Annales Academiæ Scientiarum Fennicæ Mathematica Volumen 23, 1998, 461–473

BLOCH SPACE ON THE UNIT BALL OF C^{n}

Maria Nowak

Marie Curie-Skłodowska University, Institute of Mathematics pl. Marii Curie-Skłodowskiej 1, 20-031 Lublin, Poland; nowakm@golem.umcs.lublin.pl

Abstract. In this paper we prove that for 0 < p < ∞, the norm of a function f in the Bergman space L_{a}^{p} on the unit ball B of C^{n}, n ≥ 1, is equivalent to the quantity

∫_{B} |˜∇f(z)|^{2}|f(z)|^{p−2}h(|z|) dτ(z),

where ˜∇ and τ denote the invariant gradient and invariant measure on B, respectively, and

h(|z|) = ∫_{|z|}^{1} \frac{(1 − t^{2})^{n−1}(1 − t^{2n})}{t^{2n−1}} dt.

If n > 1, this result allows us to extend the characterization J_{2} of the Bloch space obtained in [OYZ, Theorem 2] to the range 0 < p < ∞. We also get this kind of description of Bloch functions for n = 1.

Moreover, we generalize the result obtained in [CKP] and show that f ∈ H(B) is a Bloch function if and only if for some p, 0 < p < ∞, |˜∇f(z)|^{p}dv(z) is a Bergman–Carleson measure. Finally, we get some results for spaces H^{p} and BMOA, e.g. an extension of the classical Littlewood– Paley inequality to the case of the unit ball.

1. Introduction

Let B denote the unit ball in C^{n}, dv the normalized Lebesgue measure on B and dσ the normalized surface measure on the boundary S of B. Let H(B) be the space of all holomorphic functions on B. For 0 < p < ∞, the Bergman space L_{a}^{p} and the Hardy space H^{p} are defined respectively as

L_{a}^{p} = {f : f ∈ H(B), ∥f∥_{L_{a}^{p}}^{p} = ∫_{B} |f(z)|^{p} dv(z) < ∞}

and

H^{p} = {f : f ∈ H(B), ∥f∥_{H^{p}}^{p} = sup_{0<r<1} ∫_{S} |f(rζ)|^{p} dσ(ζ) < ∞}.

1991 Mathematics Subject Classification: Primary 32A37, 32A35. This work was supported in part by KBN grant no. 2 PO3A-002-08.

For f ∈ H(B), set

Q_{f}(z) = sup\{\frac{|⟨∇f(z), x̄⟩|}{(H_{z}(x, x))^{1/2}} : 0 ≠ x ∈ C^{n}\},

where ∇f(z) = (∂f/∂z_{1}, …, ∂f/∂z_{n}) is the complex gradient of f and H_{z}(x, x) is the Bergman metric on B, i.e.

H_{z}(x, x) = \frac{n + 1}{2} \frac{(1 − |z|^{2})|x|^{2} + |⟨x, z⟩|^{2}}{(1 − |z|^{2})^{2}} .

The Bloch space ℬ (introduced by R. Timoney, [T]) is the set of holomorphic functions f on B for which

∥f∥_{ℬ} = sup{Q_{f}(z) : z ∈ B} < ∞.

R. Timoney [T] has proved that the norms

\begin{aligned}∥f∥_{1} & = sup\{|∇f(z)|(1 − |z|^{2}) : z ∈ B\}, \\ ∥f∥_{2} & = sup\{|⟨∇f(z), z̄⟩|(1 − |z|^{2}) : z ∈ B\} \end{aligned}

and ∥f∥_{ℬ} are equivalent.

Let ˜∇ denote the invariant gradient on B, that is,

(˜∇f)(z) = ∇(f ∘ φ_{z})(0),

where φ_{z} denotes the involutive automorphism of B satisfying φ_{z}(0) = z, φ_{z}(z) = 0.

Recently C. Ouyang, W. Yang and R. Zhao [OYZ] gave the following characterization of the Bloch space.

Theorem A. Let n > 1, p ≥ 2; then the following quantities are equivalent:

(a) ∥f∥_{ℬ}^{p} ,

(b) J_{2} = sup_{a∈B} ∫_{B} |˜∇f(z)|^{2}|f(z) − f(a)|^{p−2}(1 − |φ_{a}(z)|^{2})^{n+1}|φ_{a}(z)|^{−2n+2} dτ(z),

(c) J_{3} = sup_{a∈B} ∫_{B} |˜∇f(z)|^{2}|f(z) − f(a)|^{p−2}[G(z, a)]^{1+(1/n)} dτ(z),

where G(z, a) is the Green function of B and dτ(z) = (n + 1)dv(z)/(1 − |z|^{2})^{n+1} .

The authors [OYZ, p. 4310, Remark 3] conjectured that Theorem A holds for all p > 0, that is, also for 0 < p < 2. Here we show that the quantities given by formulas (a) and (b) are equivalent for all positive p. Moreover, we prove

Bloch space on the unit ball of C^{n} 463

Theorem 1. Let 0 < p < ∞ and n > 1; then ∥f∥_{ℬ}^{p} and J_{2} , defined in Theorem A, and the following quantities are equivalent

\begin{aligned}J_{4} & = sup_{a∈B} ∫_{B} |˜∇f(z)|^{2}|f(z) − f(a)|^{p−2}G(z, a)(1 − |φ_{a}(z)|^{2}) dτ(z), \\ J_{5} & = sup_{a∈B} ∫_{B} |˜∇f(z)|^{2}|f(z) − f(a)|^{p−2}h(|φ_{a}(z)|) dτ(z), \end{aligned}

where the function h is given by the formula

h(|z|) = ∫_{|z|}^{1} \frac{(1 − t^{2})^{n−1}(1 − t^{2n})}{t^{2n−1}} dt.

For the case n = 1 we get

Theorem 2. Let 0 < p < ∞ and f be a holomorphic function on the unit disc D in the complex plane. Then f is a Bloch function if and only if

(1)  sup_{a∈D} ∫_{D} |f(z) − f(a)|^{p−2}|f′(z)|^{2}g(z, a)(1 − |φ_{a}(z)|^{2}) dv(z) < ∞

where g(z, a) is the Green function log|(1 − āz)/(z − a)| of D with logarithmic singularity at a ∈ D.

Our next characterization of Bloch functions is connected with a Bergman– Carleson measure. For η ∈ S and δ > 0, let

D_{δ}(η) = {z ∈ B : |1 − ⟨z, η⟩| < δ}.

A positive measure μ on B is called a Bergman–Carleson measure (Carleson measure) if and only if μ(D_{δ}(η)) = O(δ^{n+1}) (μ(D_{δ}(η)) = O(δ^{n})). It has been proved in [CKP] that a holomorphic function f on B is a Bloch function if and only if |˜∇f(z)|^{2} dv(z) is a Bergman–Carleson measure. Here we obtain the following generalization of this result.

Theorem 3. Let 0 < p < ∞. If f ∈ H(B), then the following statements are equivalent:

(a) ∥f∥_{ℬ} < ∞,

(b) |˜∇f(z)|^{p} dv(z) is a Bergman–Carleson measure,

(c) |∇f(z)|^{p}(1 − |z|^{2})^{p} dv(z) is a Bergman–Carleson measure.

Section 4 of this paper contains the proof of Theorem 3 and some results for the Hardy space H^{p} and BMOA functions in the unit ball, e.g. a higher dimensional version of the classical Littlewood–Paley result [LP]: if f ∈ H^{p}, 2 ≤ p < ∞, then

∫_{D} |f′(z)|^{p}(1 − |z|)^{p−1} dv(z) < ∞.

The author would like to express her sincere thanks to the referee for helpful comments and remarks which greatly improved the final version of this paper.

2. Preliminaries

For a ∈ B, let φ_{a} be the involutive automorphism of B given by the formula

(2)  φ_{a}(z) = \frac{a − P_{a}z − (1 − |a|^{2})^{1/2}Q_{a}z}{1 − ⟨z, a⟩} ,

where P_{a} is the orthogonal projection into the space spanned by a i.e.

P_{a}z = \frac{⟨z, a⟩}{|a|^{2}}a, P_{0}z = 0,

and

Q_{a} = I − P_{a}.

It has been shown in [R] that

(3)  φ′_{a}(0) = −(1 − |a|^{2})P_{a} − \sqrt{1 − |a|^{2}} Q_{a}.

Now, by the chain rule and the symmetry of matrix (3) we get

(4)  \begin{aligned}|˜∇f(z)|^{2} & = |∇(f ∘ φ_{z})(0)|^{2} = |∇f(z)φ′_{z}(0)|^{2} = |φ′ ; z̄(0)∇f(z)|^{2} \\ & = (1 − |z|^{2})^{2}|Pz̄∇f(z)|^{2} + (1 − |z|^{2})|Qz̄∇f(z)|^{2} \\ & = (1 − |z|^{2})(|∇f(z)|^{2} − |⟨∇f(z), z̄⟩|^{2}) \\ & = (1 − |z|^{2})(|∇f(z)|^{2} − |Rf(z)|^{2}), \end{aligned}

where Rf is the radial derivative of f.

For u ∈ C^{2}(B), let Δ̃u denote the invariant Laplacian on B, i.e.

(Δ̃u)(z) = Δ(u ∘ φ_{z})(0),

where Δ is the ordinary Laplacian.

K. Hahn and E. Youssfi [HY] proved that for f ∈ H(B)

Q_{f}(z) = \frac{1}{\sqrt{2(n + 1)}} \sqrt{Δ̃|f|^{2}(z)} = \sqrt{\frac{2}{n + 1}} |˜∇f(z)|.

Hence the following statements are equivalent:

(i) f ∈ ℬ, (ii) sup{Δ̃|f|^{2} : z ∈ B} < ∞, (iii) sup{|˜∇f(z)| : z ∈ B} < ∞.

We will say that the quantities A_{f} and B_{f} that depend on f are equivalent if there exists a positive constant C independent of f such that

A_{f}/C ≤ B_{f} ≤ CA_{f} for every f ∈ H(B).

In this paper C and C_{j} will always denote positive constants, independent of f, not necessarily the same at each occurrence.

The next lemma is the generalization of the result due to S. Axler (see also [S]) to the several variables case. For p ≥ 1, this lemma has been proved in [O2].

Bloch space on the unit ball of C^{n} 465

Lemma 1. Let 0 < p < ∞. A holomorphic function f is in the Bloch space ℬ if and only if

(5) sup_{a∈B} ∥f ∘ φ_{a} − f(a)∥_{L_{a}^{p}} < ∞.

Proof. Assume that f ∈ ℬ. Then we have

\begin{aligned}|f(z) − f(0)| & = |∫_{0}^{1}⟨∇f(tz), z̄⟩ dt| ≤ ∫_{0}^{1} |∇f(tz)| |z| dt \\ & ≤ C∥f∥_{ℬ} ∫_{0}^{1} \frac{|z|}{(1 − |zt|)} dt = C∥f∥_{ℬ} ln \frac{1}{1 − |z|}. \end{aligned}

The Möbius invariance property of the Bloch space implies

|f ∘ φ_{a}(z) − f(a)| ≤ C∥f∥_{ℬ} ln \frac{1}{1 − |z|}.

This gives

∫_{B} |f ∘ φ_{a}(z) − f(a)|^{p} dv(z) ≤ C∥f∥_{ℬ}^{p} ∫_{B}(ln \frac{1}{1 − |z|})^{p} dv(z) ≤ C∥f∥_{ℬ}^{p} < ∞.

Now, suppose that sup_{a∈B} ∥f ∘φ_{a}−f(a)∥_{L_{a}^{p}} < ∞. It was shown in [Shi, p. 623 (9)] that for f ∈ H(B) and 0 < p < ∞

(6)  |∇f(0)|^{p} ≤ C ∫_{B} |f(z)|^{p} dv(z).

Replacing f by f ∘ φ_{a} − f(a) we obtain |˜∇f(a)|^{p} = |∇(f ∘ φ_{a})(0)|^{p} ≤ C ∫_{B} |f ∘ φ_{a}(z) − f(a)|^{p} dv(z),

which implies sup_{a∈B} |˜∇f(a)|^{p} < ∞.

3. Proof of Theorems 1 and 2

For f ∈ H(B), let

f_{p}^{#}(z) = p^{2}|f(z)|^{p−2}|˜∇f(z)|^{2}.

The proof of Theorem 1 is based on the following

Lemma 2. If 0 < p < ∞, f ∈ L_{a}^{p} and f(0) = 0, then

∥f∥_{L_{a}^{p}}^{p} = \frac{1}{(2n)^{2}} ∫_{B} f_{p}^{#}(z)h(|z|) dτ(z),

where

h(|z|) = ∫_{|z|}^{1} \frac{(1 − t^{2})^{n−1}(1 − t^{2n})}{t^{2n−1}} dt.

Proof. Combining formulas (3) and (4) of [OYZ, p. 4305] we get

∫_{S} |f(rζ)|^{p} dσ(ζ) = \frac{1}{2n} ∫_{0}^{r} \frac{(1 − t^{2})^{n−1}}{t^{2n−1}} (∫_{B_{t}} f_{p}^{#}(z) dτ(z)) dt,

where B_{t} = {z ∈ C^{n} : |z| < t}.

Multiplying both sides of this equality by r^{2n−1} and integrating with respect to r give

∫_{B} |f(z)|^{p} dv(z) = \frac{1}{2n} ∫_{0}^{1} r^{2n−1} dr(∫_{0}^{r} \frac{(1 − t^{2})^{n−1}}{t^{2n−1}} dt ∫_{B_{t}} f_{p}^{#}(z) dτ(z)).

By Fubini’s theorem

\begin{aligned}∫_{B} |f(z)|^{p} dv(z) & = \frac{1}{2n} ∫_{0}^{1}(∫_{t}^{1} r^{2n−1} dr \frac{(1 − t^{2})^{n−1}}{t^{2n−1}} ∫_{B_{t}} f_{p}^{#}(z) dτ(z)) dt \\ & = \frac{1}{(2n)^{2}} ∫_{0}^{1} \frac{(1 − t^{2n})(1 − t^{2})^{n−1}}{t^{2n−1}} ∫_{B_{t}} f_{p}^{#}(z) dτ(z) dt. \end{aligned}

Let

χ_{|z|}(t) = \{ \begin{matrix}1, & |z| < t, \\ 0, & otherwise, \end{matrix}

then the right side of the last equality can be be written as

\begin{aligned}\frac{1}{(2n)^{2}} ∫_{B} ∫_{0}^{1} \frac{(1 − t^{2n})(1 − t^{2})^{n−1}}{t^{2n−1}} & χ_{|z|}(t)f_{p}^{#}(z) dt dτ(z) \\ & = \frac{1}{(2n)^{2}} ∫_{B} h(|z|)f_{p}^{#}(z) dτ(z). \end{aligned}

Proof of Theorem 1. Let n > 1 and 0 < p < ∞ be fixed. By Lemma 1 ∥f∥_{ℬ}^{p} is equivalent to sup_{a∈B} ∥f ∘ φ_{a} − f(a)∥_{L_{a}^{p}}. Next, using Lemma 2 we obtain

sup_{a∈B} ∥f ∘ φ_{a}−f(a)∥_{L_{a}^{p}}

= sup_{a∈B} \frac{p^{2}}{(2n)^{2}} ∫_{B} h(|z|)|f ∘ φ_{a}(z) − f(a)|^{p−2}|˜∇(f ∘ φ_{a})(z)|^{2} dτ(z)

= sup_{a∈B} \frac{p^{2}}{(2n)^{2}} ∫_{B} h(|φ_{a}(z)|)|f(z) − f(a)|^{p−2}|˜∇f(z)|^{2} dτ(z).

Since

lim_{|z|→1} \frac{h(|z|)}{(1 − |z|^{2})^{n+1}|z|^{−2n+2}} = \frac{n}{2(n + 1)}

and

lim_{|z|→0} \frac{h(|z|)}{(1 − |z|^{2})^{n+1}|z|^{−2n+2}} = \frac{1}{2(n − 1)},

and the function h(|z|) is continuous in the interval (0, 1) there are positive constants C_{1} and C_{2} such that for all z ∈ B \ {0}

C_{1}(1 − |z|^{2})^{n+1}|z|^{−2n+2} ≤ h(|z|) ≤ C_{2}(1 − |z|^{2})^{n+1}|z|^{−2n+2}.

Thus the quantities ∥f∥_{ℬ}^{p} , J_{2} and J_{5} are equivalent. It follows immediately from Lemma 1 of [OYZ] that also J_{4} is equivalent to each of these quantities.

Proof of Theorem 2. The proof is based on the Hardy–Stein identity [H, p. 42]

r \frac{d}{dr} (\frac{1}{2π} ∫_{0}^{2π} |f(re^{iθ})|^{p} dθ) = ∫_{|z|<r} f^{#}(z) dv(z),

where

f^{#}(z) = \frac{1}{2}p^{2}|f(z)|^{p−2}|f′(z)|^{2}.

If f(0) = 0 then we have (by integration)

\frac{1}{2π} ∫_{0}^{2π} |f(re^{iθ})|^{p} dθ = ∫_{0}^{r}(∫_{|z|<t} \frac{1}{t}f^{#}(z) dv(z)) dt.

From this

∫_{D} |f(z)|^{p} dv(z) = 2 ∫_{0}^{1} r(∫_{0}^{r}(∫_{|z|<t} \frac{1}{t}f^{#}(z) dv(z)) dt) dr

= 2 ∫_{0}^{1}(\frac{1}{t} ∫_{t}^{1} r dr ∫_{|z|<t} f^{#}(z) dv(z)) dt

= ∫_{0}^{1} \frac{1 − t^{2}}{t} (∫_{|z|<t} f^{#}(z) dv(z)) dt

= ∫_{D} ∫_{|z|}^{1} \frac{1 − t^{2}}{t} dtf^{#}(z) dv(z) = ∫_{D} h_{1}(|z|)f^{#}(z) dv(z),

where h_{1}(r) = ln(1/r) − (1 − r^{2})/2, 0 < r < 1. Now, applying Lemma 1 and a change of variables we conclude that f is a Bloch function if and only if

(7)  sup_{a∈D} ∫_{D} |f(z) − f(a)|^{p−2}|f′(z)|^{2}h_{1}(|φ_{a}(z)|) dv(z) < ∞.

Since

lim_{r→1^{−}} \frac{(1 − r^{2}) ln(1/r)}{h_{1}(r)} = 2  and  lim_{r→0^{+}} \frac{(1 − r^{2}) ln(1/r)}{h_{1}(r)} = 1,

condition (7) is equivalent to (1).

4. Carleson measure characterizations of Bloch and BMOA space

Proof of Theorem 3. The implications: (a) implies (c) and (b) implies (c) follow immediately from the descriptions of the Bloch space presented in the introduction and from Proposition 3.1 of [CKP]. In fact, for example,

sup_{a∈B} ∫_{B}(\frac{1 − |z|^{2}}{|1 − ⟨z, a⟩|^{2}} )^{n+1}|˜∇f(z)|^{p} dv(z)

≤ sup_{z∈B} |˜∇f(z)|^{p} sup_{a∈B} ∫_{B}(\frac{1 − |z|^{2}}{|1 − ⟨z, a⟩|^{2}} )^{n+1} dv(z) < ∞.

Now assume that (for some positive p) |˜∇f(z)|^{p} dv(z) is a Bergman–Carleson measure. According to Proposition 3.1 of [CKP] this means that

(7)  sup_{a∈B} ∫_{B}(\frac{1 − |z|^{2}}{|1 − ⟨z, a⟩|^{2}} )^{n+1}|˜∇f(z)|^{p} dv(z) < ∞.

Let 0 < r < 1 be arbitrarily fixed. The subharmonity of |∂f/∂z_{i}|^{p} ∘ φ_{a}, i = 1, 2, …, n, implies

(8) |\frac{∂f}{∂z_{i}} (a)|^{p} ≤ \frac{1}{r^{2n}} ∫_{B_{r}} |\frac{∂f}{∂z_{i}} |^{p} ∘ φ_{a}(z) dv(z) ≤ \frac{1}{r^{2n}} ∫_{B_{r}} |∇f|^{p} ∘ φ_{a}(z) dv(z).

In view of the inequality

|∇f(a)|^{p} ≤ \{ \begin{matrix}∑_{i=1}^{n}|\frac{∂f}{∂z_{i}} (a)|^{p} & if 0 < p ≤ 1, \\ n^{p−1}∑_{i=1}^{n}|\frac{∂f}{∂z_{i}} (a)|^{p} & if p > 1, \end{matrix}

(8) yields

|∇f(a)|^{p} ≤ C \frac{1}{r^{2n}} ∫_{B_{r}} |∇f|^{p} ∘ φ_{a}(z) dv(z).

The change of variables z = φ_{a}(w) gives

|∇f(a)|^{p} ≤ C \frac{1}{r^{2n}} ∫_{φ_{a}(B_{r})} |∇f(z)|^{p}(\frac{1 − |a|^{2}}{|1 − ⟨z, a⟩|^{2}} )^{n+1} dv(z).

The equality

1 − |φ_{a}(z)|^{2} = \frac{(1 − |a|^{2})(1 − |z|^{2})}{|1 − ⟨z, a⟩|^{2}}

implies

1 − |z|^{2} > \frac{1}{4}(1 − r)(1 − |a|^{2}) for z ∈ φ_{a}(B_{r}).

This follows that

(1 − |a|^{2})^{p}|∇f(a)|^{p} ≤ \frac{4^{p}}{(1 − r)^{p}r^{2n}} ∫_{φ_{a}(B_{r})}|∇f(z)|^{p}(1 − |z|^{2})^{p}(\frac{1 − |a|^{2}}{|1 − ⟨z, a⟩|^{2}} )^{n+1}dv(z)

≤ \frac{4^{p}}{(1 − r)^{p}r^{2n}} ∫_{B} |∇f(z)|^{p}(1 − |z|^{2})^{p}(\frac{1 − |a|^{2}}{|1 − ⟨z, a⟩|^{2}} )^{n+1} dv(z)

≤ \frac{4^{p}}{(1 − r)^{p}r^{2n}} ∫_{B} |˜∇f(z)|^{p}(\frac{1 − |a|^{2}}{|1 − ⟨z, a⟩|^{2}} )^{n+1} dv(z)

which together with (7) imply f ∈ ℬ.

Recall that f ∈ H^{2} is said to be a BMOA function if its radial limit function f^{⋆} is a function of bounded mean oscillation on S with respect to nonisotropic balls Q_{δ}(ζ) = {η ∈ S : |1 − ⟨η, ζ⟩| < δ}, ζ ∈ S, 0 < δ ≤ 2. The following Carleson measure characterizations of BMOA functions have been obtained in [J] and [CC].

Theorem B. Suppose f ∈ H^{2} . Then the following statements are equivalent:

(a) f ∈ BMOA,

(b) |˜∇f(z)|^{2}(1 − |z|^{2})^{−1} dv(z) is a Carleson measure,

(c) |∇f(z)|^{2}(1 − |z|^{2}) dv(z) is a Carleson measure.

In view of Theorem 3 it is natural to ask if there is also an analogous generalization of this theorem. Here, we will give the following necessary conditions for f to belong to BMOA.

Theorem 4. If p ≥ 2 and f ∈ BMOA then

(a) |˜∇f(z)|^{p}(1 − |z|^{2})^{−1} dv(z) is a Carleson measure,

(b) |∇f(z)|^{p}(1 − |z|^{2})^{p−1} dv(z) is a Carleson measure.

Remark 1. For n = 1, the results contained in Theorems 3 and 4 are equivalent to those stated in [S, Theorem 1, D and p. 417]. Moreover, it is shown in [M] that the function f(z) = ∑_{n=1}^{∞} n^{−1/2}z^{2^{n}}, z ∈ D, satisfies condition (a) (or equivalently (b)) given in Theorem 4 with p > 2 but f ∉BMOA. So, we do not think that in the case n > 1 and p > 2 these conditions are sufficient for f to belong to BMOA.

To prove Theorem 4 we will need the below stated characterization of the Hardy space H^{p}, p ≥ 2, which is an extension of a Littlewood–Paley classical result to the setting of the unit ball B. Similarly to the case n = 1 this can be proved by using the M. Riesz–Thorin theorem (see e.g. [Z, pp. 95, 216])

For F ∈ C^{1}(B) let

DF = (\frac{∂F}{∂x_{1}} , \frac{∂F}{∂y_{1}} , …, \frac{∂F}{∂x_{n}} , \frac{∂F}{∂y_{n}} ), z_{k} = x_{k} + iy_{k}, k = 1, …, n,

be the real gradient of F and let D̃F (z) = D(F ∘ φ_{z})(0). If, as above,

∇F = (\frac{∂F}{∂z_{1}} , …, \frac{∂F}{∂z_{n}} )

is the complex gradient of F then, as one can easily check,

2|DF |^{2} = |∇F |^{2} + |∇\overline{F} |^{2}.

As in [R] the Poisson integral P [f] of a function f ∈ L^{1}(S, dσ) is defined, for z ∈ B, by

P [f](z) = ∫_{S} \frac{(1 − |z|^{2})^{n}f(ζ)}{|1 − ⟨z, ζ⟩|^{2n}} dσ(ζ).

The next lemma is an analogue of Theorem 3.24 of [Z, p. 216].

Lemma 3. If 2 ≤ p ≤ ∞, f ∈ L^{p}(S, dσ) and F = P [f], then

(9)  ∫_{B} |D̃F (z)|^{p}(1 − |z|^{2})^{−1} dv(z) ≤ C ∫_{S} |f(ζ)|^{p} dσ.

Proof. We assume that n > 1. For f ∈ L^{p}(S, dσ), define the operator T by the formula

T f = D̃(F).

Inequality (9) can be written in the form

∥T f∥_{p} ≤ C∥f∥_{p}, 2 ≤ p ≤ ∞,

where the norm ∥T f∥_{p} is taken with respect to the measure (1 − |z|^{2})^{−1} dv(z). In view of the Riesz–Thorin theorem it is enough to show that T is both of type (2, 2) and of type (∞, ∞). Since F is an ℳ -harmonic function in B (see [R, p. 49]) Lemma 3.2 in [CC] and (4) give

(10)  \begin{aligned}Δ̃|F |^{2} & = 4(1 − |z|^{2})(|∇F |^{2} − |RF |^{2} + |∇\overline{F} |^{2} − |R\overline{F} |^{2}) \\ & = 4(|˜∇F |^{2} + |˜∇\overline{F} |^{2}) = 8|D̃F |^{2}. \end{aligned}

By Theorem A in [CC]

∫_{B} GΔ̃|F |^{2} dτ = (n + 1)^{2} ∫_{S} |f − F (0)|^{2} dσ ≤ 4(n + 1)^{2} ∫_{S} |f|^{2} dσ

where

G(z) = \frac{n + 1}{2n} ∫_{|z|}^{1} r^{−2n+1}(1 − r)^{n−1} dr.

From (10) and Lemma 1 in [OYZ]

∫_{B} |D̃F (z)|^{2}(1 − |z|^{2})^{−1} dv(z) = \frac{1}{8} ∫_{B} Δ̃|F (z)|^{2}(1 − |z|^{2})^{−1} dv(z)
≤ C_{1} ∫_{B} |˜∇F (z)|^{2}G(z) dτ(z)
≤ C_{2} ∫_{S} |f(ζ)| dσ(ζ),

which proves that T is of type (2, 2).

Let

P (z, ζ) = \frac{(1 − |z|^{2})^{n}}{|1 − ⟨z, ζ⟩|^{2n}}

denote the Poisson kernel. To prove that T is of type (∞, ∞) it is enough to show that

(11)  ∫_{S} |D̃P (z, ζ)| dσ(ζ) ≤ C < ∞.

Notice first that since P (z, ζ) =\overline{P} (z, ζ)

(12)  |D̃P (z, ζ)| = |˜∇P (z, ζ)|.

To calculate |˜∇P (z, ζ)| we will use formula (4). We have

∇P (z, ζ) = −n(\frac{(1 − |z|^{2})^{n−1}}{|1 − ⟨z, ζ⟩|^{2n}} z̄ − \frac{(1 − |z|^{2})^{n}(1 − \overline{⟨z, ζ⟩})}{|1 − ⟨z, ζ⟩|^{2n+2}} ζ̄)

and

\begin{aligned}Pz̄∇P (z, ζ) & = −n(\frac{(1 − |z|^{2})^{n−1}}{|1 − ⟨z, ζ⟩|^{2n}} − \frac{(1 − |z|^{2})^{n}(1 − \overline{⟨z, ζ⟩})⟨ζ̄, z̄⟩}{|z|^{2}|1 − ⟨z, ζ⟩|^{2n+2}})z̄, \\ Qz̄∇P (z, ζ) & = −n\frac{(1 − |z|^{2})^{n}(1 − \overline{⟨z, ζ⟩})(|z|^{2}ζ̄ − ⟨ζ̄, z̄⟩z̄)}{|z|^{2}|1 − ⟨z, ζ⟩|^{2n+2}} \end{aligned}

Hence

\begin{aligned}|˜∇P (z, ζ)| & ≤ (1 − |z|^{2})|Pz̄∇P (z, ζ)| + (1 − |z|^{2})^{1/2}|Qz̄∇P (z, ζ)| \\ & ≤ n(\frac{(1 − |z|^{2})^{n}}{|1 − ⟨z, ζ⟩|^{2n}} + \frac{(1 − |z|^{2})^{n+1}}{|1 − ⟨z, ζ⟩|^{2n+1}} + \frac{(1 − |z|^{2})^{n+(1/2)}}{|1 − ⟨z, ζ⟩|^{2n+(1/2)}} ). \end{aligned}

Now, Proposition 1.4.10. in [R, p. 17] and [12] imply [11].

If 1 ≤ p < ∞, then H^{p} is the space of the Poisson integrals of L^{p} functions that are holomorphic on B (see e.g. [R, pp. 87–88]). Therefore, Lemma 3 implies immediately the following

Corollary. If f ∈ H^{p}, 2 ≤ p < ∞, then

(a)  ∫_{B} |˜∇f(z)|^{p}(1 − |z|^{2})^{−1} dv(z) < ∞,
(b)  ∫_{B} |∇f(z)|^{p}(1 − |z|^{2})^{p−1} dv(z) < ∞.

Proof of Theorem 4. The theorem follows from the corollary and from the following characterization of BMOA functions [O1]. For f ∈ H^{2} and for any p, 0 < p < ∞

f ∈ BMOA  if and only if  sup_{a∈B} ∥f ∘ φ_{a} − f(a)∥_{H^{p}} < ∞.

Moreover, the change of variables gives

f ∈ BMOA if and only if

\begin{matrix}sup_{a∈B} ∫_{B} |˜∇(f ∘ φ_{a}) & (z)|^{p}(1 − |z|^{2})^{−1} dv(z) \\ = sup_{a∈B} ∫_{B} |˜∇f(z)|^{p}(1 − |z|^{2})^{−1} \frac{(1 − |a|^{2})^{n}}{|1 − ⟨z, a⟩|^{2n}} dv(z) < ∞. \end{matrix}

In view of Lemma 4.1 in [CC] the last condition means that |˜∇f(z)|^{p}(1−|z|^{2})^{−1} dv(z) is a Carleson measure and (a) is proved. But (a) implies (b) since, by (4), |∇f(z)|(1 − |z|^{2}) ≤ |˜∇f(z)|.

Remark 2. As in the case n = 1 (see [S]) condition (a) (or (b)) of Theorem 4 with some p, 0 < p ≤ 2, is sufficient for f ∈BMOA. To prove this show first that if 0 < p ≤ 2 then

\begin{aligned}∥f∥_{H^{p}}^{p} & ≤ C(|f(0)|^{p} + ∫_{B} |∇f(z)|^{p}(1 − |z|)^{p−1} dv(z)) \\ & ≤ C(|f(0)|^{p} + ∫_{B} |˜∇f(z)|^{p}(1 − |z|)^{−1} dv(z)) \end{aligned}

and next proceed as in the proof of Theorem 4.

References

[A] Axler, S.: The Bergman space, Bloch space, and the commutators of multiplication operators. - Duke Math. J. 53, 1986, 315–332.

[CC] Choa, J.S., and B.R. Choe: A Littlewood–Paley type identity and a characterization of BMOA. - Complex Variables Theory Appl. 17, 1991, 15–23.

[CKP] Choa, J.S., H.O. Kim, and Y.Y. Park: A Bergman–Carleson measure characterization of Bloch functions in the unit ball of C^{n}. - Bull. Korean Math. Soc. 29, 1992, 285–293.

[HY] Hahn, K.T., and E.H. Youssfi: Möbius invariant Besov p-spaces and Hankel operators in the Bergman space on the ball in C^{n}. - Complex Variables Theory Appl. 17, 1991, 89-104.

[H] Hayman, W.K.: Multivalent Functions. - Cambridge University Press, 1958.

[J] Jevtič, M.: A note on the Carleson measure characterization of BMOA functions on the unit ball. - Complex Variables Theory Appl. 17, 1992, 189–194.

[LP] Littlewood, J.E., and R.E.A.C. Paley: Theorems on Fourier series and power series (II). - Proc. London Math. Soc. (2) 42, 1937, 52–89.

[M] Miao, J.: A property of analytic functions with Hadamard gaps. - Bull. Austral. Math. Soc. 45, 1992, 105–112.

[O1] Ouyang, C.H.: Some classes of functions with exponential decay in the unit ball of C^{n}. - Publ. Res. Inst. Math. Sci. 25, 1989, 263–277.

[O2] Ouyang, C.H.: An extension theorem for Bloch functions in the unit ball. - Acta Math. Sci. 10, 1990, 455–461 (Chinese).

[OYZ] Ouyang, C., W. Yang, and R. Zhao: Characterizations of Bergman spaces and Bloch space in the unit ball of C^{n}. - Trans. Amer. Math. Soc. 374, 1995, 4301–4312.

[R] Rudin, W.: Function Theory in the Unit Ball of C^{n}. - Springer-Verlag, 1980.

[Shi] Shi Ji-Huai: Inequalities for the integral means of holomorphic functions and their derivatives in the unit ball of C^{n}. - Trans. Amer. Math. Soc. 328, 1991, 619–637.

[S] Stroethoff, K.: On Besov-type characterizations for the Bloch space. - Bull. Austral. Math. Soc. 39, 1989, 405–420.

[T] Timoney, R.M.: Bloch functions in several complex variables. - Bull. London Math. Soc. 12, 1980, 241–267.

[Z] Zygmund, A.: Trigonometric Series, Vol. II. - Cambridge University Press, 1959.

Received 7 January 1997