WEIGHTED BLOCH, LIPSCHITZ, ZYGMUND, BERS, AND GROWTH SPACES OF THE BALL: BERGMAN PROJECTIONS

AND CHARACTERIZATIONS

H. TURGAY KAPTANOĞLU AND SERDAR TÜLÜ

Abstract. We determine precise conditions for the boundedness of Bergman projections from Lebesgue classes onto the spaces in the title, which are members of the same one-parameter family of spaces. The projections provide integral representations for the functions in the spaces. We obtain many properties of the spaces as straightforward corollaries of the projections, integral representations, and isometries among the spaces. We solve the Gleason problem and an extremal problem for point evaluations in each space. We establish maximality of these spaces among those that exhibit Möbius-type invariances and possess decent functionals. We find new Hermitian non-Kählerian metrics that characterize half of these spaces by Lipschitz-type inequalities.

1. Introduction

Let 𝔹 be the unit ball in ℂ^{N} with respect to the usual hermitian inner product ⟨z, w⟩ = z_{1}\overline{w}_{1} + ⋯ + z_{N} \overline{w}_{N} and the norm |z| = \sqrt{⟨z, z⟩}. Let H(𝔹) denote the space of holomorphic functions on 𝔹 and H^{∞} its subclass of bounded functions.

We let ν be the Lebesgue measure on 𝔹 normalized so that ν(𝔹) = 1, which is the normalized area measure on the unit disc 𝔻 when N = 1. For q ∈ ℝ, we also define on 𝔹 the measures

dν_{q}(z) = (1 − |z|^{2})^{q} dν(z).

For 0 < p < ∞, we denote the Lebesgue classes with respect to ν_{q} by L_{q}^{p}. The Lebesgue class of essentially bounded functions on 𝔹 with respect to any ν_{q} is the same (see [Ka1, Proposition 2.3]); we denote it by L^{∞}. For α ∈ ℝ, we also define the weighted classes

L_{α}^{∞} = { φ measurable on 𝔹 : (1 − |z|^{2})^{α} φ(z) ∈ L^{∞} }.

Let’s also call the subspace of L_{α}^{∞} consisting of holomorphic functions H_{α}^{∞}.

Date: 21 October 2010. 2000 Mathematics Subject Classification. Primary 32A37, 32A18; Secondary 30D45, 26A16,

32A25, 46E15, 47B38, 47B34, 32M99, 32F45.

Key words and phrases. Bergman projection, Bloch, Lipschitz, Zygmund, growth, Bers, Besov space, isometry, Gleason problem, slice function, boundary growth, Taylor coefficient, extremal point evaluation, duality, interpolation, α-Möbius invariance, decent functional, maximal space, Hermitian metric, Kähler metric, geodesic completeness, Laplace-Beltrami operator, holomorphic sectional curvature.

The first author thanks Özgün Ünlü of Bilkent University for helpful discussions.

We use C_{0} to denote the space of continuous functions on the closure \overline{𝔹} and C_{00} its subspace of those that vanish on the boundary ∂𝔹. We also define

C_{α} = { φ ∈ C_{0} : (1 − |z|^{2})^{α} φ(z) ∈ C_{0} }

and

C_{α0} = { φ ∈ C_{0} : (1 − |z|^{2})^{α} φ(z) ∈ C_{00} }.

Further, the ball algebra is A(𝔹) = H(𝔹) ∩ C_{0}.

Almost all results in this work depend on certain radial differential operators D_{s}^{t} of order t ∈ ℝ for any s ∈ ℝ that map H(𝔹) to itself defined in detail in [Ka1, Definition 3.1]. 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).

Definition 1.1. For any α ∈ ℝ, we define the weighted Bloch space B_{α} to consist of all f ∈ H(𝔹) for which I_{s}^{t}f belongs to L_{α}^{∞} for some s, t satisfying

(1)  α + t > 0.

The weighted little Bloch space B_{α0} is the subspace of B_{α} consisting of those f for which I_{s}^{t}f lies in C_{α0} for some s, t satisfying (1).

Condition (1) ascertains that all B_{α} and B_{α0} contain the polynomials and therefore are nontrivial. The spaces B_{0} and B_{00} are the usual Bloch and little Bloch spaces. In notation concerning C and B, a single subscript indicates boundedness, and double subscripts, the second of which is always 0, indicate vanishing on the boundary.

Our use of α is nontraditional, follows [Ka3, Section 8], conforms with the notation of closely related Besov spaces, and is more logical in view of the operators I_{s}^{t}. Most other authors use α − 1 where we use α.

Definition 1.1 can be shown to be independent of s, t satisfying (1) using the methods of [BB, p. 41]; note that s does not affect the order of the radial differential operator D_{s}^{t}. Similarly, these spaces can be defined using other kinds of derivatives; see also [Z3, Chapter 7]. We show independence from s and t essentially under (1) and (2) in Corollaries 3.4 and 3.5 below as easy consequences of Bergman projections and other similar integral operators.

Definition 1.2. For s ∈ ℝ and z, w ∈ 𝔹, the generalized Bergman-Besov kernels are

K_{s}(z, w) = \begin{cases}\frac{1}{(1 − ⟨z, w⟩)^{N+1+s}} = ∑_{k=0}^{∞} \frac{(N + 1 + s)_{k}}{k!} ⟨z, w⟩^{k}, & if s > −(N + 1); \\ \frac{_{2}F_{1}(1, 1; 1 − N − s; ⟨z, w⟩)}{−N − s} = ∑_{k=0}^{∞} \frac{k! ⟨z, w⟩^{k}}{(−N − s)_{k+1}} , & if s ≤ −(N + 1); \end{cases}

and the extended Bergman projections are

P_{s}φ(z) = ∫_{𝔹} K_{s}(z, w) φ(w) dν_{s}(w)

for suitable φ.

Above, _{2}F_{1} is the hypergeometric function, and (a)_{b} is the Pochhammer symbol given by

(a)_{b} = \frac{Γ(a + b)}{Γ(a)}

when a and a+b are off the pole set −ℕ of the gamma function Γ. The presentations of K_{s} and P_{s} follow those in [BB, Section 1] and [Ka1]. Note that K_{s}(·, w) ∈ H(𝔹) and thus P_{s}f ∈ H(𝔹) whenever the integral exists. Throughout, s and t can take complex values too as done in [FR] and [Ka1].

The following is our first main result.

Theorem 1.3. P_{s} : L_{α}^{∞} → B_{α} is bounded if and only if

(2)  α < s + 1.

Given an s satisfying (2), if t satisfies (1), then for f ∈ B_{α},

(3)  P_{s}I_{s}^{t}f = \frac{N!}{(1 + s + t)_{N}} f =: \frac{1}{C_{s+t}} f.

Also either P_{s} : C_{α} → B_{α0} or P_{s} : C_{α0} → B_{α0} is bounded if and only if (2) holds.

Note that (1) and (2) together imply s + t > −1 so that C_{s+t} makes sense. Thus P_{s} : L_{α}^{∞} → B_{α} is surjective and I_{s}^{t} : B_{α} → L_{α}^{∞} is an imbedding. For s, t satisfying (2) and (1), each of C_{s+t}I_{s}^{t} is a right inverse for P_{s} on L_{α}^{∞}, and C_{s+t} P_{s} is a left inverse for I_{s}^{t} on B_{α}. Similar statements hold for the “little” spaces. Moreover, (3) is a family of integral representations for f ∈ B_{α} which take the form

(4) f(z) = \frac{(1 + s + t)_{N}}{N!} ∫_{𝔹} K_{s}(z, w) (1 − |w|^{2})^{s+t} D_{s}^{t}f(w) dν(w)  (z ∈ 𝔹)

when written explicitly.

The case α = 0 of Theorem 1.3 has been treated earlier in [C, Theorem 2], [Ka1, Corollary 5.3], and [KÜ, Corollary 5.3]. Bergman projections P_{s} for s > −1 from Lebesgue classes of the form L_{α}^{∞} ∩ L_{s}^{p} with 1 ≤ p < ∞ and α > 0 to B_{α}, and between B_{α} with −1 < α < 0, have been considered in [CL]. Bergman-type projections, in which K_{s_{1}} is used with ν_{s_{2}}, from L^{∞} or C_{0} to B_{α} or B_{α0} have been considered in several theorems dealing with various ranges of α in [Z3, Chapter 7]. Neither of the last two sources gives a necessary condition on the parameters or a right inverse. Only [Z3] handles s_{1} ≤ −(N + 1), but with the restriction that it is an integer. Further, [HC, Theorem 3] obtains the case s = α > −1 and t = 1 of (4) with a method that does not make use of the idea of a Bergman projection.

Having integral representations is very fruitful, and we exploit (4) and Theorem 1.3 to extract many properties of the weighted Bloch spaces in Sections 3–6, giving also easier proofs of a few known facts. But several properties of these spaces require other considerations. A recurring theme is using kernels to define a new concept on one space and then using radial differential operators to carry them to the remaining spaces.

Consider the extremal problem of determining

(5)  S_{α}(b) := sup { f(b) > 0 : f ∈ B_{α}, ∥f∥_{B_{α}} = ∥I_{s}^{t}f∥_{L_{α}^{∞}} = 1 },

for each b ∈ 𝔹 and if possible finding a function realizing it. Note that S_{α}(b) also depends on s, t satisfying (1). There is also the problem of determining S_{α0}(b) in which f is allowed to vary only in B_{α0}.

Theorem 1.4. For any α, the extremal function attaining S_{α}(b) exists and is unique. This solution is also the solution for S_{α0}(b).

The proof of Theorem 1.4 depends on the following construction. For α ≥ 0, define the linear transformations

(6)  T_{ψ}^{α}f(z) = f(ψ(z)) (Jψ(z))^{2α/(N+1)},

where ψ is a holomorphic automorphism of 𝔹, J denotes the complex Jacobian, and an appropriate fixed branch of the logarithm is used for the fractional power. We extend T_{ψ}^{α} to all α by setting

(7)  W_{ψ}^{α} = D_{s+t}^{−t}T_{ψ}^{α+t}D_{s}^{t},

where s, t satisfy (1).

Definition 1.5. Let (X, ∥ · ∥) be a Banach space of holomorphic functions on 𝔹 containing the constants. We call X an α-Möbius-invariant space if W_{ψ}^{α}f ∈ X for some s, t satisfying (1) whenever f ∈ X, ∥W_{ψ}^{α}f∥ ≤ C ∥f∥, and the action ψ ⟼ W_{ψ}^{α}f is continuous for f ∈ X and unitary ψ.

Theorem 1.6. The space B_{α} contains with continuous inclusion those α-Möbius-invariant spaces that possess a decent linear functional.

All our results mentioned so far including their consequences and applications are general and cover all real values of α. If no range for α is specified, that means α ∈ ℝ is arbitrary. Most of the results are completely new for the spaces B_{α} with α < 0, which are actually the Lipschitz spaces Λ_{−α} as explained in the next section. The original definition of these spaces for −1 < α < 0 states that Λ_{−α} is the space of holomorphic functions f on 𝔹 satisfying the so-called Lipschitz condition |f(z)−f(w)| ≤ C |z−w|^{−α} for all z, w ∈ 𝔹. For α = 0, the corressponding equivalent condition is |f(z) − f(w)| ≤ C ρ_{0}(z, w), where ρ_{0} is the Bergman metric; see [T1, Theorem 3.4 (3)]. We extend this condition to all α > 0 by finding new metrics ρ_{α} in 𝔹 in place of the Euclidean or the Bergman metrics.

Theorem 1.7. For each α > 0, there exists a complete Hermitian non-Kählerian metric ρ_{α} on 𝔹 such that if f ∈ B_{α}, then |f(z) − f(w)| ≤ C ρ_{α}(z, w). The converse also holds for N = 1.

The next section gives some preparatory material on the spaces under consideration and the tools to be used. We prove Theorem 1.3 in Section 3. In Sections 4 and 5, we apply Theorem 1.3 and (3) to a solution of the Gleason problem in B_{α}, to the growth of functions in B_{α} near ∂𝔹, and to the growth of their Taylor series coefficients. In Section 6, we exhibit pairings that yield (pre)duality relationship between the Besov spaces B_{q}^{1} and the spaces B_{α} and B_{α0}, and find the complex interpolation space between two weighted Bloch spaces, again by applying Theorem 1.3. In Section 7, we prove Theorem 1.4 by determining explicitly the extremal functions. We prove Theorem 1.6 in Section 8, where a decent linear functional is also defined. In the final Section 9, we define some new Hermitian metrics similar to the hyperbolic metric that are specific to the B_{α} spaces and prove Theorem 1.7.

2. Preliminaries

Stirling formula gives

(8)  \frac{Γ(c + a)}{Γ(c + b)} ∼ c^{a−b}  and  \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. Such constants are always independent any parameters or functions in the formulas and are all denoted by the generic upper case C.

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

We follow the notation and results of [R, Section 2.2] regarding the automorphism group Aut(𝔹). If ψ ∈ Aut(𝔹) and b = ψ^{−1}(0), then the complex Jacobian of ψ is

(9)  Jψ(z) = η_{1} (\frac{1 − |ψ(z)|^{2}}{1 − |z|^{2}} )^{(N+1)/2} = η_{2} \frac{(1 − |b|^{2})^{(N+1)/2}}{(1 − ⟨z, b⟩)^{N+1}} ,

where η_{1}, η_{2} are complex numbers of modulus 1. The group Aut(𝔹) is generated by the involutive Möbius transformations φ_{b} exchanging 0 and b ∈ 𝔹 and unitary operators U on ℂ^{N}. For these special kinds of automorphisms, we write T_{b}^{α} and T_{U}^{α} in place of T_{ψ}^{α} defined in (6). When ψ = φ_{b}, then η_{2} = (−1)^{N}. To avoid the annoying appearance of η_{2} in calculations, we redefine T_{b}^{α} as

(10)  T_{b}^{α}f(z) = \frac{(1 − |b|^{2})^{α}}{(1 − ⟨z, b⟩)^{2α}} f(φ_{b}(z)).

Let f ∈ H(𝔹) and f = ∑_{k=0}^{∞} f_{k} be its homogeneous expansion, where f_{k} is a holomorphic homogeneous polynomial of degree k. The action of D_{s}^{t} on f is that of a coefficient multiplier in the form

(11)  D_{s}^{t}f = ∑_{k=0}^{∞} d_{k}f_{k},

where d_{k} depends on s, t in such a way that d_{k} ∼ k^{t} as k → ∞ for any s. So D_{s}^{t} is a continuous operator on H(𝔹). In particular, D_{s}^{t}z^{λ} = d_{|λ|}z^{λ} for any multi-index λ. An important property of our particular d_{k} is that d_{k} ≠ 0 for all k = 0, 1, 2, …, and this makes D_{s}^{t} invertible on H(𝔹). Coupled with the facts that D_{s}^{0} = I, the identity, and D_{s+t}^{u}D_{s}^{t} = D_{s}^{u+t}, we obtain the two-sided inverse

(12)  (D_{s}^{t})^{−1} = D_{s+t}^{−t}

for any s, t ∈ ℝ.

The differential operators D_{s}^{t} relate well with the Bergman-Besov kernels K_{s} for

(13)  D_{s}^{t}K_{s}(z, w) = K_{s+t}(z, w)

for any s, t ∈ ℝ, where differentiation is performed on the holomorphic variable z. All the above properties of D_{s}^{t} are taken from [Ka1, Section 3]. Moreover, if s > −1 and f ∈ H(𝔹), then for any t,

(14)  D_{s}^{t}f(z) = C_{s} lim_{r→1−} ∫_{𝔹} K_{s+t}(z, w) (1 − |w|^{2})^{s} f(rw) dν(w);

see [Ka1, Lemma 5.1].

The space L_{α}^{∞} is normed with

∥φ∥_{L_{α}^{∞}} := ess sup_{z∈𝔹} (1 − |z|^{2})^{α} |φ(z)|.

The norm on C_{α} is given by the same formula. For any s, t satisfying (1), there is induced the norm

(15)  ∥f∥_{B_{α}} := ∥I_{s}^{t}f∥_{L_{α}^{∞}}

on B_{α}. This is a genuine norm, because D_{s}^{t} is an invertible operator. Different s, t satisfying (1) give equivalent norms as mentioned above, mainly as a consequence of Corollary 3.4. It is clear from Definition 1.1 that

(16)  B_{α} ⊂ B_{β0} ⊂ B_{β}  (α < β),

and the inclusion is continuous.

Proposition 2.1. For any α and s, t, D_{s}^{t}(B_{α}) = B_{α+t} is an isomorphism, and an isometry when appropriate norms are used in the two spaces.

Proof. Let f ∈ B_{α} and put g = D_{s}^{t}f. Take u so large that α + (t + u) > 0. Then D_{s+t}^{u}g = D_{s+t}^{u}D_{s}^{t}f = D_{s}^{t+u}f and I_{s}^{t+u}f ∈ L_{α}^{∞}. This is equivalent to I_{s+t}^{u}g ∈ L_{α+t}^{∞}. Hence g ∈ B_{α+t} and the norms ∥f∥_{B_{α}} = ∥I_{s}^{t+u}f∥_{L_{α}^{∞}} and ∥g∥_{B_{α+t}} = ∥I_{s+t}^{u}g∥_{L_{α+t}^{∞}} are equal. Since D_{s}^{−t}(B_{α+t}) = B_{α} the same way, both claims are established. □

Example 2.2. A “typical” function in B_{0} is known and can be checked by Definition 1.1 to be f_{0}(z) = log(1 − z_{1})^{−1} ∼ ∑_{k} z_{1}^{k}/k. By Proposition 2.1, a “typical” function in B_{α} is f_{α}(z) = ∑_{k} k^{α−1}z_{1}^{k}. Using the series expansion in Definition 1.2 and (8), f_{α}(z) essentially is (1−z_{1})^{−α} for α > 0, and a hypergeometric function for α < 0. The same reasoning shows that all the inclusions in (16) are strict. Letting β − α = 2ε > 0 and using a large enough t in Definition 1.1, it is easy to see that f_{β} ∈ B_{β} \ B_{β0} and f_{α+ε} ∈ B_{β0} \ B_{α}. Thus all B_{α} and B_{α0} spaces are different.

By [HKZ, Definition 4.13], the space B_{α} for α > 0 is the growth space A^{−α}. The space B_{2} is also called the Bers space. A growth space does not require any derivative in its definition since now t = 0 satisfies (1). So an f ∈ H(𝔹) belongs to B_{α} for α > 0 whenever f(z) ≤ C (1 − |z|^{2})^{−α} for all z ∈ 𝔹. Thus for α > 0, the spaces B_{α} and H_{α}^{∞} coincide.

Also by [Z3, Theorems 7.17 and 7.18], the space B_{α} for α < 0 is the holomorphic Lipschitz space Λ_{−α}. Proposition 2.1 for α < 0 and t = −α appears in [Z3, Theorems 7.19 and 7.20].

Proposition 2.1 is the usual way to extend the definition of Lipschitz spaces Λ_{−α} beyond −1 ≤ α < 0. The space B_{−1} = Λ_{1} is called the Zygmund class. It is traditionally defined via a second-order difference quotient as opposed to first-order difference quotients for −1 < α < 0; see [Kr2, Section 8.8]. This is no surprise, because the least integer value of t specified by (1) is 1 for −1 < α < 0 and is 2 for α = −1. Hence the case α = 0 and t = −1 of Proposition 2.1 is in [ACP, 3.5.2].

Remark 2.3. All the statements in this section for the B_{α} spaces, including Proposition 2.1, have obvious counterparts for the B_{α0} spaces.

Definition 2.4. For q ∈ ℝ and 0 < p < ∞, the Besov space B_{q}^{p} consists of all f ∈ H(𝔹) for which I_{s}^{t}f belongs to L_{q}^{p} for some s, t satisfying

(17)  q + pt > −1.

This definition too is independent of s, t under (17), and thus we have the equivalent norms ∥f∥_{B_{q}^{p}} = ∥I_{s}^{t}f∥_{L_{q}^{p}} on B_{q}^{p}. Bergman projections on Besov spaces have been characterized in [Ka1, Theorem 1.2].

Theorem 2.5. For 1 ≤ p < ∞, P_{s} : L_{q}^{p} → B_{q}^{p} is bounded if and only if

(18)  q + 1 < p (s + 1).

Given s satisfying (18), if t satisfies (17), then (3) holds for f ∈ B_{q}^{p}.

We use α as a subscript on B rather than the usual superscript, because this not only follows the notation for L^{∞} and C, but also follows the notation for the Besov spaces B_{q}^{p}, where the upper parameter is for the power on the function, and the lower parameter is for the power on the weight 1 − |z|^{2}. The power on a function in a B_{α} space, if anything, is ∞ and not shown.

Remark 2.6. There are close connections between the Besov and weighted Bloch families of spaces. It is explained in [Ka3, Section 8] that B_{α} is the limiting case of B_{αp}^{p} (or of B_{β+αp}^{p}) at p = ∞. This is further reflected in the inequalities; (1) is the p = ∞ case of (17) with q = αp, and (2) is the p = ∞ case of (18) with q = αp. The set of (p, q) in the right half plane satisfying q = β + αp is a ray with slope α and q-intercept β.

For another connection between the two families, see [Ka3, Theorem 8.3], where the Carleson measures of Besov spaces characterize the functions in weighted Bloch spaces, which yields different proofs of Corollaries 3.4 and 3.5 below.

3. Projections

We now prove Theorem 1.3 and indicate several immediate corollaries.

Proof of Theorem 1.3. Fix α throughout the proof.

Let φ ∈ L_{α}^{∞}. Take t to satisfy (1), and for the moment so large that also s + t > −(N + 1) holds. To show P_{s}φ ∈ B_{α}, we need to show I_{s}^{t}(P_{s}φ) ∈ L_{α}^{∞}. Using (13) and the assumptions on s, t, we obtain

(1 − |z|^{2})^{α} |I_{s}^{t}(P_{s}φ)(z)| = (1 − |z|^{2})^{α+t} |D_{s}^{t} ∫_{𝔹} K_{s}(z, w) (1 − |w|^{2})^{s} φ(w) dν(w)|

≤ ∥φ∥_{L_{α}^{∞}} (1 − |z|^{2})^{α+t} ∫_{𝔹} \frac{(1 − |w|^{2})^{s−α}}{|1 − ⟨z, w⟩|^{N+1+s+t}} dν(w),

and the last integral is finite if and only if (2) holds. Then [R, Proposition 1.4.10] yields

(1 − |z|^{2})^{α} |I_{s}^{t}(P_{s}φ)(z)| ≤ C ∥φ∥_{L_{α}^{∞}} (z ∈ 𝔹).

Thus P_{s}φ ∈ B_{α} and this proves the first claim on when P_{s} : L_{α}^{∞} → B_{α} is bounded. As noted earlier, (1) and (2) together imply s + t > −1; so the momentary assumption on t above is superfluous.

Now let f ∈ B_{α} and s, t satisfy (2) and (1). Then I_{s}^{t}f ∈ L_{α}^{∞} and P_{s}(I_{s}^{t}f) ∈ B_{α} by the first part. Also s + t > −1, and using (14) and (12), we see that

P_{s}(I_{s}^{t}f)(z) = ∫_{𝔹} K_{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),

and this proves the second claim.

Next, let s satisfy (2). To show that P_{s} maps C_{α} into B_{α0}, it suffices to consider φ(w) = (1 − |w|^{2})^{−α} w^{λ} \overline{w}^{μ} ∈ C_{α}, since polynomials in w and \overline{w} are dense in C_{0}. For t satisfying (1), we have

(1 − |z|^{2})^{α} I_{s}^{t}(P_{s}φ)(z) = (1 − |z|^{2})^{α+t} ∫_{𝔹} K_{s+t}(z, w) (1 − |w|^{2})^{s−α} w^{λ} \overline{w}^{μ} dν(w).

We use the series expansion of K_{s+t} in simplified form as K_{s+t}(z, w) = ∑_{τ} c_{τ} z^{τ} \overline{w}^{τ}. In the integral above, the only nonzero term is the one with τ = λ − μ by the orthogonality in [FR, Proposition 2.4]. Then that integral is finite by (2) and is

c_{λ−μ} z^{λ−μ} ∫_{𝔹} |w^{λ}|^{2} (1 − |w|^{2})^{s−α} dν(w) = C z^{λ−μ}

with |λ − μ| ≥ 0. Thus (1 − |z|^{2})^{α} |I_{s}^{t}(P_{s}φ)(z)| → 0 as |z| → 0 again by (1). Hence P_{s}φ ∈ B_{α0} and P_{s} is bounded from either of C_{α} and C_{α0} into B_{α0}.

If f ∈ B_{α0}, then by Definition 1.1, I_{s}^{t}f ∈ C_{α0} if t satisfies (1). Now for s satisfying (2), (3) shows that P_{s}(I_{s}^{t}f) = C f lies in B_{α0}. This shows that P_{s} is onto B_{α0} from either of the continuous function classes.

To prove the necessity of (2), put φ_{1}(w) = w_{1} (1 − |w|^{2})^{−α} [log(1 − |w|^{2})]^{−1}. Clearly φ_{1} ∈ C_{α0}. The integral

P_{s}φ_{1}(z) = ∫_{𝔹} K_{s}(z, w) (1 − |w|^{2})^{s−α} \frac{w_{1}}{log(1 − |w|^{2})} dν(w)

diverges if (2) is violated.

This completes the proof. □

Corollary 3.1. The space B_{α0} is the closure of the holomorphic polynomials in the norm ∥f∥_{B_{α}} = ∥I_{s}^{t}f∥_{L_{α}^{∞}} for any s, t satisfying (2) and (1), and thus is separable and complete.

Proof. Theorem 1.3 shows that P_{s} maps C_{α} onto B_{α0}. Similar to its proof, if φ_{λμ}(w) = (1 − |w|^{2})^{−α} w^{λ} \overline{w}^{μ} ∈ C_{α}, then P_{s}φ_{λμ}(z) = C z^{λ−μ} with |λ − μ| ≥ 0. The space C_{α} is the closure of finite linear combinations of functions of the form φ_{λμ} with λ, μ having rational components. Consequently, B_{α0} is the closure of finite linear combinations of functions of the form z^{τ}. □

Remark 3.2. The inseparability of B_{α} can also be seen using Theorem 1.3. For α > 0, take s = α, and for other α, take s = 0. Also take φ(w) = (1 − |b|^{2})^{−s} for w = b and φ(w) = 0 otherwise, where b ∈ 𝔹 is arbitrary. What Theorem 1.3 now says is that K_{s}(·, b) ∈ B_{α} for every b ∈ 𝔹. A quick estimate shows that ∥K_{s}(·, b_{1}) − K_{s}(·, b_{2})∥_{B_{α}} ≥ C |b_{1} − b_{2}|.

Remark 3.3. The operator P_{s} is not always a projection on a subspace, because B_{α} need not be a subspace of L_{α}^{∞}. However, for t satisfying (1), I_{s}^{t}(B_{α}) is an isometric copy of B_{α} in L_{α}^{∞} by (15), and hence a closed subspace of L_{α}^{∞} by Corollary 5.5 below. Then by (3),

V_{s}^{t} := I_{s}^{t}P_{s}

is a true projection from L_{α}^{∞} onto I_{s}^{t}(B_{α}) for any s satisfying (2).

Corollary 3.4. For a given s satisfying (2), any two values of the order t of the differential operator D_{s}^{t} satisfying (1) generate the same space in Definition 1.1 for the same α.

Proof. Suppose t_{1}, t_{2} satisfy (1), f ∈ H(𝔹), and φ = I_{s}^{t_{1}}f ∈ L_{α}^{∞} for a given α, where s satisfies (2). By Theorem 1.3, P_{s}I_{s}^{t_{1}}f = C f. Apply I_{s}^{t_{2}} to both sides to get V_{s}^{t_{2}}φ = C I_{s}^{t_{2}}f, where

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

when written explicitly. Since φ(w) ∈ L_{α}^{∞}, [R, Proposition 1.4.10] yields that |V_{s}^{t_{2}}φ(z)| ≤ C (1 − |z|^{2})^{−α}. In other words, I_{s}^{t_{2}}f = C V_{s}^{t_{2}}φ ∈ L_{α}^{∞}, which is the desired result.

The case for B_{α0} is entirely similar. □

Corollary 3.5. For a given t satisfying (1), any two values of the parameter s of the differential operator D_{s}^{t} satisfying (2) generate the same space in Definition 1.1 for the same α if also s > −1 for α < 0.

Proof. Given α, suppose s_{1}, s_{2} satisfy (2), f ∈ H(𝔹), and I_{s_{1}}^{t}f ∈ L_{α}^{∞}, that is, g = D_{s_{1}}^{t}f ∈ L_{α+t}^{∞}, where t satisfies (1). That is, there is a C such that

(19)  |g(z)| ≤ \frac{C}{(1 − |z|^{2})^{α+t}}  (z ∈ 𝔹).

Using (12) and (14), we write D_{s_{2}}^{t}f = D_{s_{2}}^{t}D_{s_{1}+t}^{−t}D_{s_{1}}^{t}f = D_{s_{2}}^{t}D_{s_{1}+t}^{−t} g = Eg, where

Eg(z) = C_{s_{2}}C_{s_{1}+t} ∫_{𝔹} \frac{(1 − |v|^{2})^{s_{2}}}{(1 − ⟨z, v⟩)^{N+1+s_{2}+t}} ∫_{𝔹} \frac{(1 − |w|^{2})^{s_{1}+t}}{(1 − ⟨v, w⟩)^{N+1+s_{1}}} g(w) dν(w) dν(v).

The required condition s_{1} +t > −1 follows from (1) and (2). We would like to show that also Eg = D_{s_{2}}^{t}f ∈ L_{α+t}^{∞}. Again it is not necessary to check the B_{α0} spaces.

If α > 0, using (19) and [R, Proposition 1.4.10], we obtain

|Eg(z)| ≤ C ∫_{𝔹} \frac{(1 − |v|^{2})^{s_{2}}}{|1 − ⟨z, v⟩|^{N+1+s_{2}+t}} ∫_{𝔹} \frac{(1 − |w|^{2})^{s_{1}−α}}{|1 − ⟨v, w⟩|^{N+1+s_{1}}} dν(w) dν(v)

∼ ∫_{𝔹} \frac{(1 − |v|^{2})^{s_{2}−α}}{|1 − ⟨z, v⟩|^{N+1+s_{2}+t}} dν(v) ∼ \frac{C}{(1 − |z|^{2})^{α+t}} .

If α = 0, then the above computation and [R, Proposition 1.4.10] yield

|Eg(z)| ≤ C ∫_{𝔹} \frac{(1 − |v|^{2})^{s_{2}}}{|1 − ⟨z, v⟩|^{N+1+s_{2}+t}} log \frac{1}{1 − |v|^{2}} dν(v)

≤ C ∫_{𝔹} \frac{(1 − |v|^{2})^{s_{2}+ε}}{|1 − ⟨z, v⟩|^{N+1+s_{2}+t}} dν(v) ∼ \frac{C}{(1 − |z|^{2})^{t−ε}}

for some ε > 0. We next let ε → 0.

If α < 0, (19) can be strengthened to |g(z)| ≤ C (1 − |z|^{2})^{−t}. Using this estimate in |Eg(z)| reduces this case to the case of α = 0.

In the first two cases, from (1) and (2), s + t > −1 and s > −1 are automatic, but in the last case, s > −1 has to be additionally assumed. □

[Z3, Chapter 7] contains a collection of similar results dealing with various ranges of the parameters with further limitations on their values.

We have one more type of the Bergman projections that has already been utilized in [WZZ]. Define the generalized Bergman projections P_{rs} on suitable φ by

P_{rs}φ(z) = ∫_{𝔹} K_{r}(z, w) φ(w) dν_{s}(w).

Theorem 3.6. P_{rs} : L_{α}^{∞} → B_{α} is bounded if and only if r, s satisfy (2) and

(20)  r ≤ s.

Given such r, s, if t satisfies (1), then for f ∈ B_{α},

(21)  P_{rs}I_{s}^{t}f = \frac{1}{C_{s+t}} D_{s}^{r−s}f.

Also either P_{rs} : C_{α} → B_{α0} or P_{rs} : C_{α0} → B_{α0} is bounded if and only if (2) and (20) hold.

Proof. Take φ ∈ L_{α}^{∞} and t so that (1) and r + t > −(N + 1) hold. Then

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

where ψ(z) = (1 − |z|^{2})^{α} φ(z) ∈ L^{∞}. By [Ka3, Theorem 7.2], V ψ lies in L^{∞} if and only if (2) and (20) hold since (1) is assumed anyway.

Equation (21) is obtained in exactly the same way as in the proof of Theorem 1.3, and so are the statements about B_{α0}. □

Note that the only way to obtain f itself on the right side of (21) is to have r = s. Otherwise, the right side is a primitive of order s − r > 0 of f.

4. Gleason Problem

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

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

This section is for explicit solutions to the Gleason problem in the spaces B_{α}.

Theorem 4.1. For a ∈ 𝔹, there are bounded linear operators G_{1}, …, G_{N} on B_{α} satisfying

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

The operators G_{1}, …, G_{N} are bounded on B_{α0} too.

Proof. We imitate the proof of [Ka1, Theorem 6.1] and show little detail. Let s > −(N + 1) be an integer satisfying (2), let t satisfy (1), and define

G_{m}f(z) = C_{s+t} ∫_{𝔹} \frac{K_{s}(z, w) − K_{s}(a, w)}{⟨z − a, w⟩} \overline{w}_{m} I_{s}^{t}f(w) dν_{s}(w) (f ∈ B_{α})

for m = 1, …, N. It is easy to see that G_{1}, …, G_{N} satisfy (22).

The crucial part is to show that G_{m} is bounded. Proceeding as in the proof of [Ka1, Theorem 6.1], by the fact that s is an integer, it is possible to write each G_{m} as a finite sum of operators T_{j} on B_{α} such that

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

where u satisfies (1) when substituted for t. Because I_{s}^{t}f ∈ L_{α}^{∞} and s satisfies (2), [R, Proposition 1.4.10] yields that (1−|z|^{2})^{α} |I_{s−j}^{u}(T_{j}f)(z)| is bounded on 𝔹. Hence G_{m} is a bounded operator on B_{α}.

If f ∈ B_{α0}, then given ε > 0, there is an R < 1 such that (1−|w|^{2})^{α} |I_{s}^{t}f(w)| < ε for |w| ≥ R. Split the integral in (23) into two parts, J_{1} on R𝔹 and J_{2} on 𝔹 \ R𝔹. Then |J_{1}(z)| ≤ C and |J_{2}(z)| ≤ C ε (1 − |z|^{2})^{−(α+u)} for z ∈ 𝔹. We multiply J_{1} and J_{2} by (1 − |z|^{2})^{α+u}, add, and then let |z| → 1. Because u satisfies (1), we obtain

lim_{|z|→1} (1 − |z|^{2})^{α} |I_{s−j}^{u}(T_{j}f)(z)| ≤ C ε.

Since ε > 0 is arbitrary, this shows that T_{j}f and hence G_{m}f belong to B_{α0}. □

5. Analytic Properties

In this section, we use Theorem 1.3 and in particular (4) to obtain some analytic properties of functions in B_{α} spaces. Some of these properties are known, especially for α < 0, the Lipschitz range. However the emphasis here is on how they are obtained so readily from Bergman projections and the ensuing integral representations, and on their uniformity for all real α.

But first, let’s give a result that shows the versatility of the radial differential operators D_{s}^{t}. Given N ≥ 2, f ∈ H(𝔹), and ζ ∈ ℂ^{N} with |ζ| = 1, the holomorphic slice functions f_{ζ} are defined by f_{ζ}(x) = f(xζ) for x ∈ 𝔻.

Theorem 5.1. Suppose every slice function f_{ζ} of an f ∈ H(𝔹) belongs to B_{α} of the disc with uniformly bounded norms. Then f ∈ B_{α} of the ball.

Proof. For z = xζ ∈ 𝔹, by (11) we have

D_{s}^{t}f(z) = ∑_{k=0}^{∞} d_{k}f_{k}(z) = ∑_{k=0}^{∞} d_{k}f_{k}(ζ) x^{k} = D_{r}^{t} ∑_{k=0}^{∞} f_{k}(ζ) x^{k} = D_{r}^{t}f_{ζ}(x),

where r = N −1+s by [Ka1, Definition 3.1]. By assumption, there is a C such that (1−|x|^{2})^{α+t} |D_{r}^{t}f_{ζ}(x)| ≤ C for all x ∈ 𝔻 and ζ ∈ ∂𝔹, and for some r, t satisfying (1). Then s, t satisfy (1) too, and obviously (1 − |z|^{2})^{α+t} |D_{s}^{t}f(z)| ≤ C for all z ∈ 𝔹. □

The case α = 0 of Theorem 5.1 is in [T1, Theorem 4.10] with a much more roundabout proof.

Theorem 5.2. Given α, there is a C such that for all f ∈ B_{α} and z ∈ 𝔹,

|f(z)| ≤ C ∥f∥_{B_{α}} \begin{cases}(1 − |z|^{2})^{−α}, & if α > 0; \\ log(1 − |z|^{2})^{−1}, & if α = 0; \\ 1, & if α < 0; \end{cases}

where ∥f∥_{B_{α}} = ∥I_{s}^{t}f∥_{L_{α}^{∞}} with s > −(N + 1), t satisfying (2) and (1).

Proof. We use a simple estimate on (4) and obtain

|f(z)| ≤ C ∥f∥_{B_{α}} ∫_{𝔹} \frac{(1 − |w|^{2})^{s−α}}{|1 − ⟨z, w⟩|^{N+1+s}} dν(w) (z ∈ 𝔹).

We obtain all three cases by applying [R, Proposition 1.4.10]. □

Corollary 5.3. Under the same conditions as of Theorem 5.2 and for r, u ∈ ℝ, we have

|D_{r}^{u}f(z)| ≤ C ∥f∥_{B_{α}} \begin{cases}(1 − |z|^{2})^{−(α+u)}, & if α + u > 0; \\ log(1 − |z|^{2})^{−1}, & if α + u = 0; \\ 1, & if α + u < 0. \end{cases}

Proof. Just combine Theorem 5.2 with Proposition 2.1. □

Theorem 5.2 says nothing new other than Definition 1.1 for α > 0, and Corollary 5.3 is reminiscent of the classical definition of Lipschitz spaces for u = −α > 0. Now we see how the two subfamilies for positive and negative values of α are combined in a uniform manner.

Theorem 5.4. Λ_{−α} ⊂ A(𝔹) ⊂ H^{∞} ⊂ B_{β} for α < 0 and β ≥ 0.

Proof. For α < 0, let f ∈ Λ_{−α}, and pick s > −(N + 1) and t so as to satisfy (2) and (1). Then by (4), we have

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

and

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

Since N + 1 + s − (N + 1) − (−α + s) = α < 0, by the proof of [R, Proposition 1.4.10], the last integral converges uniformly for |z| ≤ 1. Thus also f ∈ C_{0}.

The claim for β ≥ 0 is the well-known fact proved via Schwarz lemma that the classical Bloch space contains H^{∞} combined with (16). □

Corollary 5.5. Given α, r, u ∈ ℝ and a compact subset E of 𝔹, there is a C such that for all f ∈ B_{α},

sup_{z∈E} |D_{r}^{u}f(z)| ≤ C ∥f∥_{B_{α}},

where ∥f∥_{B_{α}} = ∥I_{s}^{t}f∥_{L_{α}^{∞}} with s > −(N + 1), t satisfying (2) and (1), Therefore point evaluations on B_{α} are bounded linear functionals. Consequently, every B_{α} is a Banach space.

We now set N = 1 and look at the Taylor series coefficients of f ∈ B_{α} on 𝔻.

Theorem 5.6. Given α, there is a C such that for all f ∈ B_{α},

|c_{k}| ≤ C ∥f∥_{B_{α}} k^{α},

where c_{k} = f^{(k)}(0)/k! and ∥f∥_{B_{α}} = ∥I_{s}^{t}f∥_{L_{α}^{∞}} with s > −2, t satisfying (2) and (1).

Proof. Differentiation k times puts (4) into the form

f^{(k)}(z) = C (2 + s)_{k} ∫_{𝔻} \frac{\overline{w}^{k} I_{s}^{t}f(w)}{(1 − ⟨z, w⟩)^{2+s+k}} dν_{s}(w) (z ∈ 𝔻).

Then

|c_{k}| ≤ C ∥f∥_{B_{α}} \frac{Γ(2 + s + k)}{Γ(1 + k)} ∫_{𝔻} |w|^{k} (1 − |w|^{2})^{s−α} dν(w).

Evaluating the integral using [Ka1, Proposition 2.1] yields

|c_{k}| ≤ C ∥f∥_{B_{α}} \frac{Γ(2 + s + k)}{Γ(1 + k)} \frac{Γ(1 + k/2)}{Γ(2 + s − α + k/2)} ∼ C ∥f∥_{B_{α}} k^{α},

where the final estimate follows from (8). □

Remark 5.7. The statements of most results in this section could be guessed using the general principle stated in Remark 2.6. Theorem 5.2 and Corollary 5.3 are the p = ∞ versions of [KÜ, Theorem 6.1 and Corollary 6.2], Theorem 5.6 is the p = ∞ version of [KÜ, Theorem 7.1], all after setting q = αp. Next we state without proof another result that we guess by employing the same principle. For it, we set q = αp first in [KÜ, Theorem 4.2], and then replace the l^{p} condition with the l^{∞} condition as well as p by ∞. For α > −1, it is proved in [Y, Theorem 1], while our result holds for all real α.

Theorem 5.8. A Taylor series f(z) = ∑_{k} c_{k} z^{n_{k}} with Hadamard gaps belongs to B_{α} if and only if sup_{k} n_{k}^{−α} |c_{k}| < ∞, and belongs to B_{α0} if and only if n_{k}^{−α} |c_{k}| → 0 as k → ∞.

6. Duality and Interpolation

Duality results on Besov and Bloch spaces using different pairings dealing with various ranges of the parameters appear in several places; see, for example, [Z3, Sections 7.1 and 7.7]. Here we derive them for all real α using some general pairings directly from Theorems 1.3 and 2.5, (3), and some general results on Lebesgue classes.

Theorem 6.1. The dual space of every B_{α0} can be identified with every B_{q}^{1} under each of the pairings

(24)  ∫_{𝔹} I_{s}^{t}f \overline{I_{α+q+t}^{−α−q+s}g} dν_{α+q},

where s, t are chosen to satisfy (2) and (1), f ∈ B_{α0}, and g ∈ B_{q}^{1}.

Proof. It is clear that each pairing in (24) induces a bounded linear functional on B_{α0} via Hölder inequality. Note that I_{α+q+t}^{−α−q+s}g ∈ L_{q}^{1} by (17).

Conversely, let T be a bounded linear functional on B_{α0} and M denote the operator of multiplication by (1 − |z|^{2})^{−α}. Then L = T P_{s}M is a bounded linear functional on C_{00} by Theorem 1.3. So there is a complex, hence finite, Borel measure μ on 𝔹 such that Lh = ∫_{𝔹} h dμ for all h ∈ C_{00}. Pick h = M^{−1}I_{s}^{t}f for f ∈ B_{α0}. By (3), we have Lh = T P_{s}MM^{−1}I_{s}^{t}f = T P_{s}I_{s}^{t}f = C_{s+t}^{−1}T f so that T f = C_{s+t} ∫_{𝔹} I_{s}^{t}f(z) (1 − |z|^{2})^{α} dμ(z).

Let φ ∈ L_{q}^{1} be the Radon-Nikodym derivative of μ with respect to ν_{q}. We can also replace f by C_{s+t}P_{s}I_{s}^{t}f by (3). Then T f = C_{s+t}^{2} ∫_{𝔹} V_{s}^{t}(I_{s}^{t}f) \overline{φ} dν_{α+q}. Using the form of the adjoint of V_{s}^{t} computed in the proof of [Z3, Theorem 2.10], we obtain T f = C_{s+t}^{2} ∫_{𝔹} I_{s}^{t}f \overline{(V_{s}^{t})^{∗}φ} dν_{α+q} = C_{s+t}^{2} ∫_{𝔹} I_{s}^{t}f \overline{V_{α+q+t}^{−α−q+s}φ} dν_{α+q}. Define g = C_{s+t}^{2}P_{α+q+t}φ; then g ∈ B_{q}^{1} by Theorem 2.5. This yields the desired form.

Under the conditions of the theorem, g ∈ B_{q}^{1} obtained for a given T is unique. If there were two such g’s, their difference, labeled g, would give 0 as the value of the integral in (24) for all f ∈ B_{α0}. This forces I_{α+q+t}^{−α−q+s}g = 0. Applying P_{α+q+t} to this and using (3) show that g = 0. □

Theorem 6.2. The dual space of every B_{q}^{1} can be identified with every B_{α} under each of the pairings in (24), where now s, t are chosen to satisfy (18) and (17) with p = 1, f ∈ B_{q}^{1}, and g ∈ B_{α}.

Proof. The proof is almost identical to the proof of Theorem 6.1 with the roles of Bloch and Besov spaces interchanged. □

Corollary 6.3. There exist functions in every B_{q}^{1} and in every B_{α0}, and hence in every B_{α}, whose Taylor series do not converge in norm.

Proof. See [Z1], where this result is obtained in certain other settings. □

Our next purpose is to establish interpolation relations among the B_{α} family of spaces once again using Theorem 1.3. For basic definitions and notation regarding interpolation, we refer the reader to [Z3, Section 1.8]. We start with interpolation between Lebesgue classes, where [X, Y]_{θ} is the complex interpolation space between the Banach spaces X and Y.

Lemma 6.4. Suppose −∞ < α < σ < β < ∞ with σ = (1 − θ)α + θβ for some 0 < θ < 1. Then [L_{α}^{∞}, L_{β}^{∞}]_{θ} = L_{σ}^{∞}. Similar results hold for C_{σ} and C_{σ0}.

Proof. To begin with, L_{α}^{∞} ∩ L_{β}^{∞} = L_{α}^{∞} ⊂ L_{β}^{∞} = L_{α}^{∞} + L_{β}^{∞}.

First suppose φ ∈ L_{σ}^{∞}. For ζ ∈ S := { ζ : 0 ≤ Re ζ ≤ 1 } and z ∈ 𝔹, define F_{ζ}(z) = (1 − |z|^{2})^{σ−(1−ζ)α−ζβ} φ(z), which, as a function of ζ, is continuous on S and holomorphic in its interior. Obviously, F_{θ}(z) = φ(z). We have

∥F_{ζ}∥_{L_{α}^{∞}+L_{β}^{∞}} ≤ ∥φ∥_{L_{σ}^{∞}} sup_{z∈𝔹} (1 − |z|^{2})^{(1−Re ζ)(β−α)} ≤ ∥φ∥_{L_{σ}^{∞}}

for all ζ ∈ S. Similarly, ∥F_{iy}∥_{L_{α}^{∞}} ≤ ∥φ∥_{L_{σ}^{∞}} and ∥F_{1+iy}∥_{L_{β}^{∞}} ≤ ∥φ∥_{L_{σ}^{∞}} for all y ∈ ℝ. Thus also ∥F ∥ ≤ ∥φ∥_{L_{σ}^{∞}} and φ ∈ [L_{α}^{∞}, L_{β}^{∞}]_{θ}.

Next suppose φ ∈ [L_{α}^{∞}, L_{β}^{∞}]_{θ}. Then there is an F_{ζ}(z) as above with F_{θ}(z) = φ(z). Put M_{0} = sup_{y} ∥F_{iy}∥_{L_{α}^{∞}}, M_{1} = sup_{y} ∥F_{1+iy}∥_{L_{β}^{∞}}, and M_{θ} = sup_{y} ∥F_{θ+iy}∥_{L_{σ}^{∞}}. Then

∥φ∥_{L_{σ}^{∞}} ≤ sup_{y∈ℝ,z∈𝔹}(1 − |z|^{2})^{σ} |F_{θ+iy}(z)| = M_{θ} ≤ M_{0}^{1−θ}M_{1}^{θ}

by Hadamard three lines theorem, and this shows φ ∈ L_{σ}^{∞}. □

Theorem 6.5. Suppose −∞ < α < σ < β < ∞ with σ = (1 − θ)α + θβ for some 0 < θ < 1. Then [B_{α}, B_{β}]_{θ} = B_{σ} and [B_{α0}, B_{β0}]_{θ} = B_{σ0}.

Proof. Let s satisfy (2) and t satisfy (1) both with β in place of α.

The operator P_{s} maps L_{α}^{∞} onto B_{α}, L_{β}^{∞} onto B_{β}, and L_{σ}^{∞} onto B_{σ} boundedly by Theorem 1.3. By Lemma 6.4 and interpolation theory, P_{s} maps L_{σ}^{∞} = [L_{α}^{∞}, L_{β}^{∞}]_{θ} into [B_{α}, B_{β}]_{θ}. Thus B_{σ} ⊂ [B_{α}, B_{β}]_{θ}.

On the other hand, the operator I_{s}^{t} maps B_{α} into L_{α}^{∞}, B_{β} into L_{β}^{∞}, and B_{σ} into L_{σ}^{∞} boundedly by Definition 1.1. By Lemma 6.4 and interpolation theory, I_{s}^{t} maps [B_{α}, B_{β}]_{θ} into L_{σ}^{∞} = [L_{α}^{∞}, L_{β}^{∞}]_{θ}. By Definition 1.1, the last mapping just means that any f in [B_{α}, B_{β}]_{θ} also belongs to B_{σ}. □

7. Extremal Problem

Point evaluations already considered in Corollary 5.5 are important in function spaces for many reasons. Therefore it is of interest to know how large they can be as operators in any given space. In a weighted Bloch space, their size can be measured by the quantity S_{α}(b) of (5). In this section, we provide a solution of the related extremal problem.

Lemma 7.1. Given α, pick s, t to satisfy (1). Then on B_{α} considered with the norm ∥f∥_{B_{α}} = ∥I_{s}^{t}f∥_{L_{α}^{∞}}, the operator W_{ψ}^{α} of (7) is a linear surjective isometry with inverse (W_{ψ}^{α})^{−1} = D_{s+t}^{−t}T_{ψ^{−1}}^{α}D_{s}^{t}.

Proof. For f ∈ B_{α}, let g = D_{s}^{t}f ∈ B_{α+t}, note that α + t > 0, and let w = ψ(z). By (9) and Proposition 2.1, we have

∥W_{ψ}^{α}f∥_{B_{α}} = ∥I_{s}^{t}W_{ψ}^{α}f∥_{L_{α}^{∞}} = ∥T_{ψ}^{α+t}g∥_{L_{α+t}^{∞}}

= sup_{z∈𝔹} (1 − |z|^{2})^{α+t} \frac{(1 − |ψ(z)|^{2})^{α+t}}{(1 − |z|^{2})^{α+t}} |g(ψ(z))|

= sup_{w∈𝔹} (1 − |w|^{2})^{α+t} |g(w)| = ∥g∥_{B_{α+t}} = ∥f∥_{B_{α}}.

The statement about the inverse is clear by (12) and (Jψ^{−1})(ψ(z)) = 1/Jψ(z). □

Note that W_{ψ}^{α} is involutive when ψ is; this is so in particular with T_{b}^{α}. Because JU is constant, we see that T_{U} commutes with all the radial differential operators D_{s}^{t} and hence W_{U}^{α}f = T_{U}^{α}f. When α = 0, Lemma 7.1 reduces to the Möbiusinvariance of the classical Bloch space since then the derivatives cancel out and W_{ψ}^{0}f = f ∘ ψ.

Proof of Theorem 1.4. Finding the extremal function at b = 0 is easy. First for α > 0 and t = 0, clearly f(0) ≤ sup_{z∈𝔹} (1 − |z|^{2})^{α} |f(z)| = ∥f∥_{B_{α}} = 1, and equality holds if and only if f is identically 1 by the maximum modulus principle. So the unique extremal function is f_{0} ≡ 1. Next for any α and t ≠ 0 satisfying (1), we have

d_{0} f(0) = D_{s}^{t}f(0) ≤ sup_{z∈𝔹} (1 − |z|^{2})^{α+t} |D_{s}^{t}f(z)| = ∥f∥_{B_{α}} = 1.

Now the unique extremal function is f_{0} ≡ d_{0}^{−1}.

To carry the result to other b ∈ 𝔹, we use W_{b}^{α} of Lemma 7.1. Finding S_{α}(b) is equivalent to finding sup{ (W_{b}^{α}f)(0) > 0 : f ∈ B_{α}, ∥W_{b}^{α}f∥_{B_{α}} = 1 }. The involutive property of W_{b}^{α} shows that the unique extremal function at b is f_{b} = T_{b}^{α}1 for α > 0 and t = 0, and it is f_{b} = d_{0}^{−1} W_{b}^{α}1 for any α and t ≠ 0 satisfying (1). To write the detailed form of f_{b}, we use for convenience s_{0} = 2α + t − (N + 1) when t ≠ 0. Then by (10) and (13),

f_{b}(z) = \begin{cases}\frac{(1 − |b|^{2})^{α}}{(1 − ⟨z, b⟩)^{2α}} & (α > 0, t = 0); \\ (1 − |b|^{2})^{α+t} K_{2α+t−(N+1)}(z, b) & (α + t > 0, s = s_{0}). \end{cases}

It can be checked that setting b = 0 in the second line above actually yields f_{0} using the explicit forms of D_{s}^{t} in [Ka1, Definition 3.1].

We are done with B_{α}. However, let’s see that the extremal function lies in B_{α0} in each case using Definition 1.1. When α > 0 and t = 0 in the norm, (1 − |z|^{2})^{α} f_{b}(z) → 0 as |z| → 1 obviously. When α ≤ 0,

(1 − |z|^{2})^{α} I_{s_{0}}^{t}f_{b}(z) = (1 − |z|^{2})^{α+t} (1 − |b|^{2})^{α+t} K_{2(α+t)−(N+1)}(z, b)

= \frac{(1 − |z|^{2})^{α+t} (1 − |b|^{2})^{α+t}}{(1 − ⟨z, b⟩)^{2(α+t)}} → 0

as |z| → 1, since (1) always holds. This completes the proof of the theorem. □

Remark 7.2. As expected, Theorem 1.4 and Lemma 7.1 are the p = ∞ versions of [V, Theorem] and [Ka1, Theorem 8.2] after setting q = αp.

8. Maximality

The subject of this section is α-Möbius invariance and we prove Theorem 1.6 here. For each α > 0, we give a nontrivial example of an α-Möbius-invariant space in Corollary 8.4. Similar results in the case α = 0 are presented in [Z3, Lemma 3.18] and in [T2, Theorem 0.3]. This case is different in its lack of α which causes its proofs to be more difficult and sometimes requiring stronger hypotheses. For example, the equivalent of Proposition 8.1 with α = 0 requires the existence of a nonconstant function in the space. We concentrate on α ≠ 0 here. On the other hand, for any α, only diagonal unitary matrices need to be used in the proofs.

Proposition 8.1. For α ≠ 0, an α-Möbius-invariant space X contains the polynomials.

Proof. Fix b ≠ 0 in 𝔹. Since 1 ∈ X, also W_{b}^{α}1 ∈ X. By Definition 1.2, (11), and (12),

(W_{b}^{α}1)(z) = (D_{s+t}^{−t}T_{b}^{α+t}D_{s}^{t}1)(z) = d_{0} (D_{s+t}^{−t}T_{b}^{α+t}1)(z)

= d_{0} D_{s+t}^{−t} \frac{(1 − |b|^{2})^{α+t}}{(1 − ⟨z, b⟩)^{2(α+t)}} = d_{0} (1 − |b|^{2})^{α+t} D_{s+t}^{−t} ∑_{k=0}^{∞} c_{k} ⟨z, b⟩^{k}

= d_{0} (1 − |b|^{2})^{α+t} ∑_{k=0}^{∞} d_{k}^{−1} c_{k} ⟨z, b⟩^{k} = ∑_{λ} c_{λ} z^{λ},

where c_{λ} ≠ 0 for any λ ∈ ℕ^{N}. Fix any multi-index μ, let U = diag(e^{iθ_{1}}, …, e^{iθ_{N}}) be unitary, θ = (θ_{1}, …, θ_{N}), and consider

f(z) = \frac{1}{(2π)^{N}} ∫_{[−π,π]^{N}} (W_{U}^{α}W_{b}^{α}1)(z) e^{−i⟨μ,θ⟩} dθ,

which belongs to X by the assumptions on the continuity of the W_{U}^{α}-action and the completenesss of X. Recalling the series for W_{b}^{α}1 and that W_{U}^{α} = T_{U}^{α} shows that

f(z) = (det U)^{2α/(N+1)} c_{μ} z^{μ}.

Thus X contains any monomial z^{μ}. □

Definition 8.2. A nonzero bounded linear functional on a normed space X of holomorphic functions on 𝔹 is called decent if it extends to be continuous on H(𝔹).

Proof of Theorem 1.6. First let α > 0. Let L be a decent functional on X, and let U and θ be as in the proof of Proposition 8.1. Define another linear functional by

L_{U}f = \frac{1}{(2π)^{N}} ∫_{[−π,π]^{N}} L(T_{U}^{α}f) (det U)^{−2α/(N+1)} dθ (f ∈ X).

If we expand T_{U}^{α}f into series, the decency of L shows that L_{U}f = f(0) L(1). On the other hand,

|f(0)| |L(1)| = |L_{U}f| ≤ \frac{1}{(2π)^{N}} ∫_{[−π,π]^{N}} ∥L∥ ∥T_{U}^{α}f∥_{X} dθ ≤ C ∥L∥ ∥f∥_{X}.

Replacing f by T_{z}^{α}f gives

(1 − |z|^{2})^{α} |f(z)| |L(1)| ≤ C ∥L∥ ∥T_{z}^{α}f∥_{X} ≤ C ∥L∥ ∥f∥_{X}.

Hence if L(1) ≠ 0, then ∥f∥_{B_{α}} ≤ C ∥L∥ |L(1)|^{−1} ∥f∥_{X} < ∞, and X ⊂ B_{α} with continuous inclusion.

Now we prove that a decent functional on X that is nonzero on 1 always exists. As in the proof of Proposition 8.1, for fixed b ≠ 0 in 𝔹 and by the decency of L,

L(T_{b}^{α}1) = ∑_{λ} c_{λ} L(z^{λ})

with c_{λ} ≠ 0 for any multi-index λ. If L(T_{b}^{α}1) = 0 for all b ≠ 0 in 𝔹, then L(z^{λ}) = 0 for all multi-indices λ and L = 0 because of its decency. So L(T_{b}^{α}1) ≠ 0 for some b ≠ 0 in 𝔹. For such a b, define the linear functional L_{b} = LT_{b}^{α} on X. Then L_{b}(1) ≠ 0 and ∥L_{b}∥ ≤ ∥L∥ ∥T_{b}^{α}∥ ≤ C ∥L∥. The functional L_{b} is also decent, because T_{b}^{α} is continuous on H(𝔹) since any Möbius transformation φ_{b} and Jφ_{b} take compact subsets of 𝔹 to other compact subsets. This yields the desired result.

If α < 0, pick s, t to satisfy (1), and consider the space Y = { g = D_{s}^{t}f : f ∈ X } with the norm ∥g∥_{Y} = ∥f∥_{X}. It is a matter of writing down the definitions and noting that T_{U}^{α} commutes with all D_{s}^{t} to check that Y is (α + t)-Möbius-invariant. Then Y ⊂ B_{α+t} and X ⊂ B_{α} with continuous inclusions by Proposition 2.1. □

Corollary 8.3. There is no α-Möbius-invariant closed subspace of H(𝔹) other than {0}.

Note that constants do not form an α-Möbius-invariant subspace. In the sense of Definition 1.5, H(𝔹) is not α-Möbius-invariant either, but this is a technicality.

Proof. Suppose Y is an invariant subspace that is properly contained in H(𝔹). By Hahn-Banach theorem, there is a continuous linear functional L ≠ 0 on H(𝔹) whose restriction to Y is 0. For f ∈ H(𝔹), set χ(f) = sup_{ψ} |L(T_{ψ}^{α}f)| and define X as the completion of { f ∈ H(𝔹) : χ(f) < ∞ }. It is easy to see that χ is a seminorm and that Theorem 1.6 is valid if X is given by a seminorm. Then χ(f) = 0 for all f ∈ Y and hence Y is contained in X on which L is a decent functional. By Theorem 1.6, X is contained in B_{α} continuously, that is, ∥f∥_{B_{α}} ≤ C χ(f) for all f ∈ X. So if f ∈ Y, then ∥f∥_{B_{α}} = 0 and thus Y = {0}. The conclusion also implies that χ is a true norm. □

Corollary 8.4. Suppose p > 0 and q > −1, or p ≥ 2 and −(N + 1) ≤ q ≤ 1. Suppose α > 0. Suppose further α, q, and p are related by N + 1 + q = αp. Then B_{q}^{p} is an α-Möbius-invariant space and thus B_{q}^{p} ⊂ B_{α} with continuous inclusion.

Proof. By [BB, Theorem 3.3], [Ka1, Theorem 8.2] and the given relation among the parameters, the Besov space B_{q}^{p} is α-Möbius-invariant. Then Theorem 1.6 applies. The set of (p, q) in the right half plane satisfying N + 1 + q = αp is a ray with slope α and q-intercept −(N + 1). □

The inclusion part of this result appears in [BB, Corollary 5.5] with a totally different proof, and is in fact a Sobolev-type imbedding.

9. Metrics and Lipschitz Property

In this section, we consider α > 0 and develop the Hermitian metrics ρ_{α} with respect to which the weighted Bloch spaces B_{α} have the Lipschitz property. We start with their infinitesimal forms. For a different point of view regarding Hermitian metrics and Bloch spaces, see [Z4].

For z ∈ 𝔹, we define the matrix g_{α}(z) by

g_{αij} (z) = \frac{1}{(1 − |z|^{2})^{2(1+α)}} ((1 − |z|^{2}) δ_{ij} + z_{i}\overline{z}_{j}) (1 ≤ i, j ≤ N),

where δ_{ij} is the Kronecker delta. The only difference of g_{α} from the infinitesimal Bergman metric g_{0} is the presence of α in the power of the denominator. Clearly g_{αji} = \overline{g}_{αij}. Further, g_{α} is unitarily invariant in that g_{α}(Uz) = U g_{α}(z) U^{−1} for a unitary transformation U of ℂ^{N}. We compute easily that

det g_{α}(z) = \frac{1}{(1 − |z|^{2})^{N+1+2Nα}} = K_{2Nα}(z, z) > 0 (z ∈ 𝔹),

which also shows the form of g_{α} when N = 1. The leading principal minors of g_{α}(z) are just det g_{α}(z) of all the dimensions from 1 through N, which are all positive. Thus g_{α}(z) is a positive definite matrix on 𝔹. By the same reason, g_{α}(z) is invertible with its inverse given by

g_{α}^{ij}(z) = (1 − |z|^{2})^{1+2α} (δ_{ij} − z_{i}\overline{z}_{j}) (1 ≤ i, j ≤ N).

Therefore g_{α} is an infinitesimal Hermitian metric on 𝔹. It gives rise to a distance on 𝔹 in the usual manner. If γ = (γ_{1}, …, γ_{N}) is a curve in 𝔹 joining z and w in 𝔹, its α-length is

l_{α}(γ) = ∫_{0}^{1}⟨g_{α}(γ(t))γ′(t), γ′(t)⟩^{1/2}dt.

By taking the infimum of l_{α}(γ) over all curves joining z and w, we obtain a distance ρ_{α}(z, w) between z and w. If there exists a curve on which the infimum is attained, it is a geodesic of ρ_{α}.

Let z = 0 and w = (r, 0, …, 0) with 0 < r < 1. The unitary invariance of g_{α} entails that the line segment γ_{r} joining z and w is a geodesic. Using the parametrization γ_{r}(t) = (t, 0, …, 0) for 0 ≤ t ≤ r, we compute

(25)  l_{α}(γ_{r}) = ∫_{0}^{r} \frac{dt}{(1 − t^{2})^{1+α}} ∼ ∫_{0}^{r} \frac{dt}{(1 − t)^{1+α}} = \frac{1}{α}(\frac{1}{(1 − r)^{α}} − 1).

As w → 1, lim_{r→1^{−}} l_{α}(γ_{r}) = ∞; that is, 𝔹 is unbounded in the metric ρ_{α}. This

simple result has important implications for ρ_{α}. By the Hopf-Rinow theorem (see [d, Theorem 7.2.8]), (𝔹, ρ_{α}) is geodesically complete, complete as a metric space, its closed and bounded subsets are compact, and there exists a geodesic of g_{α} joining any two points in 𝔹.

Associated to any Hermitian metric, there is defined a Laplace-Beltrami operator (see [Sto, Section 3.1]), which in our case is

Δ̃_{α} = \frac{2}{det g_{α}} ∑_{i,j=1}^{N}[\frac{∂}{∂\overline{z}_{j}} ((det g_{α}) g_{α}^{ij} \frac{∂}{∂z_{i}} ) + \frac{∂}{∂z_{i}} ((det g_{α}) g_{α}^{ij} \frac{∂}{∂\overline{z}_{j}} )]

= 4 (1 − |z|^{2})^{1+2α}[ ∑_{i,j=1}^{N}(δ_{ij} − z_{i}\overline{z}_{j}) \frac{∂^{2}}{∂z_{i} ∂\overline{z}_{j}} + (N − 1) α (R + \overline{R})]

= 4 ∑_{i,j=1}^{N} g_{α}^{ij}(z) \frac{∂^{2}}{∂z_{i} ∂\overline{z}_{j}} + 4 (N − 1) α (1 − |z|^{2})^{1+2α}(R + \overline{R}),

where Rf(z) = ⟨∇f(z), \overline{z}⟩ is the classical radial derivative of f, and ∇ denotes the complex gradient. The operators Δ̃_{α} are reminiscent of the variants of Laplace-Beltrami operators defined in a different context in [G, (1.11)], but there is a difference. Our Δ̃_{α} have no constant terms and annihilate constants. This can have interesting connections; see [Ka2, Section 4]. Here also (Δ̃_{α}f)(0) = (Δf)(0), where Δ is the usual Laplacian. Further, Δ̃_{α} = (1 − |z|^{2})^{2(1+α)}Δ when N = 1, so Δ̃_{α} and Δ annihilate the same functions on 𝔻.

The infinitesimal Bergman metric is obtained as

(26)  g_{0ij} = \frac{1}{N + 1} \frac{∂^{2} log K_{0}(z, z)}{∂z_{i} ∂\overline{z}_{j}}.

The presence of the first-order terms in Δ̃_{α} for α > 0 and N > 1 imply that holomorphic functions are not annihilated by it, and more importantly, the corresponding g_{α} is not a Kähler metric; see [Ste, p. 26]. This is equivalent to the fact that g_{α} cannot be obtained by differentiation as in (26) for α > 0 and N > 1. For N = 1 and small positive integer α, we can use integration by partial fractions to find formulas similar to (26). For example, let

L_{1}(z, w) = \frac{1}{3}(log \frac{1}{1 − z \overline{w}} + \frac{1}{1 − z \overline{w}} + \frac{1}{2 (1 − z \overline{w})^{2}} ) (z, w ∈ 𝔻);

then

\frac{∂^{2}L_{1}(z, z)}{∂z ∂\overline{z}} = \frac{1}{(1 − |z|^{2})^{4}} = g_{1}(z) (z ∈ 𝔻).

Each Hermitian metric additionally gives rise to a gradient field; see [Sto, Section 3.4]. For f ∈ H(𝔹) and our g_{α}, it takes the form

| ˜∇_{α}f(z)|^{2} = 2 ∑_{i,j=1}^{N} g_{α}^{ij} \frac{∂f}{∂z_{i}} \overline{\frac{∂f}{∂z_{j}}} = 2 (1 − |z|^{2})^{1+2α} (|∇f(z)|^{2} − |Rf(z)|^{2})
= 2 (1 − |z|^{2})^{2α} | ˜∇f(z)|^{2},

where ˜∇ = ˜∇_{0}/\sqrt{2} and is called the invariant gradient since ˜∇f(z) = ∇(f ∘φ_{z})(0).

Before relating B_{α} to ρ_{α}, we find an equivalent definition of B_{α} much the same way as the early definition of B_{0} in [T1, Definition 3.1]. To this end, set

Q_{α}f(z) = sup_{w≠0} \frac{|⟨∇f(z), \overline{w}⟩|}{\sqrt{⟨g_{α}(z)w, w⟩}} (z ∈ 𝔹),

and f ∈ H(𝔹).

Lemma 9.1. If α > 0 and f ∈ H(𝔹), then f ∈ B_{α} if and only if Q_{α}f ∈ L^{∞} and if and only if | ˜∇_{α}f| ∈ L^{∞}.

Proof. The proof of [Z3, Theorem 3.1] with straightforward modifications for the presence of α yields that Q_{α}f(z) = (1 − |z|^{2})^{α} | ˜∇f(z)| = | ˜∇_{α}f(z)|/\sqrt{2}. Then [Z3, Theorem 7.2 (a)] gives us what we want. □

When N = 1, Rf(z) = zf′(z), ∇f(z) = f′(z), ˜∇f(z) = (1 − |z|^{2}) f′(−z), and Q_{α}f(z) = (1 − |z|^{2})^{1+α} |f′(z)|.

For f ∈ H(𝔹), a direct computation shows that

Δ̃_{α}(|f|^{2})(z) = 2 | ˜∇_{α}f(z)|^{2} + 2 (N − 1) α Re(\overline{f}(z) Rf(z)).

Thus Δ̃_{α}(|f|^{2}) = C | ˜∇_{α}f|^{2} for f ∈ H(𝔹) if and only if α = 0 or N = 1.

Proof of Theorem 1.7. Let z, w ∈ 𝔹 and γ_{α} be a geodesic of ρ_{α} joining them, which we know exists. Then

|f(z) − f(w)| ≤ ∫_{γ_{α}} |⟨∇f(u), \overline{du}⟩| ≤ ∫_{γ_{α}} Q_{α}f(u) \sqrt{⟨g_{α}(u)du, du⟩}

≤ C ∥f∥_{B_{α}} ∫_{0}^{1} \sqrt{⟨g_{α}(γ_{α}(t))γ′_{α}(t), γ′_{α}(t)⟩} dt = C ∥f∥_{B_{α}} ρ_{α}(z, w).

In the converse direction, N = 1. Consider the curve γ(t) = t for 0 ≤ t ≤ r < 1 between 0 and r. Let z ∈ 𝔻 and ξ(t) = φ_{z}(γ(t)). Then ξ(t) is a curve between z and w = φ_{z}(r), ξ′(t) = φ′_{z}(t), and

l_{α}(ξ) = ∫_{0}^{r} \frac{|φ′_{z}(t)|}{(1 − |φ_{z}(t)|^{2})^{1+α}} dt = ∫_{0}^{r} \frac{1}{(1 − t^{2})^{1+α}} \frac{|1 − \overline{z}t|^{1+α}}{(1 − |z|^{2})^{α}} dt

≤ \frac{(1 + |z|)^{2α}}{(1 − |z|^{2})^{α}} ∫_{0}^{r} \frac{dt}{(1 − t)^{1+α}} = \frac{(1 + |z|)^{2α}}{(1 − |z|^{2})^{α}} \frac{1}{α}(\frac{1}{(1 − r)^{α}} − 1).

by [R, Theorem 2.2.2 (iv)] and (25). The same is true if also φ_{z} is composed with rotations in obtaining ξ from γ. In all cases r = |φ_{z}(w)|. Then

ρ_{α}(z, w) ≤ \frac{2^{2α}}{α} \frac{1}{(1 − |z|^{2})^{α}} (\frac{1}{(1 − |φ_{z}(w)|)^{α}} − 1).

Now let w = z + h with |h| small. Then

φ_{z}(w) ∼ \frac{|h|}{1 − |z|^{2}}  and  (1 − |φ_{z}(w)|)^{α} ∼ 1 − \frac{α|h|}{1 − |z|^{2}} ,

and thus

ρ_{α}(z, w) ≤ \frac{C}{(1 − |z|^{2})^{α}} \frac{|h|}{1 − |z|^{2} − α|h|}.

Using the assumption,

C ≥ \frac{|f(z) − f(w)|}{ρ_{α}(z, w)} ≥ C \frac{|f(z) − f(z + h)|}{|h|} (1 − |z|^{2})^{α} (1 − |z|^{2} − α|h|).

Letting h → 0, we obtain (1 − |z|^{2})^{α+1} |f′(z)| ≤ C for all z ∈ 𝔻. Therefore f ∈ B_{α}. □

The case of the classical Bloch space has some extra properties which we do not know for α > 0. The metric ρ_{0} is invariant under Möbius transformations, but we do not know of any isometries of ρ_{α} other than unitary transformations. The invariance gives rise to the well-known explicit logarithmic formula for ρ_{0}(z, w) which we do not have for general α > 0. This lack of explicit formula for ρ_{α} is the main obstacle to obtaining the converse in Theorem 1.7 for N > 1. The computations involved for the style of proof presented above for N > 1 are prohibitively complicated. For N = 1 and small positive integer α, it is possible to obtain explicit expressions for ρ_{α}(z, w) using integration by partial fractions and the unitary invariance of g_{α}. For example, with α = 1, a tedious computation yields that

ρ_{1}(z, w) = \frac{1}{4} log \frac{1 + |φ_{z}(w)|}{1 − |φ_{z}(w)|} + \frac{1}{2} \frac{|φ_{z}(w)| (1 − 2 Re(\overline{z} φ_{z}(w)) + |z|^{2})}{1 − |φ_{z}(w)|^{2}} (z, w ∈ 𝔻).

As a final note, let’s compute the holomorphic sectional curvatures (see [Kr1, Section 2.1]) of the new metrics. When N = 1, they are

κ_{α}(z) = −\frac{Δ log g_{α}(z)}{g_{α}(z)} = −4 (1 + α) (1 − |z|^{2})^{2α} (z ∈ 𝔻).

Clearly κ_{α}(z) ≤ 0, and κ_{α}(z) → 0 as z → ∂𝔻. This curvature is more difficult to compute exactly for higher N, but it is clear that the factor 1−|z|^{2} will persist with a positive power. So the new metrics for α > 0 have curvatures that are neither constant nor bounded away from 0 unlike the Bergman metric.

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Department of Mathematics, Bi̇lkent University, Ankara 06800, Turkey

   E-mail address: kaptan@fen.bilkent.edu.tr

Department of Mathematics, Bi̇lkent University, Ankara 06800, Turkey

   E-mail address: serdartulu@iyte.edu.tr

Current address: Department of Mathematics, İzmir Institute of Technology (İYTE), Urla, İzmir 35437, Turkey