 335 (2002) 729–732

for f ∈ L¹(ν_s). It is clear that Psf ∈ 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 Ps maps L^p(ν_q) boundedly onto Bq^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 Ps((1− |z|²)^tD_N^t _+1+sf (z))= C^f (z) for all f ∈ Bq^p.

COROLLARY. – The Bergman projection Ps maps L^∞(νq)= L^∞(ν) boundedly ontoB and each of C(B) and C0(B) onto B0 if and only if Re s > −1. Given such an s, if t satisfies Re t > 0, then P_s[(1 − |z|²)^tD_N^t _+1+sf (z)] = C^f (z) for all f ∈ B and hence for all f ∈ B0too.

5. Duality and interpolation

THEOREM 4. – Let 1
p < ∞ and q
−1. The dual space (Bq^p)^∗can be identified with B_q^p under the pairing (·, ·)N+1,−qof (1), where 1/p+ 1/p^= 1. In particular, the Bloch space B is the dual space of all B_q¹. The dual spaceB0∗can be identified with each of B_q¹under the pairings (·, ·)N+1,−qof (1).

THEOREM 5. – Suppose 1
p0< p < p₁
∞ with 1/p = (1 − θ)/p0+ θ/p1 for some θ ∈ (0, 1).

Then the complex interpolation space[Bq^p0, Bq^p1]θis Bq^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 ofB. It is shown in [9] using (1) that the Bergman spaces A^pq for 0 < p
∞ and q > −1 are invariant under each of the isometries

U_ψ^p,qf (z)= f ψ(z) 

J ψ(z) (2/p)(1+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 spaceP.

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