BERGMAN PROJECTIONS ON BESOV SPACES ON BALLS

H. TURGAY KAPTANOĞLU

Abstract. Extended Bergman projections from Lebesgue classes onto Besov spaces on the unit ball are defined and characterized. Right inverses and adjoints of the projections share the property that they are imbeddings of Besov spaces into Lebesgue classes via certain combinations of radial derivatives. Applications to the Gleason problem at arbitrary points in the ball, duality, and complex interpolation in Besov spaces are obtained. Results apply, in particular, to the Hardy space H^{2}, the Arveson space, the Dirichlet space, and the Bloch space.

1. Introduction

The inner product and the norm in ℂ^{N} are ⟨z, w⟩ = z_{1}\overline{w}_{1} + ⋯ + z_{N} \overline{w}_{N} and |z| = \sqrt{⟨z, z⟩}, where \overline{( )} denotes the complex conjugate (or the closure of a set if context requires). Let ν be the Lebesgue (volume) measure on the unit ball 𝔹 of ℂ^{N} normalized with ν(𝔹) = 1, which is the area measure on the unit disc 𝔻 when N = 1. We define on 𝔹 also the measures

dν_{c}(z) = (1 − |z|^{2})^{c} dν(z) (c ∈ ℝ),

which are finite only when c > −1. Unless otherwise specified or restricted, our main parameters are the following:

q ∈ ℝ,  0 < p ≤ ∞,  s ∈ ℂ,  σ = Re s,  t ∈ ℂ,  τ = Re t.

Let H(𝔹) denote the space of holomorphic functions on 𝔹. For q > −1, an f ∈ H(𝔹) belongs to the (weighted) Bergman space A_{q}^{p} whenever f lies in the Lebesgue class L^{p}(ν_{q}). The norm ∥f∥_{A_{q}^{p}} is simply the L^{p}(ν_{q}) norm of f, where we use the term norm even when 0 < p < 1. So the inclusion map i : A_{q}^{p} → L^{p}(ν_{q}) is an isometric imbedding.

Date: Received September 2, 2004; received in final form February 24, 2005. 1991 Mathematics Subject Classification. Primary 32A37, 47B38; Secondary 46E15,

46E20, 46E22, 32A36, 32A18, 32A35, 32A25, 32W99, 46B70.

Key words and phrases. Besov, Bergman, Bloch, Hardy, Arveson, Dirichlet space; reproducing kernel Hilbert space; radial derivative; Bergman projection; adjoint; Gleason problem; duality; complex interpolation.

The research of the author is partially supported by a Fulbright grant.

Bergman projections are the linear operators P_{s} defined for σ > −(N + 1) by

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

for suitable f. It is clear that P_{s}f is a member of H(𝔹). Complex powers are always understood to be principal branches.

The following result is classical; see [FR], [C], and [HKZ, §1.2] for example.

Theorem 1.1. For 1 ≤ p < ∞, P_{s} is a bounded operator from L^{p}(ν_{q}) onto A_{q}^{p} if and only if

(1)  q + 1 < p (σ + 1).

For such an s,

(2)  (P_{s} ∘ i)f = \frac{N!}{(1 + s)_{N}} f  (f ∈ A_{q}^{p}).

The inequality (1) necessitates σ > −1 since q > −1 for Bergman spaces. The expression (a)_{b} in (2) is the Pochhammer symbol given by

(3)  (a)_{b} = \frac{Γ(a + b)}{Γ(a)}

when a and a + b are off the pole set −ℕ of the gamma function Γ.

Besov spaces extend weighted Bergman spaces to all q. To define them, we first take a radial differential operator D_{s}^{t} of order t, and consider the linear transformation I_{s}^{t} defined for f ∈ H(𝔹) by

I_{s}^{t}f(z) = (1 − |z|^{2})^{t} D_{s}^{t}f(z).

We say an f ∈ H(𝔹) belongs to the Besov space B_{q}^{p} whenever I_{s}^{t}f lies in L^{p}(ν_{q}) for some s, t satisfying

(4)  \begin{cases}q + p τ > −1, & (0 < p < ∞); \\ τ > 0, & (p = ∞). \end{cases}

The L^{p}(ν_{q}) norm of any one of I_{s}^{t}f can be used as an equivalent norm of ∥f∥_{B_{q}^{p}}. It turns out that B_{q}^{p} = A_{q}^{p} for q > −1.

We also need an extended notion of Bergman projections to be able to handle all q. Consider the kernel

(5) H_{s}(λ) = \begin{cases}\frac{1}{(1 − λ)^{N+1+s}} = ∑_{k=0}^{∞} \frac{(N + 1 + s)_{k}}{k!} λ^{k}, & if σ >−(N +1); \\ \frac{_{2}F_{1}(1, 1; 1−N −s; λ)}{−N − s} = ∑_{k=0}^{∞} \frac{k! λ^{k}}{(−N −s)_{k+1}} , & if σ ≤−(N +1); \end{cases}

where _{2}F_{1} is the hypergeometric function; see [BB, p. 13]. With no restriction on s, we introduce the (extended) Bergman projections, also named P_{s}, by

P_{s}f(z) = ∫_{𝔹} H_{s}(⟨z, w⟩) (1 − |w|^{2})^{s} f(w) dν(w) (z ∈ 𝔹).

Our main result is the following generalization of Theorem 1.1.

Theorem 1.2. For 1 ≤ p ≤ ∞, P_{s} is a bounded operator from L^{p}(ν_{q}) onto B_{q}^{p} if and only if

(6)  \begin{cases}q + 1 < p (σ + 1), & (1 ≤ p < ∞); \\ σ > −1, & (p = ∞). \end{cases}

Given an s satisfying (6), if t satisfies (4), then

(7)  (P_{s} ∘ I_{s}^{t})f = \frac{N!}{(1 + s + t)_{N}} f = C_{st} f  (f ∈ B_{q}^{p}).

Note that (6) does not necessitate σ > −1 any more. On the other hand,

(6) and (4) together imply σ + τ > −1 so that (1 + s + t)_{N} never hits a pole of Γ.

The best partial result in this direction is [P, Theorem 3.11], in which s is restricted to s > −1; then s > −(N + 1) trivially and only the binomial part of the kernel (5) is used. Consequently the only-if part is also missing. The same restriction on s carries also to the right inverses given for P_{s}. A very special case of (7) is [Z2, Lemma 4.2.8]. Although H_{s} appears in [BB] in its full generality, this source considers projections only from differentiable classes onto Besov spaces.

We prove Theorem 1.2 in Section 5. Our proof is entirely different from that of [P]. We emphasize that Theorem 1.2 generalizes Theorem 1.1 also in the direction that i in (2) can be replaced by the more general I_{s}^{t} even when q > −1. We also find the adjoint of P_{s} in this section.

Section 4 is devoted to properties of Besov spaces that place the extended Bergman projections in context. This is required partly because the q we use in the definition of B_{q}^{p} is not standard. It is shown that H_{q}(⟨z, w⟩) is the reproducing kernel of B_{q}^{2}. The independence of B_{q}^{p} of the parameters s, t under (4) and their relation to Bergman and other spaces are established. Equation (2) in [Kap], where some of the results in this paper are announced, graphically shows the classification of B_{q}^{p} with our q.

The operators D_{s}^{t} are developed in Section 3. The are defined by using coefficient multipliers on the homogeneous expansions of functions in H(𝔹).

Section 2 is for notation and some preliminary formulas.

Sections 6, 7, and 8 are for applications of Theorem 1.2. First we solve the Gleason problem at an arbitrary point a ∈ 𝔹 in Besov spaces. Then we study the duality of B_{q}^{p} spaces under some new pairings. Finally we investigate complex interpolation in these spaces and identify some linear operators that leave them invariant.

Proofs of a few technical results are postponed to the Appendix. Particular cases of our results refer to the Hardy space H^{2} = B_{−1}^{2}, the Arveson space B_{−N}^{2}, and the Dirichlet space B_{−(N+1)}^{2}.

2. Notation and Preliminaries

Constants appearing in formulas are all denoted C although each might have a different value as context requires. They may depend on various parameters, but never on the functions in the formula in which they appear.

For fixed a, b, Stirling formula and (3) give

(8) \frac{Γ(c + a)}{Γ(c + b)} ∼ c^{a−b},  (c)_{a} ∼ c^{a},  \frac{(a)_{c}}{(b)_{c}} ∼ c^{a−b}  (Re c → ∞),

where x ∼ y means that |x/y| is bounded above and below by two positive constants that are independent of any parameter present (c here).

For 1 ≤ p < ∞, the symbol p′ denotes the exponent conjugate to p; that is, 1/p + 1/p′ = 1. The dual space X^{∗} of a Banach space X is the space of all bounded linear functionals on X.

We use multi-index notation in which α = (α_{1}, …, α_{N}) ∈ ℕ^{N} is an N-tuple of nonnegative integers, |α| = α_{1}+⋯+α_{N}, α! = α_{1}! ⋯ α_{N}!, z^{α} = z_{1}^{α_{1}} ⋯ z_{N}^{α_{N}}, and 0^{0} = 1. Then

(9) H_{s}(⟨z, w⟩) = \begin{cases}∑_{α} \frac{(N + 1 + s)_{|α|}}{α!} z^{α}\overline{w}^{α}, & if σ > −(N + 1); \\ ∑_{α} \frac{(|α|!)^{2}}{α! (−N − s)_{|α|+1}} z^{α}\overline{w}^{α}, & if σ ≤ −(N + 1). \end{cases}

When s = −(N + 1), (9) sums to

(10) H_{−(N+1)}(⟨z, w⟩) = ∑_{k=0}^{∞} \frac{1}{k + 1} ⟨z, w⟩^{k} = \frac{1}{⟨z, w⟩} log \frac{1}{1 − ⟨z, w⟩}.

By (8), the coefficient of λ^{k} in (5) is ∼ k^{N+σ} for large k. Thus (5) converges with λ = ⟨z, w⟩, in particular, when z, w ∈ 𝔹.

Let Σ be the Lebesgue (surface) measure on the boundary ∂𝔹 of 𝔹 normalized so that Σ(∂𝔹) = 1. The following result extends [R, Proposition 1.4.9] to p ≠ 2 and q ≠ 0. Its proof follows similar lines and is omitted.

Proposition 2.1. For a multi-index α, 0 < p < ∞, and σ > −1, we have

∫_{∂𝔹} |ζ^{α}|^{p} dΣ(ζ) = \frac{(N − 1)! ∏_{j=1}^{N} Γ(1 + α_{j}p/2)}{Γ(N + |α|p/2)}

and

∫_{𝔹} |z^{α}|^{p} (1 − |z|^{2})^{s} dν(z) = \frac{N! Γ(1 + s) ∏_{j=1}^{N} Γ(1 + α_{j}p/2)}{Γ(N + 1 + s + |α|p/2)}.

Remark 2.2. The case p = 2 of the second integral is [AK1, Lemma 1]. The similar orthogonality result ∫_{𝔹} z^{α}\overline{z}^{β} (1 − |z|^{2})^{s} dν(z) = 0 if α ≠ β appears in [FR, Proposition 2.4].

Proposition 2.3. For any c ∈ ℝ, L^{∞}(ν_{c}) = L^{∞}(ν).

Proof. It suffices to show that the null sets of ν_{c} and ν are the same. Note that ν_{c} is σ-finite. We have dν(z) = (1 − |z|^{2})^{−c} dν_{c}(z) with z ↦ (1 − |z|^{2})^{−c} integrable with respect to ν_{c}, and we have dν_{c}(z) = (1 − |z|^{2})^{c} dν(z) with z ↦ (1 − |z|^{2})^{c} locally integrable with respect to ν. Since neither measure has atoms, it follows that either measure is absolutely continuous with respect to the other. □

Now for a, b ∈ ℂ and suitable g, consider the operator

V_{b}^{a}g(z) = (1 − |z|^{2})^{a} ∫_{𝔹} \frac{(1 − |w|^{2})^{b}}{(1 − ⟨z, w⟩)^{N+1+a+b}} g(w) dν(w).

Theorem 2.4. (a) For 1 ≤ p ≤ ∞, V_{b}^{a} is bounded on L^{p}(ν_{c}) if and only if

− Re a < \frac{c + 1}{p} < Re b + 1.

(b) For 0 < p ≤ 1, if

0 < Re b + 1 − N(\frac{1}{p} − 1) and − Re a < \frac{c + 1}{p} < Re b + 1 − N(\frac{1}{p} − 1) ,

then V_{b}^{a} is a continuous map from L^{p}(ν_{c}) ∩ H(𝔹) to L^{p}(ν_{c}).

Proof. (a) This is essentially [HKZ, Theorem 1.9] for 1 ≤ p < ∞ and contained in [Z3, Theorem 9] for p = ∞.

(b) See the Appendix. □

For σ > −(N + 1), clearly P_{s} is the operator V_{s}^{0}. Then the first part of Theorem 1.1 follows immediately from Theorem 2.4 (a).

Remark 2.5. Theorem 2.4 is true also for operators of type V_{b}^{a} that have the form

Mg(z) = ∫_{𝔹} k(z, w) |g(w)| dν(w),

where k(z, w) is a measurable kernel satisfying

|k(z, w)| ≤ C \frac{(1 − |z|^{2})^{Re a} (1 − |w|^{2})^{Re b}}{|1 − ⟨z, w⟩|^{N+1+Re a+Re b}} .

3. Radial Differential Operators

Let f ∈ H(𝔹) be given by its homogeneous expansion f(z) = ∑_{k=0}^{∞} f_{k}(z), where f_{k} is a homogeneous polynomial of degree k. The radial derivative at z of f is

Rf(z) = ∑_{m=1}^{N} z_{m} \frac{∂f}{∂z_{m}} (z) = ∑_{k=1}^{∞} k f_{k}(z).

In particular, R(z^{α}) = |α|z^{α} and R(⟨z, w⟩) = ⟨z, w⟩, where R acts on the holomorphic variable z. What is nice about Rf is that it is also holomorphic and dominates the derivatives of f in tangential directions; see [R, §6.4]. By imitating the passage across σ = −(N + 1) in (5) and following [AU, §3], we extend R to arbitrary orders.

Definition 3.1. Let f ∈ H(𝔹). We define D_{s}^{t}f = ∑_{k=0}^{∞} _{s}^{t}d_{k} f_{k}, where

_{s}^{t}d_{k} = \begin{cases}\frac{(N + 1 + s + t)_{k}}{(N + 1 + s)_{k}} , & if σ >−(N +1), σ+τ >−(N +1); \\ \frac{(N +1+s+t)_{k} (−(N +s))_{k+1}}{(k!)^{2}} , & if σ ≤−(N +1), σ+τ >−(N +1); \\ \frac{(k!)^{2}}{(N +1+s)_{k} (−(N +s+t))_{k+1}} , & if σ >−(N +1), σ+τ ≤−(N +1); \\ \frac{(−(N + s))_{k+1}}{(−(N + s + t))_{k+1}} , & if σ ≤−(N +1), σ+τ ≤−(N +1). \end{cases}

Clearly, D_{s}^{t}f ∈ H(𝔹), D_{s}^{0} = I, D_{s}^{t}(1) = _{s}^{t}d_{0},

(11)  D_{−N}^{1} = R + I,  _{s}^{t}d_{k} ≠ 0,  and  D_{s}^{t}(z^{α}) = _{s}^{t}d_{|α|}z^{α}.

Moreover, by (8),

(12)  _{s}^{t}d_{k} ∼ k^{τ}  (k → ∞).

Theorem 3.2. Any D_{s}^{t} is a continuous operator on H(𝔹).

Proof. This is one direction of [Ara, Theorem 5] using the estimate (12). □

Hence identities for D_{s}^{t} can be proved by checking their action on z^{α} since {z^{α}} generates H(𝔹). So by (5) we have the important identity

(13)  D_{s}^{t}H_{s}(⟨z, w⟩) = H_{s+t}(⟨z, w⟩),

where D_{s}^{t} acts on the holomorphic variable z. The properties (11), (12), and (13) allow us to state the following.

Proposition 3.3. Any D_{s}^{t} is a radial differential operator of order t.

The parameter s does not affect the order and is there for convenience in proofs. These operators are truly differential when t is a positive integer, and integral when t is negative. By (11), any D_{s}^{t} is a bijection on H(𝔹) and thus invertible. A case by case checking as in Definition 3.1 yields that

(14)  D_{s+t}^{r}D_{s}^{t} = D_{s}^{t+r}.

This formula for s > −1, s + t > −1, and r + s + t > −(N + 1) appears in [P, Remark 3.6 (b)]. The two-sided inverse of D_{s}^{t} is obtained by having D_{s}^{0} = I on the right in (14). Therefore

(15)  (D_{s}^{t})^{−1} = D_{s+t}^{−t}.

Consequently, if f ∈ H(𝔹) and D_{s}^{t}f ≡ 0 or I_{s}^{t} ≡ 0 for some s, t, then f ≡ 0.

Let’s note in passing that the operators R_{a} = (R − aI)^{−1} = −a^{−1}D_{−(N+a)}^{−1} satisfy the resolvent equation R_{a} − R_{b} = (a − b)R_{a}R_{b}.

Theorem 3.4. Suppose b satisfies Re b > −1 and Re(u − t + b) > −(N + 1). Let f ∈ H(𝔹). Then (1 − |z|^{2})^{u−t}D_{r}^{u}(f) = M(D_{s}^{t}(f)) for an operator M of type V_{b}^{u−t}. In particular, D_{r}^{t}(f) = M(D_{s}^{t}(f)) for an operator M of type V_{b}^{0}.

Proof. See the Appendix. □

The parameter b can be chosen at will as long as it satisfies the two inequalities stated. This provides great flexibility as we show next.

4. Besov Spaces

We first make sure that B_{q}^{p} spaces are well-defined.

Theorem 4.1. Suppose f ∈ H(𝔹), q ∈ ℝ, and r, s, t, u ∈ ℂ.

(a) Let 0 < p < ∞. For q+p Re t > −1 the function I_{s}^{t}f belongs to L^{p}(ν_{q}) if and only if for some r and u satisfying q +p Re u > −1 the function I_{r}^{u}f belongs to L^{p}(ν_{q}), and the L^{p}(ν_{q}) norms of these two functions are equivalent.

(b) Let p = ∞. For Re t > 0 the function I_{s}^{t}f is bounded on 𝔹 if and only if for some r and Re u > 0 the function I_{r}^{u}f is bounded on 𝔹, and the supremums of these two functions on 𝔹 are equivalent norms for f.

Note that there is absolutely no restriction on the lower parameters r, s of the differential operators.

Proof. (a) The belongings I_{s}^{t}f ∈ L^{p}(ν_{q}) and I_{r}^{u}f ∈ L^{p}(ν_{q}) can be restated in the forms D_{s}^{t}f ∈ L^{p}(ν_{q+p τ}) and (1 − |z|^{2})^{u−t} D_{r}^{u}f(z) ∈ L^{p}(ν_{q+p τ}), respectively. So we apply Theorem 2.4 (a) or (b) with c = q + p τ and a = u − t. These values satisfy the first inequalities there since q + p Re u > −1. We take a sufficiently large real b in Theorem 3.4 which also satisfies the second inequality in Theorem 2.4 (a) or (b). What Theorem 2.4 says then is that if I_{s}^{t}f ∈ L^{p}(ν_{q}), then I_{r}^{u}f ∈ L^{p}(ν_{q}).

In the opposite direction, we interchange the roles of the pairs s, t and r, u, and use the condition q + p Re t > −1.

(b) By choosing a large enough b in Theorem 3.4 and then replacing b by b − t, we can write I_{r}^{u}f = (M_{1} ∘ I_{s}^{t})f for an operator M_{1} of type V_{b−t}^{u}. Interchanging the pair s, t with r, u, we get an operator M_{2} that is of type V_{b−u}^{t}. We take once c = 0, a = u, and b large, and again c = 0, a = t, and b large in Theorem 2.4 (a). The inequalities on a and c are satisfied by Re u > 0 and Re t > 0 in turn. Then I_{r}^{u}f is uniformly bounded on 𝔹 if and only if I_{s}^{t}f is uniformly bounded on 𝔹. □

Corollary 4.2. The space B_{q}^{p} is independent of the particular choice of s, t as long as (4) holds. The L^{p}(ν_{q}) norms of I_{s_{1}}^{t_{1}}f and I_{s_{2}}^{t_{2}}f are equivalent as long as (4) is satisfied by t_{1} and t_{2}.

Hence, if (4) holds, the map I_{s}^{t} : B_{q}^{p} → L^{p}(ν_{q}) is an isometric imbedding modulo the equivalence of norms much like i is for Bergman spaces.

The reference [BB] uses the differential operators (R + sI)^{t} instead of D_{s}^{t}, and as remarked there (p. 41), the corresponding spaces are the same. Thus our B_{q}^{p} spaces are the holomorphic Sobolev spaces A_{q+pt+1,t}^{p} of [BB], which places the restriction q + pt + 1 > 0 as in (4). In fact, our Corollary 4.2 is contained in [BB, Theorem 5.12 (i)], and we could have refered to it instead of proving Theorems 3.4 and 4.1.

However, we work through the details, because our relatively restricted approach makes the exposition simpler, and our definition of Besov spaces uses the same parameters as those of weighted Bergman spaces, but on I_{s}^{t}f rather than f, making the roles of various functions and parameters clearer. Also Corollary 4.2 precedes Theorem 1.2 unlike in many treatments of the subject; see [HKZ, Proposition 1.11] for comparison.

When q > −1 and 0 < p < ∞, (4) is satisfied by t = 0 independently of p; then the spaces B_{q}^{p} and the weighted Bergman spaces A_{q}^{p} coincide. On the other hand, for such q and p, (4) is satisfied for certain t with τ < 0 too. Then Corollary 4.2 gives a new characterization of weighted Bergman spaces using integrals of the functions contained in them rather than their derivatives.

In contrast, when q ≤ −1 and 1 ≤ p < ∞, t = −q always satisfies (4) independently of p. Then we see the holomorphic Besov spaces B_{p}(𝔹) of [Z3] are our B_{−(N+1)}^{p} spaces for such p. This value of t is also used in the pairings of Theorems 7.1 and 7.2 when identifying the dual of B_{q}^{p}.

When p = ∞, Corollary 4.2 combined with Proposition 2.3 says that the spaces B_{q}^{∞} are the same for all q. Using t = 1, by [AFJP, Theorem 2], we see that this space is the Bloch space B. The subspace B_{0} of B consisting of f for which I_{s}^{t}f restricts to 0 on ∂𝔹 for some t with τ > 0 is called the little Bloch space. These results are in [Z4, Theorem 5] for t > N.

Let’s denote by BSV_{p}^{v} the diagonal Besov spaces as defined in [AFJP, Remark 5.2], [P, Definition 1.1], or [AC], where the expression diagonal refers to the equality of two parameters out of three in the full Besov-space family. A straightforward checking of parameters yields that B_{q}^{p} and BSV_{p}^{−(q+1)/p}, and conversely BSV_{p}^{v} and B_{−(vp+1)}^{p} coincide.

Each B_{q}^{2} space, being a Hilbert space (see [BB]), is equipped with several equivalent inner products

(16)  _{q}[f, g]_{s}^{t} = ∫_{𝔹} I_{s}^{t}f \overline{I_{s}^{t}g} dν_{q},

one for each s, t satisfying q+2τ > −1. Using_{q}[·, ·]_{0}^{0} is standard with Bergman spaces (q > −1). The monomials {z^{α}} form an orthogonal set with respect to each of these inner products by Remark 2.2.

The following result clearly explains our choice of kernel in defining the extended Bergman projections. Hypergeometric kernels are not rare; see [Kar].

Theorem 4.3. Each B_{q}^{2} space is a reproducing kernel Hilbert space. The reproducing kernel of B_{q}^{2} is K_{q}(z, w) = H_{q}(⟨z, w⟩).

Proof. See [BB, pp. 13–14]. □

The spaces B_{q}^{2} are known as Dirichlet-type spaces, with B_{−(N+1)}^{2} being the Dirichlet space D by (10), B_{−1}^{2} the Hardy space H^{2}, and B_{−N}^{2} the Arveson space A. The space A is important in operator theory (see, for example, [Arv], [AM], [AK2]) due to a universal property of its kernel in Nevanlinna-Pick interpolation. A slightly different description of B_{q}^{2} spaces for q ≥ −(N + 1) that do not involve any derivatives on the functions is given in [AK3, Proposition 2.1]. For similar results on bounded symmetric domains essentially for q > −(N + 1), see [Y].

5. Bergman Projections

We start by deriving an integral formula for D_{s}^{t}.

Lemma 5.1. If σ > −1 and f ∈ H(𝔹), then for any t,

D_{s}^{t}f(z) = \frac{(s + 1)_{N}}{N!} lim_{r→1^{−}} ∫_{𝔹} H_{s+t}(⟨z, w⟩) (1 − |w|^{2})^{s} f(rw) dν(w).

Proof. This is a direct computation using f(z) = z^{α}, (9), and Proposition 2.1. To finish the proof, we invoke Theorem 3.2. □

Hence I_{s}^{t} is a constant multiple of V_{s}^{t} on suitable f for σ > −1. The more precise relationship (17) below complements this. The restriction on s can be weakened using the method of [AK1, §5].

As a matter of fact, our differential operators are defined in other sources using such integrals with binomial kernel and hence for limited t. Up to constant multiples, R_{s}^{μ} of [P] is our D_{s}^{μ} for s > −1 and μ + s > −(N + 1); D^{s} and D_{s} of [Z3] are our D_{0}^{s} and D_{s}^{−s} for s > −1; D^{α,β} and D_{α,β} of [Z4] are our D_{β}^{α} and D_{α+β}^{−α} for α > −1 and β > −1.

Proof of Theorem 1.2. Let φ ∈ L^{p}(ν_{q}); it is clear that P_{s}φ ∈ H(𝔹). We pick a t satisfying (4) and σ +τ > −(N +1), and apply I_{s}^{t} to P_{s}φ. By differentiating under the integral sign and employing (13), we obtain

I_{s}^{t}(P_{s}φ)(z) = (1 − |z|^{2})^{t} ∫_{𝔹} \frac{(1 − |w|^{2})^{s}}{(1 − ⟨z, w⟩)^{N+1+s+t}} φ(w) dν(w) = V_{s}^{t}φ(z).

By Theorem 2.4 (a) and Corollary 4.2, P_{s}φ lies in B_{q}^{p} if and only if (6) holds. Now (6) and (4) together give σ + τ > −1, so that the extra assumption σ + τ > −(N + 1) above is not necessary.

Further, ∥P_{s}φ∥_{B_{q}^{p}} = ∥V_{s}^{t}φ∥_{L^{p}(ν_{q})} ≤ ∥V_{s}^{t}∥ ∥φ∥_{L^{p}(ν_{q})}. So ∥P_{s}∥ ≤ ∥V_{s}^{t}∥ whenever P_{s} is bounded.

Now let s be as above, and pick a possibly different t satisfying (4). Then

P_{s}(I_{s}^{t}f)(z) = ∫_{𝔹} H_{s}(⟨z, w⟩) (1 − |w|^{2})^{s+t} D_{s}^{t}f(w) dν(w)

= \frac{N!}{(1 + s + t)_{N}} D_{s+t}^{−t}D_{s}^{t}f(z) = \frac{N!}{(1 + s + t)_{N}} f(z)

by Lemma 5.1 and (15). Lemma 5.1 applies, because (6) and (4) together imply σ + τ > −1, which also makes sure (1 + s + t)_{N} is always defined. □

Remark 5.2. The proof of Theorem 1.2 reveals the following interesting fact. When 1 ≤ p ≤ ∞, if V_{s}^{t} is bounded on L^{p}(ν_{q}), then it factors through B_{q}^{p} as

(17)  V_{s}^{t} = I_{s}^{t} ∘ P_{s}.

Theorem 1.2 and (17) can be summarized in a commutative diagram:

\xymatrix{L^{p}(ν_{q}) \ar[d]_{P_{s}} \ar[r]^{V_{s}^{t}} & L^{p}(ν_{q}) \ar[d]^{P_{s}} \\ B_{q}^{p} \ar[ur]^{I_{s}^{t}} \ar[r]_{C_{st}I} & B_{q}^{p}}

Right inverses similar to I_{s}^{t} appear in limited cases also in [C] and [BB, Corollary 6.5], the latter for a different kind of projection.

The case p = ∞ is covered by Theorem 1.2, but deserves separate mention. It is more general than [C, Theorem 2], because it provides the only-if part and a whole family of right inverses.

Corollary 5.3. The Bergman projection P_{s} maps L^{∞}(ν) boundedly onto B if and only if σ > −1. If also τ > 0, then (7) holds for all f ∈ B.

Let’s isolate one other important case when N = 1. The operator P_{s} maps L^{2}(ν_{−1}) boundedly onto H^{2} = A = B_{−1}^{2} if and only if σ > −1, and if τ > 0, then (P_{s} ∘ I_{s}^{t})f = Cf for f ∈ H^{2}. Letting s = 0 and t = 1, for f ∈ H^{2} and f(0) = 0, (7) amounts to the representation

f(z) = C ∫_{𝔻} \frac{1 − |w|^{2}}{(1 − ⟨z, w⟩)^{2}} f′(w) dν(w).

Our purpose now is to compute the adjoint of P_{s}. But first, for 1 ≤ p < ∞, (L^{p}(ν_{q}))^{∗} = L^{p′}(ν_{q}) under the pairing _{q}[·, ·]_{0}^{0}, and it is shown in Theorem 7.1 below that (B_{q}^{p})^{∗} = B_{q}^{p′} under the general pairings given in (20). Given s satisfying (6) and 1 ≤ p < ∞, by an adjoint of P_{s} : L^{p}(ν_{q}) → B_{q}^{p}, we mean a linear operator P_{s}^{∗} : B_{q}^{p′} → L^{p′}(ν_{q}) such that _{q}[P_{s}f, g]_{s,q+t}^{t,−q+s} = _{q}[f, P_{s}^{∗}g]_{0}^{0} for some t satisfying (4) and for all f ∈ L^{p}(ν_{q}) and g ∈ B_{q}^{p′}.

Theorem 5.4. The adjoint of P_{s} as defined above is P_{s}^{∗} = C_{st} I_{q+t}^{−q+s}.

Proof. Expanding the definition of P_{s}^{∗} in integrals and using (13) and Fubini theorem, we obtain

P_{s}^{∗}g(z) = (1 − |z|^{2})^{−q+s} ∫_{𝔹} H_{s+t}(⟨z, w⟩) (1 − |w|^{2})^{s+t} D_{q+t}^{−q+s}g(w) dν(w),

which is bounded if and only if (6) and (4) because of P_{s}. But for such s, t, the kernel H_{s+t} is binomial. Thus when P_{s}^{∗} is bounded, we have

P_{s}^{∗}g(z) = (1 − |z|^{2})^{−q+s} ∫_{𝔹} \frac{(1 − |w|^{2})^{q+t}}{(1 − ⟨z, w⟩)^{N+1+s+t}} I_{q+t}^{−q+s}g(w) dν(w)
= V_{q+t}^{−q+s}(I_{q+t}^{−q+s}g)(z).

This form of P_{s}^{∗} is entirely similar to and generalizes even for q > −1 the one given in [C, Corollary 7]. The only notable difference is the presence of I_{q+t}^{−q+s}, which is expected by our definition of Besov spaces and which can be checked to imbed B_{q}^{p′} into L^{p′}(ν_{q}) under (6) and (4) with p′ in place of p. The boundedness condition of P_{s}^{∗} can now be read off also directly from Theorem 2.4 (a). By factoring V_{q+t}^{−q+s} as in (17) and using Theorem 1.2, we obtain the desired result. □

Note that no matter what t is used in the pairing of (20) to define the adjoint, P_{s}^{∗} turns out to be essentially the same since the lower parameter of radial derivatives is a mere technicality. We can take t = −q+s for symmetry.

Bounded projections from Lebesgue classes keep on playing important roles in the theory of Bergman-type spaces; see [CKY] for example.

6. Gleason Problem

Let X be a space of functions defined, say, on 𝔹. Given a ∈ 𝔹 and f ∈ X, the Gleason problem is to determine whether f_{1}, …, f_{N} ∈ X exist such that

f(z) − f(a) = ∑_{m=1}^{N}(z_{m} − a_{m}) f_{m}(z) (z ∈ 𝔹).

The point here is that f_{1}, …, f_{N} must be in the same space as f. Explicit solutions are given in [Z1] and [C] in Bergman spaces A_{q}^{p} for 1 ≤ p < ∞ at a = 0. In [AK3, §3], it is proved that solutions exist in Dirichlet-type spaces B_{q}^{2} at arbitrary a ∈ 𝔹. For further recent results on the Gleason problem and their applications to interpolation, see also [AK2], [CKY], and [AD].

In this section, we give explicit solutions to the Gleason problem in B_{q}^{p} for all q and 1 ≤ p ≤ ∞, including p = ∞, at an arbitrary point a ∈ 𝔹. Our solutions take the modification in [AK3] of the Ahern-Schneider solution (see [R, §6.6.2] and [AS]) one step further by employing Theorems 1.2 and 2.4.

We need integer values of s that satisfy (6). If q > −1, then s = ⌈q + 1⌉, the least integer greater than or equal to q + 1, works for all 1 ≤ p < ∞. If q ≤ −1, then s = 0 works for all 1 ≤ p ≤ ∞, including p = ∞. In any case, s ≥ 0 and H_{s} is binomial.

Theorem 6.1. Given q, 1 ≤ p ≤ ∞, and a ∈ 𝔹, there exist bounded linear operators _{a}G_{1}, …,_{a}G_{N} : B_{q}^{p} → B_{q}^{p} satisfying

(18)  f(z) − f(a) = ∑_{m=1}^{N}(z_{m} − a_{m})_{a}G_{m}f(z)  (f ∈ B_{q}^{p}, z ∈ 𝔹).

Proof. Let s > −(N + 1) be an integer satisfying (6), t satisfy (4), and define

aG_{m}f(z) = \frac{1}{C_{st}} ∫_{𝔹} \frac{H_{s}(⟨z, w⟩) − H_{s}(⟨a, w⟩)}{⟨z − a, w⟩} \overline{w}_{m} (1 − |w|^{2})^{s} I_{s}^{t}f(w) dν(w)

for m = 1, …, N and f ∈ B_{q}^{p}. The crucial difference from the Ahern-Schneider solution is the presence of the imbedding I_{s}^{t}. Then the right side of (18) is

\frac{1}{C_{st}} ∫_{𝔹}(H_{s}(⟨z, w⟩) − H_{s}(⟨a, w⟩)) (1 − |w|^{2})^{s} I_{s}^{t}f(w) dν(w)

= \frac{1}{C_{st}} (P_{s}(I_{s}^{t}f)(z) − P_{s}(I_{s}^{t}f)(a)) = f(z) − f(a)

by Theorem 1.2. Hence _{a}G_{1}f, …,_{a}G_{N}f satisfy (18). It remains to show that aG_{m} is bounded.

Using that s is an integer and the finite binomial expansion, we can write

aG_{m}f(z) = \frac{1}{C_{st}} ∑_{j=0}^{N+s} ∫_{𝔹} \frac{\overline{w}_{m} (1 − |w|^{2})^{s}}{(1 − ⟨z, w⟩)^{N+1+s−j} (1 − ⟨a, w⟩)^{1+j}} I_{s}^{t}f(w) dν(w).

Take a u satisfying (4) with Re u in place of τ, and apply I_{s−j}^{u} to the jth term in the sum, which we call T_{j}f(z), j = 0, …, N + s ≥ 1. By (13), the result is

\frac{1}{C_{st}} (1 − |z|^{2})^{u} ∫_{𝔹} \frac{\overline{w}_{m} (1 − |w|^{2})^{s}}{(1 − ⟨z, w⟩)^{N+1+s+u−j} (1 − ⟨a, w⟩)^{1+j}} I_{s}^{t}f(w) dν(w)

= \frac{1}{C_{st}} (1 − |z|^{2})^{u} ∫_{𝔹} \frac{(1 − |w|^{2})^{s}}{(1 − ⟨z, w⟩)^{N+1+s+u}} \frac{\overline{w}_{m} (1 − ⟨z, w⟩)^{j}}{(1 − ⟨a, w⟩)^{1+j}} I_{s}^{t}f(w) dν(w)

The second fraction is bounded for all z, w ∈ 𝔹 for fixed a ∈ 𝔹. Hence

|I_{s−j}^{u}(T_{j}f)(z)| ≤ C(1 − |z|^{2})^{u} ∫_{𝔹} \frac{(1 − |w|^{2})^{s}}{|1 − ⟨z, w⟩|^{N+1+s+u}} |I_{s}^{t}f(w)| dν(w)
= T (I_{s}^{t}f)(z),

where T is an operator of type V_{s}^{u} and is bounded on L^{p}(ν_{q}), which contains I_{s}^{t}f, if and only if (4) and (6) with Re u in place of τ by Theorem 2.4 (a). Thus T_{j} is bounded on B_{q}^{p}. Therefore _{a}G_{m} is a bounded operator on B_{q}^{p} being a finite sum of the T_{j}. □

7. Duality

It should be well-known that (BSV_{p}^{v})^{∗} = BSV_{p′}^{v}, which is equivalent to that (B_{q}^{p})^{∗} = B_{q}^{p′}. Here we derive this relationship from Theorem 1.2 and give a whole family of pairings that realize it. Some of these pairings have already been used in Section 5 in finding P_{s}^{∗}. Our results give some new pairings also for the classical duality (A_{q}^{p})^{∗} = A_{q}^{p′} of weighted Bergman spaces.

Theorem 7.1. Let q ≤ −1 and 1 ≤ p < ∞. The dual space (B_{q}^{p})^{∗} can be identified with B_{q}^{p′} under the pairing _{q}[·, ·]_{0}^{−q}. In particular, the Bloch space B is the dual space of all B_{q}^{1}. Explicitly, every g ∈ B_{q}^{p′} induces a bounded linear functional M_{g} on B_{q}^{p} via

(19)  M_{g}(f) = ∫_{𝔹} I_{0}^{−q}f \overline{I_{0}^{−q}g} dν_{q},

and every bounded linear functional M on B_{q}^{p} is M_{g} for a unique g ∈ B_{q}^{p′}.

Proof. First let g ∈ B_{q}^{p′}. Apply Hölder inequality to the right side of (19) with p and p′ to obtain |M_{g}(f)| ≤ ∥f∥_{B_{q}^{p}} ∥g∥_{B_{q}^{p′}} for all f ∈ B_{q}^{p}. This gives

that B_{q}^{p′} ⊂ (B_{q}^{p})^{∗} and ∥M_{g}∥ ≤ ∥g∥_{B_{q}^{p′}}.

Next let M be a bounded linear functional on B_{q}^{p}. By (4) , I_{0}^{−q} imbeds B_{q}^{p} in L^{p}(ν_{q}). Let Q be the restriction of P_{0} to I_{0}^{−q}(B_{q}^{p}). Then M ∘ Q is a bounded linear functional on I_{0}^{−q}(B_{q}^{p}) by Theorem 1.2. By the Hahn-Banach theorem, M ∘ Q extends to a bounded linear functional L on L^{p}(ν_{q}) with ∥L∥ = ∥M ∘Q∥. By the Riesz representation theorem, there exists a unique φ in L^{p′}(ν_{q}) such that L(h) = ∫_{𝔹} h \overline{φ} dν_{q} for all h ∈ L^{p}(ν_{q}) and ∥L∥ = ∥φ∥_{B_{q}^{p′}}.

Taking h = I_{0}^{−q}f = F for f ∈ B_{q}^{p} gives M(f) = ∫_{𝔹} I_{0}^{−q}f \overline{φ} dν_{q}. We can replace f by CP_{0}(I_{0}^{−q}f) = CP_{0}F by Theorem 1.2. Put g = CP_{0}φ. Then g is unique and clearly g ∈ B_{q}^{p′}. Now we have

M(f) = C ∫_{𝔹} I_{0}^{−q}(P_{0}F) \overline{φ} dν_{q} = C ∫_{𝔹} \overline{φ(z)} ∫_{𝔹} \frac{F (w)}{(1 − ⟨z, w⟩)^{N+1−q}} dν(w) dν(z)

= C ∫_{𝔹} F (w) (1 − |w|^{2})^{−q} ∫_{𝔹} \frac{\overline{φ(z)}}{(1 − ⟨z, w⟩)^{N+1−q}} dν(z) dν_{q}(w)

= C ∫_{𝔹} F \overline{V_{0}^{−q}φ} dν_{q} = C ∫_{𝔹} F \overline{I_{0}^{−q}(P_{0}φ)} dν_{q} = ∫_{𝔹} I_{0}^{−q}f \overline{I_{0}^{−q}g} dν_{q}

by (13), (17), and the Fubini theorem. The norms satisfy

∥g∥_{B_{q}^{p′}} ≤ C∥P_{0}∥ ∥φ∥_{L^{p′} (ν_{q})} = C∥P_{0}∥ ∥L∥ ≤ C∥P_{0}∥ ∥M∥ ∥Q∥ ≤ C∥P_{0}∥^{2} ∥M∥.

So the norms of g and M need not be equal; in other words, the identification of dual spaces may not be isometric. □

We similarly have the following duality whose proof is omitted.

Theorem 7.2. Let q ≤ −1. The dual space B_{0}^{∗} of the little Bloch space can be identified with each of B_{q}^{1} under the pairing _{q}[·, ·]_{0}^{−q}.

The cases q = −(N + 1) of Theorems 7.1 and 7.2 are with respect to the invariant measure and in [Z3, Theorems 17 and 18]. The corresponding identifications for q > −1 concern the weighted Bergman spaces and can be found essentially in [HKZ, Theorem 1.16, Theorem 1.21, p. 23].

Remark 7.3. Retracing the proofs of Theorems 7.1 and 7.2, we see that the stated dualities are realized under each of the pairings

(20)  _{q}[f, g]_{s,q+t}^{t,−q+s} = ∫_{𝔹} I_{s}^{t}f \overline{I_{q+t}^{−q+s}g} dν_{q}

now for all q, where s, t satisfy (6) and (4).

8. Complex Interpolation

Another application of Theorem 1.2 is that we can apply complex interpolation between B_{q}^{p_{0}} and B_{q}^{p_{1}}. These are compatible spaces, because they are contained in A_{q}^{1} if q > −1 and in A^{1} if q ≤ −1 as seen in [Kap, (2)]. So for q ≤ −1, for example, we have B_{q}^{p_{0}} ∩B_{q}^{p_{1}} ⊂ B_{q}^{p} ⊂ B_{q}^{p_{0}} ∪B_{q}^{p_{1}} ⊂ B_{q}^{p_{0}} +B_{q}^{p_{1}} ⊂ A^{1}, and the inclusions are dense since polynomials are dense in all B_{q}^{p}; see [BB, Lemma 5.2]. For relevant definitions and notation such as [·, ·]_{θ}, ∥ · ∥_{θ}, F, or ∥ · ∥_{F}, see [Z2, §2.2]. The notation [X, Y]_{θ} here denotes the complex interpolation space between the Banach spaces X and Y, and must not be confused with the inner products in (16).

Theorem 8.1. Suppose 1 ≤ p_{0} < p < p_{1} ≤ ∞ with 1/p = (1 − θ)/p_{0} + θ/p_{1} for some θ ∈ (0, 1). Then [B_{q}^{p_{0}}, B_{q}^{p_{1}}]_{θ} = B_{q}^{p}.

Proof. Given f ∈ B_{q}^{p}, we pick positive s, t satisfying (6) and (4) with p_{0} (the smallest), and set φ = I_{s}^{t}f ∈ L^{p}(ν_{q}). We know by Theorem 1.2 that P_{s}φ = Cf and ∥φ∥_{L^{p}(ν_{q})} = C ∥f∥_{B_{q}^{p}}. For ζ in the strip \overline{S} = \{ ζ ∈ ℂ : 0 ≤ Re ζ ≤ 1 \} and z ∈ 𝔹, we define

Φ_{ζ}(z) = \frac{φ(z)}{|φ(z)|} |φ(z)|^{p(\frac{1−ζ}{p_{0}} + \frac{ζ}{p_{1}} )}

and F_{ζ} = P_{s}Φ_{ζ} as in the proof of [Z2, Theorem 5.3.8], which takes care of the case q = −(N + 1). Both Φ and F are continuous and bounded for ζ ∈ \overline{S}, holomorphic for ζ ∈ S, Φ_{θ} = φ, and F_{θ} = f. On the left boundary of S, |Φ_{iy}(z)| = |φ(z)|^{p/p_{0}}, ∥Φ_{iy}∥_{L^{p_{0}} (ν_{q})}^{p_{0}} = ∥φ∥_{L^{p}(ν_{q})}^{p}, and I_{s}^{t}F_{iy}(z) = MΦ_{iy}(z), where M is an operator of type V_{s}^{t} by (17) and bounded on L^{p_{0}}(ν_{q}). Thus ∥F_{iy}∥_{B_{q}^{p_{0}}} ≤ ∥M∥ ∥φ∥_{L^{p}(ν_{q})}^{p/p_{0}} for all y ∈ ℝ. Similarly on the right boundary of

S, ∥F_{1+iy}∥_{B_{q}^{p_{1}}} ≤ ∥M∥ ∥φ∥_{L^{p}(ν_{q})}^{p/p_{1}} for all y ∈ ℝ. Thus ∥f∥_{θ} ≤ ∥F ∥_{F} ≤ C ∥f∥_{B_{q}^{p}} and f ∈ [B_{q}^{p_{0}}, B_{q}^{p_{1}}]_{θ}.

Conversely let f ∈ [B_{q}^{p_{0}}, B_{q}^{p_{1}}]_{θ}. There is an F_{ζ} such that F_{iy} ∈ B_{q}^{p_{0}}, F_{1+iy} ∈ B_{q}^{p_{1}}, and F_{θ} = f. Put Φ_{ζ} = I_{s}^{t}F_{ζ}. But then Φ_{iy} ∈ L^{p_{0}}(ν_{q}) and Φ_{1+iy} ∈ L^{p_{1}}(ν_{q}). Applying [Z2, Theorem 2.2.6] yields Φ_{θ} ∈ L^{p}(ν_{q}). Finally Theorem 1.2 gives P_{s}Φ_{θ} = P_{s}(I_{s}^{t}F_{θ}) = CF_{θ} = Cf ∈ B_{q}^{p}. □

Note that the interpolating space between B_{q}^{p_{0}} and B_{q}^{p_{1}} is not the same as the interpolating space between BSV_{p_{0}}^{v} and BSV_{p_{1}}^{v}.

Let Aut(𝔹) be the group of all automorphisms of 𝔹, that is, one-to-one holomorphic maps of 𝔹 onto 𝔹. We recall that Aut(𝔹) acts transitively on 𝔹, and for each ψ ∈ Aut(𝔹), there is a unique unitary transformation U of ℂ^{N} such that ψ = U ∘ ϕ_{a}, where a = ψ^{−1}(0) and ϕ_{a} is an involutive Möbius transformation as explained in detail in [R, §2.2].

The measures ν_{q} have certain invariance properties. For ψ ∈ Aut(𝔹), define the operators

(Z_{ψ}^{q}f)(z) = f(ψ(z)) |Jψ(z)|^{2(1+ \frac{q}{N+1})},

where Jψ is the complex Jacobian of ψ. Then

(21)  ∫_{𝔹}(Z_{ψ}^{q}f) dν_{q} = ∫_{𝔹} f dν_{q}

for f ∈ L^{1}(ν_{q}). This is stated in [BB, (3.5)] for q > −1, but it holds for all q. It reduces to the well-known invariance under compositions with members of Aut(𝔹) (Möbius-invariance) of ν_{−(N+1)}. Further, it is shown in [BB, Theorem 3.3] using (21) 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})} (ψ ∈ Aut(𝔹)).

Lastly we apply interpolation methods to extend this result to certain other Besov spaces.

Theorem 8.2. Suppose 2 ≤ p ≤ ∞, −(N + 1) ≤ q ≤ −1, and ψ ∈ Aut(𝔹). Then U_{ψ}^{p,q} is a bounded linear transformation on B_{q}^{p}.

Proof. We know U_{ψ}^{2,q} is a unitary transformation on B_{q}^{2} for q > −(N + 1) by [BB, Theorem 1.10]. This holds also for q = −(N + 1) since it is equivalent to the Möbius invariance of the Dirichlet space D. We also know U_{ψ}^{∞,q} maps B onto itself isometrically, which is actually Möbius invariance. To interpolate between these two ends, for ζ ∈ \overline{S}, we let

T_{ζ}f(z) = f(ψ(z)) (Jψ(z))^{(1+ \frac{q}{N+1})(1−ζ)}

and 1/p = (1 − θ)/2. Then T_{θ}f = U_{ψ}^{p,q}f. Let ω(z) = (arg(Jψ(z)))^{1+ \frac{q}{N+1}} . The Jacobian Jψ(z), being the determinant of a linear map on ℂ^{N}, the Jacobian matrix, has its argument bounded as z varies in 𝔹. So |ω(z)| ≤ C for all z ∈ 𝔹. Then ∥T_{iy}f∥_{B_{q}^{2}} ≤ e^{Cy} ∥f∥_{B_{q}^{2}} and ∥T_{1+iy}f∥_{B} ≤ e^{Cy} ∥f∥_{B} for all real y.

Now we proceed as in the proof of [Z2, Theorem 2.2.4]. Given f ∈ B_{q}^{p}, there is an F ∈ F such that F_{θ} = f and ∥F ∥_{F} ≤ ∥f∥_{B_{q}^{p}} + ε by Theorem 8.1. Put G(ζ) = e^{iCζ} T_{ζ}F_{ζ} so that |G(θ)| = |T_{θ}f|. Then on the boundary of S, we have ∥G(iy)∥_{B_{q}^{2}} = e^{−Cy} ∥T_{iy}F_{iy}∥_{B_{q}^{2}} ≤ ∥F_{iy}∥_{B_{q}^{2}} and ∥G(1+iy)∥_{B} ≤ ∥F_{1+iy}∥_{B} for all y ∈ ℝ by above. This shows that ∥G∥_{F} = ∥F ∥_{F} = ∥f∥_{B_{q}^{p}} + ε. Since ε can be made arbitrarily small, ∥U_{ψ}^{p,q}f∥_{B_{q}^{p}} = ∥T_{θ}f∥_{B_{q}^{p}} = ∥G(θ)∥_{θ} ≤ ∥G∥_{F} = ∥f∥_{B_{q}^{p}} by Theorem 8.1. It also follows that ∥U_{ψ}^{p,q}∥ ≤ 1. □

So, in particular, if N = 1 and ψ ∈ Aut(𝔹), then the map f ↦ (f ∘ψ) \sqrt{Jψ} is unitary on H^{2} = A.

Appendix

Proof of Theorem 2.4 (b). For notational simplicity, with no loss of generality, we take a, b ∈ ℝ. We apply [BB, Corollary 3.8 (iii)] with p_{2} = 1, q_{2} = b + 1, p_{1} = p, and q_{1} = p(N + 1 + b) − N to the holomorphic function

w ↦ \frac{f(w)}{(1 − ⟨w, z⟩)^{N+1+a+b}} .

The first condition on b is a result of the requirement q_{1} > 0. The requirement q_{2} > 0, that is, b > −1, is implied by this. (Actually, [BB] considers q = 0 too, but with a different kind of measure. Also note that the q of [BB] corresponds to our q − 1.) This lemma is valid for 0 < p_{1} ≤ p_{2}, which is equivalent to 0 < p ≤ 1 as we assumed. We obtain

|V_{b}^{a}f(z)|^{p} ≤ C (1 − |z|^{2})^{pa} ∫_{𝔹} \frac{(1 − |w|^{2})^{p(N+1+b)−(N+1)}}{|1 − ⟨z, w⟩|^{p(N+1+a+b)}} |f(w)|^{p} dν(w).

Then ∥V_{b}^{a}f(z)∥_{L^{p}(ν_{c})}^{p} is

= ∫_{𝔹} |V_{b}^{a}f(z)|^{p} (1 − |z|^{2})^{c} dν(z)

≤ C ∫_{𝔹}(1 − |z|^{2})^{c+pa} ∫_{𝔹} \frac{(1 − |w|^{2})^{(p−1)(N+1)+pb}}{|1 − ⟨z, w⟩|^{p(N+1+a+b)}} |f(w)|^{p} dν(w) dν(z)

= C ∫_{𝔹} |f(w)|^{p} (1 − |w|^{2})^{(p−1)(N+1)+pb} ∫_{𝔹} \frac{(1 − |z|^{2})^{c+pa} dν(z)}{|1 − ⟨z, w⟩|^{p(N+1+a+b)}} dν(w),

where we used Fubini theorem. The condition on a amounts to c + pa > −1 and the second condition on b to (p − 1)(N + 1) + pb − c > 0, and when these hold, [R, Proposition 1.4.10] gives us

(1 − |w|^{2})^{(p−1)(N+1)+pb} ∫_{𝔹} \frac{(1 − |z|^{2})^{c+pa}}{|1 − ⟨z, w⟩|^{p(N+1+a+b)}} dν(z) ≤ C (1 − |w|^{2})^{c},

which in turn yields that ∥V_{b}^{a}f(z)∥_{L^{p}(ν_{c})}^{p} ≤ C ∥f(z)∥_{L^{p}(ν_{c})}^{p}. □

Proof of Theorem 3.4. We show the details only for the case when s, s + t, r, and r + u are real and exceed −(N + 1); the other cases are similar. Following [BB, p. 41], for b > −1, we consider

h(λ) = ∑_{k=0}^{∞} \frac{(N + 1 + r + u)_{k}}{(N + 1 + r)_{k}} \frac{(N + 1 + s)_{k}}{(N + 1 + s + t)_{k}} \frac{(b + 1)_{N+k}}{N! k!} λ^{k},

which belongs to H(𝔻). Computing with f(z) = z^{α} and Proposition 2.1, we see that

D_{r}^{u}f(z) = ∫_{𝔹} D_{s}^{t}f(w) h(⟨z, w⟩) (1 − |w|^{2})^{b} dν(w)

for f ∈ H(𝔹) by virtue of Theorem 3.2. By (8),

h(λ) ∼ ∑_{k=0}^{∞} \frac{(N + 1 + b)_{k}}{k! (k + 1)^{t−u}} λ^{k} = g_{N+1+b,t−u}(λ),

where g_{x,y} is defined in [BB, §0.5]. Then by the assumption on b and [BB, Corollary 2.4 (a)],

h(λ) = \frac{h_{1}(λ)}{(1 − λ)^{N+1+u−t+b}}

for a holomorphic h_{1} in the Lipschitz class Λ_{N+1+u−t+b} on \overline{𝔻}. This finally implies, since Arg(1 − λ) is bounded, that

|h(λ)| ≤ \frac{C}{|1 − λ|^{N+1+u−t+b}}

as in the proof of [BB, Lemma 5.6]. The proof is now complete. □

Acknowledgments

The author expresses his gratitude to Daniel Alpay of Ben-Gurion University of the Negev for his extensive support and for getting him involved in the Arveson space. This paper was completed during the author’s sabbatical visit to the University of Virginia. The author thanks the Department of Mathematics, the operator theory group, and especially James Rovnyak for their hospitality.

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Mathematics Department, Middle East Technical University, Ankara 06531, Turkey

Current address: Mathematics Department, Bilkent University, Ankara 06800, Turkey

   E-mail address: kaptan@fen.bilkent.edu.tr
   URL: http://www.fen.bilkent.edu.tr/∼kaptan/