����������� ������ 49 (1997), 229–233

UDK 517.55

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BESOV SPACES ON BOUNDED SYMMETRIC DOMAINS

Miroljub Jevtić

Abstract. We define and study a class of holomorphic Besov type spaces B^{p}, 0 < p < 1, on bounded symmetric domains Ω. We show that the dual of holomorphic Besov space B^{p}, 0 < p < 1, on bounded symmetric domain Ω can be identified with the Bloch space B^{∞}.

1. Introduction

Let Ω be an irreducible symmetric domain in C^{n} in its Harish-Chandra realization. In [11] and [13] K. Zhu defined and studied a class of holomorphic Besov-type spaces B^{p} on Ω for 1 ≤ p ≤ ∞. In [4] analogous holomorphic Besov spaces B^{p} are defined for 0 < p < 1 and some of the results presented in [11] and [13] are extended to the case 0 < p < 1. The main purpose of this paper is to show that the dual of B^{p}, 0 < p < 1, can be indentified with the Bloch space B^{∞}.

It is well known [5] that the domain Ω is uniquely determined (up to a biholomorphic mapping among standard irreducible bounded symmetric domains) by three analytic invariants; r, a and b, all of which are nonnegative integers. The invariant r is called the rank of Ω, which is of course always positive. See [5] for the definition of a and b. We shall make extensive use of the following invariant of Ω: N = a(r − 1) + b + 2.

Let ν be Lebesgue measure on Ω normalized so that ν(Ω) = 1. For 0 < p < ∞ the Bergman space L_{a}^{p}(Ω) is the closed subspace of L^{p}(Ω, dν) consisting of holomorphic functions. The Bergman projection P (namely, the orthogonal projection from L^{2}(Ω, dν) onto L_{a}^{2}(Ω)) is an integral operator

P f(z) = ∫_{Ω} K(z, w)f(w) dν(w), z ∈ Ω, f ∈ L^{2}(Ω, dν).

By [3] there exists a polynomial h(z, w) in z and w̄ such that the Bergman kernel of Ω is given by

K(z, w) = h(z, w)^{−N}, z, w ∈ Ω.

AMS Subject Classification: 32 A 37

Communicated at the 4th Symposium on Mathematical Analysis and Its Applications, Aran - delovac 1997

Throughout this paper we assume α is a real number satisfying α > −1. Let c_{α} be a positive normalizing constant such that the measure dν_{α}(z) = c_{α}h(z, z)^{α}dν(z) has total mass 1 on Ω.

Let H(Ω) be the space of all holomorphic functions in Ω. We equip H(Ω) with the topology of uniform convergence on compact sets. In [13] it is shown that the operator

D^{m,α}: H(Ω) → H(Ω), m ≥ 0, α > −1,

defined by

D^{m,α}f(z) = lim_{r→1} ∫_{Ω} \frac{f(rw) dν_{α}(w)}{h(z, w)^{N+α+m}} , f ∈ H(Ω),

is continuous and invertible on H(Ω). The inverse of D^{m,α} admits the following integral representation:

D_{m,α}f(z) = c_{m+α} lim_{r→1} ∫_{Ω} \frac{h(w, w)^{m+α}f(rw) dν(w)}{h(z, w)^{N+α}}, f ∈ H(Ω), z ∈ Ω.

We note that if

f ∈ L_{a}^{1,α}(Ω) = L^{1}(Ω, dν_{α}) ∩ H(Ω)

then

D^{m,α}f(z) = ∫_{Ω} \frac{f(w) dν_{α}(w)}{h(z, w)^{N+α+m}} .

The above formula extends the domain of D^{m,α} to L^{1}(Ω, dν_{α}). We write

V_{m,α}f(z) = h(z, z)^{m}D^{m,α}f(z), for f ∈ L^{1}(Ω, dν_{α}), m ≥ 0, α > −1,

and

E_{m,α}f(z) = h(z, z)^{m}D^{m,α}f(z), m ≥ 0, α > −1, for f ∈ H(Ω).

Thus, if f ∈ L_{a}^{1,α}(Ω), then E_{m,α}f = V_{m,α}f.

We begin with a result from [4].

Theorem 1.1. Let k, m > \frac{N−1}{p} and α, β > max\{\frac{a(r−1)}{2p} −N, −1\}. If 0 < p ≤ 1 and f ∈ H(Ω) then ∫_{Ω} |E_{m,α}f(z)|^{p} dτ(z) < ∞ if and only if ∫_{Ω} |E_{k,β}f(z)|^{p} dτ(z) < ∞, where dτ(z) = h(z, z)^{−N} dν(z) is the Möbious invariant measure on Ω.

Recall that the holomorphic Besov space B^{p}, 1 ≤ p ≤ ∞ consists of functions in H(Ω) such that E_{N,0}f is in L^{p}(Ω, dτ) (see [11] and [13]). For any irreducible bounded symmetric domain Ω we have \frac{a(r−1)}{2} − N ≤ −2. Thus, a holomorphic function f in Ω belongs to B^{1} if and only if E_{m,α}f is in L^{1}(Ω, dτ) for some (any) m > N − 1 and some (any) α > −1. This is also proved in [13], Theorem 4, by a different method.

Definition 1.2. For 0 < p ≤ 1 the holomorphic Besov space B^{p} = B^{p}(Ω) consists of holomorphic functions f ∈ H(Ω) such that

∥f∥_{B^{p}} = |f(0)| + ||E_{m,α}f||_{L^{p}(Ω,dτ)} < ∞,

for some (any) m > \frac{N−1}{p} and α > max\{\frac{a(r−1)}{2p} − N, −1\}.

For every z in Ω let E_{r}(z) be the closed Bergman metric ball with center z and radius r > 0, i.e.,

E_{r}(z) = {w : β(z, w) ≤ r},

where β(·, ·) is the Bergman metric on Ω.

For a complex measurable function f on B we define

M_{∞,r} = esssup{ |f(w)| : w ∈ E_{r}(z) }

and

M_{p,r}f(z) = [\frac{1}{τ(r)} ∫_{E_{r}(z)} |f(w)|^{p} dτ(w)]^{\frac{1}{p}} , 0 < p < ∞,

where τ(r) = τ(E_{r}(z)).

For 0 < p, q ≤ ∞, we define L_{r}^{p,q}(Ω, dτ) to be the space of all measurable functions f on Ω for which

∥f∥_{L_{r}^{p,q}(Ω,dτ)} = ∥M_{p,r}f∥_{L^{q}(Ω,dτ)} < ∞.

Since the definition is independent of r, 0 < r < 1, we will write L^{p,q}(Ω, dτ) instead of L_{r}^{p,q}(Ω, dτ) (see [1]).

We let P_{α} denote the orthogonal projection from L^{2}(Ω, dν_{α}) onto L_{a}^{2,α}(Ω) = L^{2}(Ω, dν_{α}) ∩ H(Ω). It can be shown that (see [9], for instance)

P_{α}f(z) = ∫_{Ω} \frac{f(w) dν_{α}(w)}{h(z, w)^{N+α}}, z ∈ Ω, f ∈ L^{2}(Ω, dν_{α}).

The above formula extends the domain of P_{α} to L^{1}(Ω, dν_{α}). Note that P_{α}f = D^{0,α}f for f ∈ L^{1}(Ω, dν_{α}).

If 1 ≤ p ≤ ∞ and α > −1 then B^{p} = P_{α}L^{p}(Ω, dτ), see [13]. In [4] it is shown that the analytic Besov space B^{p}, 0 < p < 1, can be naturally embedded as a complemented subspace of L^{1,p}(Ω, dτ) by a topological embedding

E_{m,α}: B^{p} → L^{1,p}(Ω, dτ).

It is also shown that E_{m,α} ^{∘} P_{α} is projection on this embedded copy and that B^{p} = P_{α}L^{1,p}(Ω, dτ).

More precicely the following theorem is proven.

Theorem 1.3. Let 0 < p < 1. Then for any α > max\{\frac{a(r−1)}{2p} − N, −1\},

P_{α}: L^{1,p}(Ω, dτ) → B^{p}

is a continuous linear map. Moreover if m > \frac{N−1}{p} and α > max\{\frac{a(r−1)}{2p} − N, −1\} then

E_{m,α}: B^{p} → L^{1,p}(Ω, dτ)

is a topological embedding.

Now we apply Theorem 1.3. to obtain a result about duality.

Theorem 1.4. Let 0 < p < 1, m > \frac{N−1}{p} and α = m−N. The integral pairing

⟨f, g⟩_{τ} = ∫_{Ω} E_{m,α}f(z) \overline{E_{m,α}g(z)} dτ(z)

induces the following duality (B^{p})^{⋆} = B^{∞}.

2. Duality

A linear functional λ on B^{p}, 0 < p < 1, is said to be bounded if

∥λ∥ = sup{|λ(f)| : ∥f∥_{B^{p}} ≤ 1} < ∞.

The dual space of B^{p}, denoted (B^{p})^{⋆}, is then the space of all bounded linear functionals on B^{p}. In [13] it is shown that each (L_{a}^{p,α})^{⋆}, 0 < p ≤ 1, can be identified with B^{∞} via volume integral pairing

⟨f, g⟩_{β} = lim_{r→1} ∫_{Ω} f(rz)g(z) dν_{β}(z), where β = \frac{N + α}{p} − N.

Our Theorem 1.4 shows that (B^{p})^{⋆} can also be identified with B^{∞}, but via a different integral pairing ⟨ · , · ⟩_{τ}.

Proof of Theorem 1.4. First, assume that g is a function in B^{∞}. Then

∥g∥_{B^{∞}} ≥ sup_{z∈Ω} h(z, z)^{m}|D^{m,α}g(z)| = sup_{z∈Ω} |E_{m,α}g(z)|, (see [13]).

We show that g gives rise to a bounded linear functional on B^{p} under the pairing ⟨ · , · ⟩_{τ}. By Theorem 1.3 if f ∈ B^{p} then

∥f∥_{B^{p}} ≥ ∥E_{m,α}f∥_{L^{1,p}(Ω,dτ)} ≥ C∥E_{m,α}f∥_{L^{1}(Ω,dτ)}.

Thus if f ∈ B^{p}, then we have

|⟨f, g⟩_{τ} | ≤ sup_{z∈Ω} |E_{m,α}g(z)|∥E_{m,α}f∥_{L^{1}(Ω,dτ)} ≤ C∥f∥_{B^{p}}∥g∥_{B^{∞}}.

Conversely, assume that λ is a bounded linear functional on B^{p}; we show that λ arises from a function in B^{∞}. Since E_{m,α} is a topological embedding of B^{p} into L^{1,p}(Ω, dτ), λ ^{∘} E_{m,α}^{−1} extends to a bounded linear functional on L^{1}(Ω, dτ). Thus there exists a function φ ∈ L^{∞}(Ω, dτ) such that

λ^{∘} E_{m,α}^{−1}(ψ) = ∫_{Ω} ψ(z)\overline{φ(z)} dτ(z), ψ ∈ L^{1}(Ω, dτ).
When f ∈ B^{p}, E_{m,α}f ∈ L^{1,p}(Ω, dτ) ⊂ L^{1}(Ω, dτ). Therefore
λ(f) = ∫_{Ω} E_{m,α}f(z)\overline{φ(z)} dτ(z) = ∫_{Ω} V_{m,α}f(z)\overline{φ(z)} dτ(z), f ∈ B^{p}.

As was noted in Introduction E_{m,α}f = V_{m,α}f if f ∈ L_{a}^{1,α}(Ω). Since B^{p} ⊂ B^{∞}, we have that ∫_{Ω} |f(ξ)|h(ξ, ξ)^{α} dν(ξ) < ∞. (See the remark following Lemma 7 in [11] ). Let h = P_{α}φ. Then h ∈ B^{∞} by Theorem 6 ([13])

E_{m,α}h(z) = V_{m,α}h(z) = h(z, z)^{m}D^{m,α}P_{α}φ(z) = h(z, z)^{m}D^{m,α}(D^{0,α}φ)(z)
= h(z, z)^{m}D^{m,α}φ(z) = V_{m,α}φ(z).

To finish the proof of Theorem 1.4 it remains to show that

⟨V_{m,α}f, V_{m,α}φ⟩_{τ} = ⟨V_{m,α}^{2}f, φ⟩_{τ} and that c_{m+α}V_{m,α}^{2}f = c_{α}V_{m,α}f.

This follows easily from Fubini‘s theorem and the reproducing property of P_{α}. Note that α = m − N. We leave the details to the interested reader. Thus,

λ(f) = ∫_{Ω} V_{m,α}f(z)\overline{V_{m,α}g(z)} dτ(z) = ∫_{Ω} E_{m,α}f(z)\overline{E_{m,α}g(z)} dτ(z)

for all f ∈ B^{p}, where g = c_{m+α}c_{α}^{−1}h ∈ B^{∞}.

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(received 04.09.1997.)

Faculty of Mathematics, Studentski trg 16, P.O.Box 550, 11000 Beograd, Yugoslavia e-mail: jevtic@matf.bg.ac.yu