Annales Academiæ Scientiarum Fennicæ Series A. I. Mathematica Volumen 19, 1994, 35–46

CARLESON MEASURE, ATOMIC DECOMPOSITION AND FREE INTERPOLATION FROM BLOCH SPACE

Jie Xiao

Peking University, Department of Mathematics Beijing 100871, P.R. China

Abstract. Several characterizations, Carleson measures and atomic decomposition for the Bloch space B are given. For their applications, free interpolations from B are also discussed.

1. Introduction

Let D = { z : |z| < 1 } be the unit disk in the finite complex plane C and dm_{α}(z) = (1 − |z|^{2})^{α} dm(z) the two-dimensional Lebesgue measure with weight (1 − |z|^{2})^{α}, α > −1. Denote by A and H^{∞} the sets of functions analytic and boundedly analytic on D, respectively. For f ∈ A we say f ∈ B if

(1.1)  ∥f∥_{B} = |f(0)| + sup_{z∈D}(1 − |z|^{2})|f′(z)| < ∞;

also f ∈ A_{α}^{1} if

(1.2)  ∥f∥_{1,α} = ∫_{D}|f(z)| dm_{α}(z) < ∞.

B and A_{α}^{1} are the so-called Bloch space and the Bergman space weighted by (1 − |z|^{2})^{α}, [6], [15].

It is well known that the dual space of A_{0}^{1} is identified with B under the following inner product:

(1.3)  \begin{aligned}⟨f, g⟩ & = \frac{1}{π} lim_{t→1} ∫_{tD} f(z)\overline{g(z)} dm(z) \\ & = \frac{1}{π} ∫_{D}(∨f)(z)(1 − |z|^{2})\overline{g′(z)} dm(z) + f(0)\overline{g(0)} \end{aligned}

for f ∈ A_{0}^{1} and g ∈ B, where t ∈ (0, 1), tD = { z : |z| < t } and (∨f)(z) = [f(z) − f(0)]/z; see [3].

1991 Mathematics Subject Classification: Primary 30C55, 31A20, 46G10.

Research supported by the National Science Foundation of China.

In [11] we discussed the atomic decomposition and the free interpolation on the Bergman space A_{0}^{1} . Since (A_{0}^{1})^{∗} = B, it is very natural to consider similar problems on the Bloch space. As far as we know, these questions have not been thoroughly dealt with yet ([8], [13]), which is what we try to do in this paper. First, in Section 2, we give several characterizations of B as well as relations between B and Carleson measure. Next, in Section 3, we obtain an atomic decomposition of B by means of the pseudohyperbolic metric. Finally, in Section 4, we study the free interpolations by functions from B by means of the direct construction and the operator theory.

We wish to express our deepest gratitude to Professor X.C. Shen for helpful suggestions, especially during his illness. Also, we are grateful to Professors R. Aulaskari and O. Martio for their kind help. Besides, we would like to thank the referee for his/her comments, and the secretary of Department Mathematics of the Joensuu University for her dedicated typing.

2. Bloch space and Carleson measure

There are many works on the Bloch space, [1], e.g. [2], [7], [10]. Here we will give several interesting characterizations, some of which are new.

For z and w in D, let φ_{w}(z) = (w − z)/(1 − \overline{w}z), ϱ(w, z) = |φ_{w}(z)| and d(w, z) = \frac{1}{2} log\{[1+ϱ(w, z)]/[1−ϱ(w, z)]\}. Here ϱ( · , ·) and d( · , ·) are called the pseudohyperbolic and hyperbolic distances, respectively. Also, denote the measure of set E ⊂ D, relative to dm_{α}(z), by m_{α}(E) = ∫_{E} dm_{α}(z) = ∫_{E}(1−|z|^{2})^{α} dm(z). Then we have the following result.

Theorem 2.1. Let f ∈ A. Then the following statements are equivalent:

(i) f ∈ B; (ii) sup_{w,z∈D}|f(w) − f(z)|/d(w, z) < ∞; (iii) there is a constant C > 0 such that

sup_{w∈D} ∫_{D} exp[C|(f ∘ φ_{w})(z) − f(w)|] dm_{α}(z) < ∞.

Proof. We will show this fact according to (i) ⟹ (ii) ⟹ (iii) ⟹ (i). Firstly, (i) ⟹ (ii). Let f ∈ B, and g_{w}(λ) = (f ∘ φ_{w})(λ) − f(w), λ, w ∈ D.

Then g_{w}(0) = 0 and ∥g_{w}∥_{B} = sup_{z∈D}(1 − |z|^{2})|f′(z)| ≤ ∥f∥_{B} < ∞. Further,

|g_{w}(λ)| = |∫_{0}^{λ} g′_{w}(ζ) dζ| ≤ \frac{1}{2}∥f∥_{B} log \frac{1 + |λ|}{1 − |λ|}.

Setting z = φ_{w}(λ) we obtain

|f(z) − f(w)| ≤ \frac{1}{2}∥f∥_{B} log \frac{1 + ϱ(z, w)}{1 − ϱ(z, w)} = ∥f∥_{B}d(z, w),

i.e., (ii) holds.

Carleson measure, atomic decomposition and free interpolation 37 Secondly, (ii) ⟹ (iii). Suppose that

0 < ∥f∥′_{B} = sup_{w,z∈D}|f(w) − f(z)|/d(w, z) < ∞;

then for t ≥ 0,

\begin{aligned}\{ z : z ∈ D, & |g_{w}(z)| > t \} \\ & ⊂ \{ z : z ∈ D, |z| > [exp(\frac{2t}{∥f∥′_{B}} ) − 1]/[exp(\frac{2t}{∥f∥′_{B}} ) + 1] \}. \end{aligned}

Moreover, when 0 < C < [2(α + 1)/∥f∥′_{B}],

∫_{D} exp[C|g_{w}(z)|] dm_{α}(z) = C ∫_{0}^{∞}(exp Ct) · m_{α}({ z : z ∈ D, |g_{w}(z)| > t }) dt

≤ C ∫_{0}^{∞}(exp Ct)\frac{4π}{α + 1} exp(−\frac{2(α + 1)t}{∥f∥′_{B}}) dt

= \frac{4πC · ∥f∥′_{B}}{(α + 1)[2(α + 1) − C∥f∥′_{B}]}.

Thirdly, (iii) ⟹ (i). Let

∥f∥′′_{B} = sup_{w∈D} ∫_{D} exp[C|(f ∘ φ_{w})(z) − f(w)|] dm_{α}(z) < ∞

for some constant C > 0. Then

∥(f ∘ φ_{w})(z) − f(w)∥_{1,α} ≤ \frac{∥f∥′′_{B}}{C} < ∞.

Since g_{w} has a Taylor series ∑_{n} a_{n}z^{n} which converges uniformly on tD (0 < t < 1), a simple calculation gives

a_{1} = g′_{w}(0) = \frac{(α + 1)(α + 2)}{1 − [1 + (α + 1)t^{2}](1 − t^{2})^{α+1}} ∫_{tD} g_{w}(z)\overline{z} dm_{α}(z).

By letting t → 1 we get

|g′_{w}(0)| ≤ (α + 1)(α + 2) ∫_{D}|g_{w}(z)| dm_{α}(z),

i.e.,

(1 − |w|^{2})|f′(w)| ≤ [\frac{(α + 1)(α + 2)}{C}] · ∥f∥′′_{B}.

So, f ∈ B.

Remark. This theorem tells us that B is Lipschiz’s class, relative to the hyperbolic metric d( · , ·). However, we know that B can be identified with the Zygmund class (see [1], [5]). Hence our result is much clearer than the one in [1].

In what follows we characterize connection between the Bloch space and Carleson measure.

For w ∈ D let D(w, r) = { z : z ∈ D, ϱ(w, z) < r }, r ∈ (0, 1). D(w, r) is called the pseudohyperbolic disk. It is more convenient to use D(w, r) (not Carleson square) for discussing Borel measure on the Bergman space A_{α}^{1} ; see [6], [12]. Similarly, we have the following theorem.

Theorem 2.2. Let p ∈ (0, ∞) and r ∈ (0, 1), and let μ be a nonnegative Borel measure on D. Then the following statements are equivalent:

(i) sup_{{w∈D \atop 0≢f∈B}} [\frac{1}{∥f∥_{B}^{p}} ∫_{D}|f(z) − f(w)|^{p} \frac{(1 − |w|^{2})^{2+α}}{|1 − \overline{w}z|^{4+2α}} dμ(z)]^{1/p} < ∞;

(ii) sup_{w∈D}[\frac{μ(D(w, r))}{m_{α}(D(w, r))}] < ∞;

(iii) sup_{w∈D}[∫_{D} \frac{(1 − |w|^{2})^{2+α}}{|1 − \overline{w}z|^{4+2α}} dμ(z)] < ∞.

Proof. (ii) ⇔ (iii) has been derived in [13], so we only need to claim (i)⇔(ii). On the one hand, if (ii) is true, it follows by Theorem 2.1 that

[∫_{D}|(f ∘ φ_{w})(z) − f(w)|^{p} dm_{α}(z)]^{1/p} ≤ C∥f∥_{B}

for f ∈ B, where C > 0 is a constant independent of f. Further, by [15], [12] and [6] it yields another constant C_{0} depending on the condition (ii) such that

[∫D|f(z) − f(w)|^{p} \frac{(1 − |w|^{2})^{2+α}}{|1 − \overline{w}z|^{4+2α}} dμ(z)]^{1/p}

≤ C_{0}[∫_{D}|f(z) − f(w)|^{p} \frac{(1 − |w|^{2})^{2+α}}{|1 − \overline{w}z|^{4+2α}} dm_{α}(z)]^{1/p}

= C_{0}[∫_{D}|(f ∘ φ_{w})(z) − f(w)|^{p} dm_{α}(z)]^{1/p} ≤ C_{0}C∥f∥_{B}.

On the other hand, let (i) hold. Taking f_{0}(z) = [1/(1 − \overline{w}_{0}z)] − 1 for w_{0} = (−\frac{1}{2}(r + 1) + w)/(1 − \frac{1}{2}(r + 1) · \overline{w}), \frac{1}{2}(r + 1) ≠ w ∈ D, r ∈ (0, 1), we get ∥f_{0}∥_{B} = |w_{0}|/(1−|w_{0}|^{2}), |f_{0}(z)−f_{0}(w_{0})| = |w_{0}||z −w_{0}|/|1−\overline{w}_{0}z|(1−|w_{0}|^{2}) and

\begin{aligned}sup_{z∈D(w,r)} |1 − \overline{w}_{0}z| & = sup_{λ∈rD}|1 − \overline{(\frac{w − \frac{1}{2}(r + 1)}{1 − \overline{w} \frac{1}{2}(r + 1)})} · (\frac{w − λ}{1 − \overline{w}λ})| \\ & ≤ \frac{(2 + r^{2} + r)(1 − |w|^{2})}{(1 − r)^{2}}. \end{aligned}

Also, there are two constants C_{1} > 0 and C_{2} > 0 depending only on α and r such that (see [14])

C_{1} · (1 − |w|^{2})^{2+α} ≤ m_{α}(D(w, r)) ≤ C_{2} · (1 − |w|^{2})^{2+α}.

We also have

∞ > sup_{{λ∈D \atop 0≢f∈B}} [\frac{1}{∥f∥_{B}^{p}} · ∫_{D}|f(z) − f(λ)|^{p} · \frac{(1 − |λ|^{2})^{2+α}}{|1 − \overline{λ}z|^{4+2α}} dμ(z)]^{1/p}

≥ [\frac{1}{∥f_{0}∥_{B}^{p}} · ∫_{D}|f_{0}(z) − f_{0}(w_{0})|^{p} · \frac{(1 − |w_{0}|^{2})^{2+α}}{|1 − \overline{w}_{0}z|^{4+2α}} dμ(z)]^{1/p}

≥ [(\frac{1 − |w_{0}|^{2}}{|w_{0}|} )^{p} · ∫_{D(w,r)}(\frac{|w_{0}|}{1 − |w_{0}|^{2}} )^{p}[ϱ(z, w_{0})]^{p} · \frac{(1 − r)^{4(2+α)}}{(2 + r + r^{2})^{4+2α}}

·\frac{1}{(1 − |w|^{2})^{2+α}} dμ(z)]^{1/p}

≥ [\frac{(1 − r)^{4(2+α)}}{4^{4+2α}} · ∫_{D(w,r)}[ϱ(w_{0}, w) − ϱ(z, w)]^{p} · \frac{1}{(1 − |w|^{2})^{2+α}} dμ(z)]^{1/p}

≥ \frac{(1 − r)^{(4α+8+p)/p}}{2^{(4α+8+p)/p}} C_{2}^{1/p} · [\frac{μ(D(w, r))}{m_{α}(D(w, r))}]^{1/p}.

Therefore

sup_{w∈D}[\frac{μ(D(w, r))}{m_{α}(D(w, r))}] < ∞.

The measure μ satisfying one of the three statements in Theorem 2.2 is said to be α-Carleson measure. The following fact is interesting.

Theorem 2.3. Let f ∈ A. Then the following statements are equivalent:

(i) f ∈ B; (ii) |f′(z)|^{2}(log 1/|z|)^{2} dm(z) is 0-Carleson measure; (iii) |f′(z)|^{2}(1 − |z|^{2})^{2} dm(z) is 0-Carleson measure.

Proof. We will give the whole claim in accordance with the order (i) ⟹ (ii) ⟹ (iii) ⟹ (i).

First of all, (i) ⟹ (ii). Under f ∈ B, we consider the integral below:

I_{1} = ∫_{D} \frac{(1 − |w|^{2})^{2}}{|1 − \overline{w}z|^{4}} ·|f′(z)|^{2}(log \frac{1}{|z|})^{2} dm(z) = (∫_{\{|z|> \frac{1}{4} \}}+ ∫_{\{|z|≤ \frac{1}{4} \}})\{⋯\} dm(z).

Since log(1/|z|) ≤ C_{1}(1 − |z|^{2}) when |z| > \frac{1}{4} ,

\begin{aligned}∫_{\{|z|> \frac{1}{4} \}}\{⋯\} dm(z) & ≤ C_{1}^{2} ∫_{\{|z|> \frac{1}{4} \}} \frac{|f′(z)|^{2}(1 − |w|^{2})^{2}(1 − |z|^{2})^{2}}{|1 − \overline{w}z|^{4}} dm(z) \\ & ≤ C_{1}^{2} · ∥f∥_{B}^{2} ∫_{D} \frac{(1 − |w|^{2})^{2}}{|1 − \overline{w}z|^{4}} dm(z) ≤ πC_{1}^{2}∥f∥_{B}^{2}, \end{aligned}

where C_{1} > 0 is an absolute constant. At the same time

\begin{aligned}∫_{\{|z|≤ \frac{1}{4} \}}\{⋯\} dm(z) & ≤ (\frac{16}{15})^{2}∥f∥_{B}^{2} ∫_{\{|z|≤ \frac{1}{4} \}} \frac{(1 − |w|^{2})^{2}}{|1 − \overline{w}z|^{4}} (log \frac{1}{|z|})^{2} dm(z) \\ & ≤ (\frac{16}{15})^{2} · \frac{4^{4}}{3^{4}} ∥f∥_{B}^{2} ∫_{\{|z|≤ \frac{1}{4} \}}(log \frac{1}{|z|})^{2} dm(z) = C_{2}∥f∥_{B}^{2}, \end{aligned}

where C_{2} > 0 is an absolute constant. Consequently

I_{1} ≤ (πC_{1}^{2} + C_{2})∥f∥_{B}^{2}.

So, from Theorem 2.2 (iii) we see that |f′(z)|^{2}(log(1/|z|))^{2} dm(z) is 0-Carleson measure.

Next (ii) ⟹ (iii). This is obvious, since (1 − |z|^{2})^{2} ≤ 4(log(1/|z|))^{2} for all z ∈ D.

Finally (iii) ⟹ (i). Assuming that |f′(z)|^{2}(1 − |z|^{2})^{2} dm(z) is 0-Carleson measure, we have

I_{2} = sup_{w∈D} ∫_{D}|f′(z)|^{2} · \frac{(1 − |z|^{2})^{2}(1 − |w|^{2})^{2}}{|1 − \overline{w}z|^{4}} dm(z) < ∞,

and obviously ∞ > I_{2} ≥ ∫_{D}|f′(z)|^{2}(1 − |z|^{2})^{2} dm(z). Moreover,

(f′(w))^{2} = \frac{3}{π} ∫_{D}(f′(λ))^{2} · \frac{(1 − |λ|^{2})^{2}}{(1 − w\overline{λ})^{4}} dm(λ).

Hence

(1 − |w|^{2})^{2}|f′(w)|^{2} ≤ \frac{3}{π} ∫_{D}|f′(λ)|^{2} · \frac{(1 − |λ|^{2})^{2}(1 − |w|^{2})^{2}}{|1 − \overline{w}λ|^{4}} dm(λ) ≤ I_{2} < ∞,

i.e., f ∈ B.

Supposing g_{D}(z, w) = log |(1 − \overline{w}z)/(w − z)| (the Green’s function on D), we just have

Corollary 2.4. Let f ∈ A. Then f ∈ B if and only if

(2.1)  sup_{w∈D} ∫_{D}|f′(z)|^{2}g_{D}^{2}(z, w) dm(z) < ∞.

Proof. This fact is readily derived from the equivalence between ∥f ∘ φ_{w}∥_{B} and ∥f∥_{B}, and Theorem 2.3 (ii). Nevertheless, the result can also be shown by Theorem 2.1 and 2.3.

3. Atomic decomposition

To begin with, we let 1^{1} and 1^{∞} stand for the usual sequence spaces as follows:

(3.1)  1^{1} = { {c_{n}} : {c_{n}} ⊂ C, ∥{c_{n}}∥_{1} = ∑_{n} |c_{n}| < ∞ },
(3.2)  1^{∞} = { {c_{n}} : {c_{n}} ⊂ C, ∥{c_{n}}∥_{∞} = sup_{n} |c_{n}| < ∞ }.

Both are Banach spaces. Also, suppose that {z_{n}} is a sequence of points on D. A sequence of points {z_{n}} is called δ -weakly separated if δ = inf_{m≠n} ϱ(z_{m}, z_{n}) > 0 and η -uniformly separated if η = inf_{n} ∏_{m≠n} ϱ(z_{m}, z_{n}) > 0. Clearly an η - uniformly separated sequence must be δ -weakly separated. A sequence of points {z_{n}} is said to be ε-dense if D = ∪_{n}D(z_{n}, ε), where D(z_{n}, ε) = { z : z ∈ D, ϱ(z_{n}, z) < ε } and ε ∈ (0, 1).

Luecking [6] and Xiao [12] proved the quasi-atomic decomposition theorem of A_{α}^{1} as follows.

Lemma 3.1. Let {z_{n}} be a sequence of points on D, α > −1 and f ∈ A_{α}^{1} . If {z_{n}} is δ -weakly separated, there is a constant C_{1} > 0 depending only on δ and α so that

(3.3)  ∥f∥_{1,α} ≥ C_{1} · ∑_{n}(1 − |z_{n}|^{2})^{2+α}|f(z_{n})|.

Furthermore, there are an ε_{0} > 0 and a constant C_{2} > 0 depending only on δ and α so that

(3.4)  ∥f∥_{1,α} ≤ C_{2} ∑_{n}(1 − |z_{n}|^{2})^{2+α}|f(z_{n})|

if {z_{n}} is also ε-dense with 0 < ε ≤ ε_{0} .

After the above lemma, we can state an atomic decomposition theorem on the Bloch space.

Theorem 3.2. Let {z_{n}} be a sequence of points on D. If {z_{n}} is δ -weakly separated, the function of the form (3.5) is in B for any {c_{n}} ∈ 1^{∞}

(3.5)  f(z) = ∑_{n} c_{n} · (\frac{1 − |z_{n}|^{2}}{1 − \overline{z}_{n}z} )^{2}.

Moreover, there is an ε_{0} > 0 such that every f ∈ B has the form (3.5) for some {c_{n}} ∈ 1^{∞} if {z_{n}} is also ε-dense with 0 < ε ≤ ε_{0} .

Proof. Let {z_{n}} be δ -weakly separated. Then T, defined as follows, is a bounded linear operator from A_{0}^{1} to 1^{1} ,

(3.6)  Tf = {(Tf)_{n}} = {(1 − |z_{n}|^{2})^{2}f(z_{n})},

in that (3.3) holds under {z_{n}} being δ -weakly separated. Thus T^{∗}, the adjoint operator of T given by (3.7), is a bounded linear operator from 1^{∞} (= (1^{1})^{∗}) to B (= A_{0}^{1})^{∗}),

(3.7)  ⟨Tf, y⟩ = ⟨f, T^{∗}y⟩,  f ∈ A_{0}^{1}, y ∈ 1^{∞},

where the left ⟨·, ·⟩ is just the usual inner product between 1^{1} and 1^{∞}. To compute T^{∗}, we take

y = e_{n}, (e_{n})_{m} = \{ \begin{matrix}1, & m = n \\ 0, & m ≠ n, \end{matrix}

so

⟨Tf, e_{n}⟩ = (Tf)_{n} = (1 − |z_{n}|^{2})^{2}f(z_{n}) = (1 − |z_{n}|^{2})^{2}⟨f, K_{z_{n}}⟩,

where K_{z_{n}}(z) = 1/(1 − \overline{z}_{n}z)^{2} is the reproducing kernel for A_{0}^{1} . Hence

T^{∗}e_{n} = (1 − |z_{n}|^{2})^{2}K_{z_{n}}(z)

and

T^{∗}y = ∑_{n} c_{n} · \frac{(1 − |z_{n}|^{2})^{2}}{(1 − \overline{z}_{n}z)^{2}} for y = \{c_{n}\} ∈ 1^{∞},

i.e., the function in the form (3.5) is in B. Indeed, it is easy to derive T^{∗}y ∈ B by means of the direct computation.

Now we turn to showing the second part of Theorem 3.2. In fact, it is only necessary to claim T^{∗} to be surjective. However, T^{∗} is onto if and only if T is bounded below. By Lemma 3.1, there exists an ε_{0} > 0 such that T is bounded below if {z_{n}} is ε-dense with 0 < ε ≤ ε_{0} . That is to say, there is an ε_{0} > 0 such that every f ∈ B has the form (3.5) for some {c_{n}} ∈ 1^{∞} as {z_{n}} is ε-dense with 0 < ε ≤ ε_{0} . Therefore the proof is completed.

4. Free interpolation

As is well-known, a given sequence of points {z_{n}} on D is called an H^{∞} - interpolating sequence if for any {c_{n}} ∈ 1^{∞} there exists f ∈ H^{∞} satisfying f(z_{n}) = c_{n} for all n. Carleson stated in [4] that {z_{n}} is an H^{∞} -interpolating sequence if and only if {z_{n}} is η -uniformly separated. Here we want to extend this fact to the Bloch space. Yet, it is unfortunate that the η -uniformly separated property is only a sufficient condition for B. A sequence of points {z_{n}} is said to be a B -interpolating sequence if there is f ∈ B such that f(z_{n}) = c_{n} for all n and any {c_{n}} ∈ 1^{∞}.

Theorem 4.1. Let {z_{n}} be a sequence of points on D. If {z_{n}} is a B - interpolating sequence, {z_{n}} is δ -weakly separated. Conversely, if {z_{n}} is δ - weakly separated and (4.1) or (4.2) is true, then {z_{n}} is a B -interpolating sequence where

sup_{n} ∑_{m≠n} \frac{(1 − |z_{m}|^{2})(1 − |z_{n}|^{2})}{|1 − \overline{z}_{n}z_{m}|^{2}} < ∞, (4.1)

sup_{n} ∑_{m≠n} \frac{(1 − |z_{m}|^{2})^{2}}{|1 − \overline{z}_{n}z_{m}|^{2}} < 1. (4.2)

Proof. Firstly, if {z_{n}} is a B -interpolating sequence, then 1^{∞} ⊂ T_{∞}B, where T_{∞}f = {f(z_{n})}. Since B is a Banach space, relative to ∥ · ∥_{B}, it follows from the open mapping theorem that there is a uniform constant C_{1} > 0 and f ∈ B so that ∥f∥_{B} ≤ C_{1} with f(z_{n}) = w_{n} for all n and ∥{w_{n}}∥_{∞} ≤ 1. Picking w_{m} = 0, m ≠ n; w_{m} = 1, m = n, there exist f_{n} ∈ B, ∥f_{n}∥_{B} ≤ C_{1} satisfying f_{n}(z_{n}) = 1; f_{n}(z_{m}) = 0, m ≠ n. Theorem 2.1 yields

\frac{|f_{n}(z_{n}) − f_{n}(z_{m})|}{d(z_{n}, z_{m})} ≤ C_{1}, m ≠ n,

and so inf_{m≠n} d(z_{n}, z_{m}) ≥ 1/C_{1} > 0, i.e.,

δ = inf_{m≠n} ϱ(z_{m}, z_{n}) ≥ (e^{2/C_{1}} − 1)/(e^{2/C_{1}} + 1) > 0.

Conversely, let {z_{n}} be δ -weakly separated. If (4.1) is true, {z_{n}} is η - uniformly separated and hence 1^{∞} = T_{∞}H^{∞} ⊂ T_{∞}B since H^{∞} is a proper subspace of B. Furthermore, if (4.2) holds, we consider the linear operator T^{∗}, given by T^{∗}(\{c_{n}\}) = ∑_{n} c_{n} · ((1 − |z_{n}|^{2})/(1 − \overline{z}_{n}z))^{2} , \{c_{n}\} ∈ 1^{∞}. Clearly, T^{∗} is bounded from 1^{∞} to B (by Theorem 3.2), while

∥(T_{∞}T^{∗} − I)\{c_{n}\}∥_{∞} = sup_{n}| ∑_{m≠n} c_{m} · (\frac{1 − |z_{m}|^{2}}{1 − \overline{z}_{m}z_{n}} )^{2}|

≤ ∥\{c_{n}\}∥_{∞} · sup_{n} ∑_{m≠n}(\frac{1 − |z_{m}|^{2}}{|1 − \overline{z}_{n}z_{m}|})^{2}.

So, ∥(T_{∞}T^{∗} − I)∥ < 1, where I is the identify operator, i.e., T_{∞}T^{∗} has an inverse, denoted by (T_{∞}T^{∗})^{−1} . Further, T_{∞} has a right inverse T^{∗}(T_{∞}T^{∗})^{−1} , that is to say, T_{∞}(T^{∗}(T_{∞}T^{∗})^{−1}) = I, and thus 1^{∞} ⊂ T_{∞}B. So, {z_{n}} is a B -interpolating sequence.

Note that T_{∞}H^{∞} ⊊ T_{∞}B. In general, it is necessary to take into consideration the generic free interpolation problem from B. That is, for which {w_{n}} ⊂ C there is f ∈ B satisfying {f(z_{n})} = T_{∞}f = {w_{n}}. For this we obtain the following fact.

Theorem 4.2. Let {z_{n}} be a δ -weakly separated sequence of points on D. If {f(z_{n})} = T_{∞}f = {w_{n}} is solvable in B for {w_{n}} ⊂ C, the following assertions (i) and (ii) hold:

(i) there are a constant C_{1} > 0 and a function β(z) such that

(4.3)  sup_{z∈D} ∑_{n}[1 − ϱ^{2}(z, z_{n})]^{γ} exp[C_{1}|w_{n} − β(z)|] < ∞

for γ > 1; (ii) there is a constant C_{2} > 0 such that

(4.4)  sup_{z∈D} ∑_{n}[1 − ϱ^{2}(z, z_{n})]^{γ} exp[C_{1}|w_{n} − h(z)|] < ∞

for γ > 1, where h(z) = {∑_{n} w_{n}[1 − ϱ^{2}(z, z_{n})]^{γ}}/ ∑_{n}[1 − ϱ^{2}(z, z_{n})]^{γ}.

Conversely, if (i) or (ii) holds for γ = 1, then {f(z_{n})} = T_{∞}f = {w_{n}} is solvable in B.

Proof. First we consider the case (i). If {f(z_{n})} = T_{∞}f = {w_{n}} is solvable in B, then Theorem 2.1 yields

sup_{z∈D} ∫_{D} exp[C_{1}|g_{z}(w)|] dm_{α}(w) < ∞

for C_{1} < 2(α + 1)/∥f∥_{B} (∥f∥_{B} > 0 is naturally assumed), where g_{z}(w) = (f ∘ φ_{z})(w) − f(z). The above statement means that exp(C_{1}g_{z}) is in A_{α}^{1} . Consequently, by Lemma 3.1,

(4.5)  sup_{z∈D} ∑_{n} exp[C_{1}|g_{z}(z̃_{n})|](1 − |z̃_{n}|^{2})^{α+2} ≤ C sup_{z∈D} ∥ exp C_{1}g_{z}∥_{1,α} < ∞,

where {z̃_{n}} = {φ_{z}(z_{n})}, C > 0 is a constant independent of g_{z}, and {z̃_{n}} is also a δ -weakly separated sequence of points on D since {z_{n}} is such a sequence. Thus (4.5) means that (i) holds for γ = α + 2 > 1 and β(z) = f(z).

Now let us consider (ii).

Because {z_{n}} is δ -weakly separated, we get ∑_{n}[1 − ϱ^{2}(z, z_{n})]^{α+2} < ∞ by Lemma 3.1. By (4.5) we further have

∑_{{n:|w_{n}−f(z)|>t}}[1 − ϱ^{2}(z, z_{n})]^{α+2} ≤ C_{2} exp(−C_{1}t)

for t ≥ 0, where C_{1} and C_{2} are constants with C_{1} < 2(α + 1)/∥f∥_{B}, ∥f∥_{B} > 0, and f is the interpolating function for T_{∞}f = {w_{n}} in B.

Thus, for γ = α + 2 > 1

|h(z) − f(z)| ≤ \frac{1}{∑_{n}[1 − ϱ^{2}(z, z_{n})]^{γ}} ∑_{n}|w_{n} − f(z)|[1 − ϱ^{2}(z, z_{n})]^{γ}

= \frac{1}{∑_{n}[1 − ϱ^{2}(z, z_{n})]^{γ}} ∫_{0}^{∞}\{ ∑_{\{n:|w_{n}−f(z)|>t\}}[1 − ϱ^{2}(z, z_{n})]^{γ}\} dt

≤ \frac{1}{∑_{n}[1 − ϱ^{2}(z, z_{n})]^{γ}} ∫_{0}^{∞} min\{∑_{n}[1 − ϱ^{2}(z, z_{n})]^{γ}, C_{2} exp(−C_{1}t)\} dt

≤ \frac{1}{C_{1}} \{1 + log \frac{C_{2}}{∑_{n}[1 − ϱ^{2}(z, z_{n})]^{γ}} \}

and, consequently,

sup_{z∈D} ∑_{n}[1 − ϱ^{2}(z, z_{n})]^{γ} exp[\frac{C_{1}}{2} |w_{n} − h(z)|]

≤ sup_{z∈D} ∑_{n}[1 − ϱ^{2}(z, z_{n})]^{γ} exp[\frac{C_{1}}{2} |w_{n} − f(z)|] exp[\frac{C_{1}}{2} |f(z) − h(z)|]

≤ sup_{z∈D}{{∑_{n}[1 − ϱ^{2}(z, z_{n})]^{γ} exp[C_{1}|w_{n} − f(z)|]}^{1/2}·{∑_{n}[1 − ϱ^{2}(z, z_{n})]^{γ}}^{1/2}

·(exp \frac{1}{2})\{\frac{C_{2}}{∑_{n}[1 − ϱ^{2}(z, z_{n})]^{γ}} \}^{1/2}\}

= \sqrt{C_{2}e} · \{sup_{z∈D} ∑_{n}[1 − ϱ^{2}(z, z_{n})]^{γ} exp[C_{1}|w_{n} − f(z)|]\}^{1/2} < ∞.

That is to say, (ii) is true for γ = α + 2 > 1.

Next we show the contrary assertion. If (i) or (ii) holds for γ = 1, then it follows from (4.4) that

sup_{z∈D} ∑_{n}[1 − ϱ^{2}(z, z_{n})] < ∞.

46 Jie Xiao

This, together with {z_{n}} being δ -weakly separated, shows that {z_{n}} is η - uniformly separated. Also, it follows by [9] that there is f ∈ BMOA(∂D) to make T_{∞}f = {f(z_{n})} = {w_{n}} for {w_{n}} ⊂ C which is satisfied with (4.3) or (4.4) for γ = 1. Since BMOA(∂D) ⊊ B, there exists f ∈ B such that T_{∞}f = {f(z_{n})} = {w_{n}} under the previous assumption. Thus the theorem is proved.

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[11] Xiao, J.: Atomic decomposition and sequence interpolation on Bergman space A^{1} . - Approx. Theory Appl. 8:1, 1992, 40–49.

[12] Xiao, J.: Carleson measure and atomic decomposition on Bergman space A^{p}(φ) (1 < p < ∞). - J. Liaoning Normal Univ. (Natural Science) 15:3, 1992, 188–194.

[13] Xiao, J.: Interpolating sequences for A^{∞}(φ)-functions. - Sci. China Ser. A 35:8, 1992, 907–916.

[14] Xiao, J.: Compact Toeplitz operators on Bergman spaces A^{p}(φ) (1 ≤ p < ∞). - Acta Math. Sci. (B) 13:1, 1993, 56–64.

[15] Xiao, J.: Dual space, Carleson measure and sequence interpolation on A^{p}(φ) (0 < p < 1). - Acta Math. Sinica (N.S.) (to appear).

Received 13 October 1992