Annales Academiæ Scientiarum Fennicæ Mathematica Volumen 31, 2006, 391–404

COMPOSITION OPERATORS IN HYPERBOLIC Q-CLASSES

Xiaonan Li, Fernando Pérez-González, and Jouni Rättyä

University of Joensuu, Department of Mathematics P.O. Box 111, FI-80101 Joensuu, Finland; xiaonan.li@joensuu.fi Universidad de La Laguna, Departamento de Análisis Matemático ES-38271 La Laguna, Tenerife, Spain; fernando.perez.gonzalez@ull.es University of Joensuu, Department of Mathematics P.O. Box 111, FI-80101 Joensuu, Finland; jouni.rattya@joensuu.fi

Abstract. Function theoretic characterizations are given of when a composition operator mapping from a weighted Dirichlet space 𝒟_{q} into a holomorphic Q_{s} -space is bounded or compact. If X^{∗} stands for the hyperbolic class corresponding to the space X, it is shown that a composition operator mapping from 𝒟_{q} into Q_{s} is bounded if and only if it is bounded from 𝒟_{q}^{∗} into Q_{s}^{∗}, provided q ≤ 0 and s ≤ 1.

1. Introduction and statements of results

Let H(D) denote the space of all analytic functions in the open unit disc D of the complex plane, and let B(D) be the subset of H(D) consisting of those h ∈ H(D) for which |h(z)| < 1 for all z ∈ D. Every φ ∈ B(D) induces a linear composition operator C_{φ}(f) = f ∘ φ from H(D) or B(D) into itself. For the general theory of composition operators in analytic function spaces, see [4] and [13].

A function f ∈ H(D) belongs to the α-Bloch space ℬ_{α}, 0 < α < ∞, if

∥f∥_{ℬ_{α}} = sup_{z∈D} |f′(z)|(1 − |z|^{2})^{α} < ∞.

The little α-Bloch space ℬ_{α,0} consists of those f ∈ H(D) for which |f′(z)|(1 − |z|^{2})^{α} → 0 as |z| → 1. Denoting h^{∗}(z) = |h′(z)|/(1 − |h(z)|^{2}), the hyperbolic derivative of h ∈ B(D), the hyperbolic α-Bloch classes ℬ_{α}^{∗} and ℬ_{α,0}^{∗} are defined as the sets of those h ∈ B(D) for which

∥h∥_{ℬ_{α}^{∗}} = sup_{z∈D} h^{∗}(z)(1 − |z|^{2})^{α} < ∞

and lim_{|z|→1} h^{∗}(z)(1−|z|^{2})^{α} = 0, respectively. If α = 1, it is simply denoted ℬ^{∗} = ℬ_{1}^{∗} and ℬ_{0}^{∗} = ℬ_{1,0}^{∗}. Clearly ℬ_{α}^{∗} and ℬ_{α,0}^{∗} are not linear spaces. Moreover, the Schwarz–Pick lemma implies ℬ_{α}^{∗} = B(D) if α ≥ 1, and therefore the hyperbolic α-Bloch classes are only considered when 0 < α ≤ 1.

2000 Mathematics Subject Classification: Primary 47B38; Secondary 30D45, 30D50, 46E15.

For s > −1, the weighted Dirichlet space 𝒟_{s} (respectively weighted hyperbolic Dirichlet class 𝒟_{s}^{∗}) consists of those f ∈ H(D) (respectively h ∈ B(D)) for which

∥f∥_{𝒟_{s}} = (∫_{D} |f′(z)|^{2}(log \frac{1}{|z|})^{s} dA(z))^{1/2} < ∞

(respectively

∥h∥_{𝒟_{s}^{∗}} = (∫_{D}(h^{∗}(z))^{2}(log \frac{1}{|z|})^{s} dA(z))^{1/2} < ∞),

where dA(z) denotes the element of the Lebesgue area measure on D. The Schwarz–Pick lemma implies 𝒟_{s}^{∗} = B(D) for s > 1, and therefore the class 𝒟_{s}^{∗} is considered only when −1 < s ≤ 1. In this range the class 𝒟_{s}^{∗} contains no inner functions by [14, Theorem 1.1].

Let the Green’s function of D be defined as g(z, a) = − log |φ_{a}(z)|, where φ_{a}(z) = (a − z)/(1 − āz) is the automorphism of D which interchanges the points zero and a ∈ D. For 0 ≤ s < ∞, the Möbius invariant subspace (respectively subclass) Q_{s} (respectively Q_{s}^{∗}) of 𝒟_{s} (respectively 𝒟_{s}^{∗}) consists of those f ∈ H(D) (respectively h ∈ B(D)) for which

(1.1)  ∥f∥_{Q_{s}} = (sup_{a∈D} ∫_{D} |f′(z)|^{2}g^{s}(z, a) dA(z))^{1/2} < ∞

(respectively

(1.2)  ∥h∥_{Q_{s}^{∗}} = (sup_{a∈D} ∫_{D}(h^{∗}(z))^{2}g^{s}(z, a) dA(z))^{1/2} < ∞).

The space Q_{s,0} (respectively class Q_{s,0}^{∗}) consists of those f ∈ H(D) (respectively h ∈ B(D)) for which the integral expression in (1.1) (respectively (1.2)) tends to zero as |a| → 1. If s = 0, then Q_{0} is the classical Dirichlet space 𝒟 = 𝒟_{0} . If s > 1, then, by [2, Theorem 1], the spaces Q_{s} and Q_{s,0} coincide with the Bloch space ℬ and the little Bloch space ℬ_{0} , respectively, and the class Q_{s}^{∗} reduces to B(D) by the Schwarz–Pick lemma.

The following characterization of bounded composition operators mapping from ℬ_{α} into Q_{s} can be found in [17, Theorem 2.2.1(i)].

Theorem A. Let 0 < α < ∞, 0 ≤ s < ∞ and φ ∈ B(D). Then the following statements are equivalent:

Composition operators in hyperbolic Q-classes 393

(1) C_{φ}: ℬ_{α} → Q_{s} is bounded;

(2) C_{φ}: ℬ_{α,0} → Q_{s} is bounded;

(3) sup_{a∈D} ∫_{D} \frac{|φ′(z)|^{2}}{(1 − |φ(z)|^{2})^{2α}} g^{s}(z, a) dA(z) < ∞;

(4) sup_{a∈D} ∫_{D} \frac{|φ′(z)|^{2}}{(1 − |φ(z)|^{2})^{2α}} (1 − |φ_{a}(z)|^{2})^{s} dA(z) < ∞.

To be precise, the case s = 0 of Theorem A does not appear in [17, Theorem 2.2.1(i)], but it has been included above since the same proof works also in this case.

A composition operator C_{φ}: ℬ_{α}^{∗} → Q_{s}^{∗} is said to be bounded if there exists a positive constant C such that ∥C_{φ}(h)∥_{Q_{s}^{∗}} ≤ C∥h∥_{ℬ_{α}^{∗}} for all h ∈ ℬ_{α}^{∗}. Hereafter it is agreed the same meaning for the boundedness of C_{φ} mapping from one hyperbolic class X^{∗} into another hyperbolic class Y^{∗}. This definition is of the same spirit as the definition in [9] of a bounded composition operator mapping from one meromorphic function class into another. However, it would be of interest to find metrics in ℬ_{α}^{∗} and Q_{s}^{∗} such that these classes would become complete metric spaces and the continuity of C_{φ} would be equivalent to the natural definition of a bounded composition operator given above.

The first result of this paper extends Theorem A to the corresponding hyperbolic classes.

Theorem 1.1. Let 0 < α ≤ 1, 0 ≤ s ≤ 1 and φ ∈ B(D). Then the following statements are equivalent:

(1) C_{φ}: ℬ_{α} → Q_{s} is bounded;

(2) C_{φ} : ℬ_{α}^{∗} → Q_{s}^{∗} is bounded;

(3) C_{φ} : ℬ_{α,0}^{∗} → Q_{s}^{∗} is bounded.

Keeping the consideration mostly out of meromorphic function classes, it is settled to point out a somewhat surprising phenomenon which occurs here. Namely, by Theorem A, Theorem 1.1 and the theorem in [9], a composition operator C_{φ} mapping from ℬ into Q_{s} is bounded, if and only if, it is bounded from ℬ^{∗} into Q_{s}^{∗}, if and only if, it is bounded from 𝒩 into Q_{s}^{#} , where 𝒩 denotes the class of normal functions and Q_{s}^{#} is the meromorphic Q_{s} -class. See [2] and [9] for necessary definitions. Since the functions in ℬ_{α} are bounded if 0 < α < 1, it is easy to see, by using functions with Hadamard gaps as in the proof of Theorem 1.1, that this result remains also valid when the domain space and classes are ℬ_{α}, ℬ_{α}^{∗} and 𝒩_{α}^{#} , 0 < α < 1, respectively.

Two quantities a and b are said to be comparable, denoted by a ≃ b, if there exists a positive constant C such that C^{−1}a ≤ b ≤ Ca.

Example 1.2. For 0 < β < 1, define ϕ_{β}(z) = 1 − (1 − z)^{β}. Then ϕ_{β}(z) is a conformal mapping which fixes the points zero and one, and maps D onto a lens-type region. Since ϕ_{β}^{∗}(z) ≃ |1 − z|^{−1} in D, Theorem 1.1 and Theorem A imply that C_{ϕ_{β}}: ℬ^{∗} → Q_{s}^{∗} is bounded if and only if s is positive. It is now shown that C_{ϕ_{β}}: Q_{s_{1}}^{∗} → Q_{s_{2}}^{∗}, 0 < s_{1} ≤ 1, is bounded if and only if s_{2} is positive. Since Q_{s_{1}}^{∗} ⊂ ℬ^{∗} with ∥h∥_{ℬ^{∗}} ≤ C∥h∥_{Q_{s_{1}}^{∗}} , where C is a positive constant, it suffices to show that C_{ϕ_{β}}: Q_{s_{1}}^{∗} → 𝒟 is not bounded. But this easily follows by the fact (ϕ_{β} ∘ϕ_{β})^{∗}(z) ≃ |1−z|^{−1} in D, since Fatou’s lemma and (3.2) below yield ϕ_{β} ∈ Q_{s_{1}}^{∗} for 0 < s_{1} ≤ 1.

If 0 < α < ∞, 0 < s < ∞ and φ ∈ B(D), then C_{φ}: Q_{s} → ℬ_{α} is bounded if and only if φ ∈ ℬ_{α}^{∗} by [17, Theorem 2.2.1(iii)]. This result is extended for the corresponding hyperbolic classes in the following theorem.

Theorem 1.3. Let 0 < α ≤ 1, 0 ≤ s ≤ 1 and φ ∈ B(D). Then the following statements are equivalent:

(1) C_{φ}: ℬ → ℬ_{α} is bounded;

(2) C_{φ}: ℬ^{∗} → ℬ_{α}^{∗} is bounded;

(3) C_{φ}: Q_{s}^{∗} → ℬ_{α}^{∗} is bounded;

(4) φ ∈ ℬ_{α}^{∗}.

Theorem 1.4 generalizes [16, Theorem 4.1(i)] since the term |φ′_{a}(φ(z))| in conditions (2) and (3) below can be replaced by (1 − |a|^{2})^{τ} /|1 − āφ(z)|^{1+τ}, 0 < τ < ∞, by Lemma B below (in Section 2).

Theorem 1.4. Let −1 < s_{1} < ∞, 0 < s_{2} < ∞ and φ ∈ B(D). Then the following statements are equivalent:

(1) C_{φ}: 𝒟_{s_{1}} → Q_{s_{2}} is bounded;

(2) sup_{a,b∈D} ∫_{D}|φ′_{a}(φ(z))|^{2+s_{1}}|φ′(z)|^{2}g^{s_{2}}(z, b) dA(z) < ∞;

(3) sup_{a,b∈D} ∫_{D}|φ′_{a}(φ(z))|^{2+s_{1}}|φ′(z)|^{2}(1 − |φ_{b}(z)|^{2})^{s_{2}} dA(z) < ∞.

Moreover, if s_{1} ≤ 0 and s_{2} ≤ 1, then (1)–(3) are equivalent to

(4) C_{φ}: 𝒟_{s_{1}}^{∗} → Q_{s_{2}}^{∗} is bounded.

By Theorem 1.3, C_{φ}: 𝒟 → ℬ is bounded for all φ ∈ B(D). Moreover, since Q_{s} = ℬ for s > 1, Theorem 1.4 implies that C_{φ}: 𝒟 → ℬ is bounded if and only if

(1.3)  sup_{a,b∈D} ∫_{D} |(φ_{a} ∘ φ)′(z)|^{2}g^{s}(z, b) dA(z) < ∞, 1 < s < ∞.

However, (1.3) is equivalent to

sup_{a,z∈D} |(φ_{a} ∘ φ)′(z)|(1 − |z|^{2}) = ∥φ∥_{ℬ^{∗}} < ∞,

which is, of course, satisfied for all φ ∈ B(D).

Example 1.5. For 0 ≤ p < ∞, define ψ_{p}(z) = (p + z)/(p + 1). Then ψ_{p} is a conformal mapping which maps D onto the disc centered at p/(p + 1) with radius 1/(p + 1). Clearly,

(1.4)  ∥f ∘ φ∥_{Q_{s_{2}}} ≤ C∥f ∘ φ∥_{𝒟} ≤ C∥f∥_{𝒟}

for all f ∈ 𝒟 and φ ∈ B(D), thus, in particular, C_{ψ_{p}}: 𝒟 → Q_{s_{2}} is bounded. A similar reasoning shows that C_{ψ_{p}}: 𝒟^{∗} → Q_{s_{2}}^{∗} is also bounded. However, a geometric argument or a straightforward calculation based on the identity

1 − āψ_{p}(z) = \frac{p(1 − ā) + 1 − āz}{p + 1}

shows that |1 − aψ_{p}(z)| ≤ |1 − az| for all z ∈ D and a ∈ (0, 1), and therefore

sup_{a,b∈D} ∫_{D}|φ′_{a}(ψ_{p}(z))|^{2+s_{1}}|ψ′_{p}(z)|^{2}(1 − |φ_{b}(z)|^{2})^{s_{2}} dA(z)

≥ lim_{a→1} \frac{(1 − a^{2})^{2+s_{1}+s_{2}}}{(p + 1)^{2}} ∫_{D} \frac{(1 − |z|^{2})^{s_{2}}}{|1 − aψ_{p}(z)|^{2(2+s_{1})}|1 − az|^{2s_{2}}} dA(z)

≥ lim_{a→1} \frac{(1 − a^{2})^{2+s_{1}+s_{2}}}{(p + 1)^{2}} ∫_{D} \frac{(1 − |z|^{2})^{s_{2}}}{|1 − az|^{2(2+s_{1}+s_{2})}} dA(z) ≃ lim_{a→1}(1 − a^{2})^{−s_{1}},

from which it follows by Theorem 1.4 that C_{ψ_{p}}: 𝒟_{s_{1}} → Q_{s_{2}} is not bounded if s_{1} is positive.

Example 1.6. Let 0 < β < 1, −1 < s_{1} < ∞ and 0 ≤ s_{2} < ∞, and consider the map ϕ_{β}(z) = 1 − (1 − z)^{β}. It is proved that C_{ϕ_{β}} admits the same behavior as C_{ψ_{p}} does in the sense that C_{ϕ_{β}}: 𝒟_{s_{1}} → Q_{s_{2}} is bounded if and only if −1 < s_{1} ≤ 0. In view of (1.4) it suffices to show that C_{ϕ_{β}}: 𝒟_{s_{1}} → Q_{s_{2}} is not bounded if s_{1} > 0. To this end, choose f_{a}(z) = (1 − z)^{−a}, 0 < a < ∞. Now, by [5, Lemma on p. 65], there is a positive constant C_{1} such that

∥f_{a}∥_{𝒟_{s_{1}}}^{2} = a^{2} ∫_{D} \frac{(1 − |z|^{2})^{s_{1}}}{|1 − z|^{2(1+a)}} dσ(z) ≤ C_{1}a^{2} ∫_{0}^{1} \frac{r dr}{(1 − r)^{2a+1−s_{1}}} ,

and therefore f_{a} ∈ 𝒟_{s_{1}} for 0 < a < \frac{1}{2}s_{1} . Moreover, denote ω_{a,β}(z) = (1−z)^{−aβ} = f_{a} ∘ ϕ_{β}. Then, by [2, Proposition 1], there is a positive constant C_{2} such that

∥f_{a} ∘ ϕ_{β}∥_{Q_{s_{2}}} = ∥ω_{a,β}∥_{Q_{s_{2}}} ≥ C_{2}∥ω_{a,β}∥_{ℬ} = ∞,

and therefore f ∘ ϕ_{β} ∉ Q_{s_{2}} . Thus C_{ϕ_{β}}: 𝒟_{s_{1}} → Q_{s_{2}} is not bounded if s_{1} > 0. The following result generalizes [16, Theorem 4.1(ii)].

Theorem 1.7. Let −1 < s_{1} < ∞, 0 ≤ s_{2} < ∞ and φ ∈ B(D). Then the following statements are equivalent:

(1) C_{φ}: 𝒟_{s_{1}} → Q_{s_{2}} is compact;

(2) lim_{|a|→1} sup_{b∈D} ∫_{D} |φ′_{a}(φ(z))|^{2+s_{1}}|φ′(z)|^{2}g^{s_{2}}(z, b) dA(z) = 0;

(3) lim_{|a|→1} sup_{b∈D} ∫_{D} |φ′_{a}(φ(z))|^{2+s_{1}}|φ′(z)|^{2}(1 − |φ_{b}(z)|^{2})^{s_{2}} dA(z) = 0.

Since, by the general definition of a bounded (respectively compact) operator mapping from one Banach space into another, C_{φ}: 𝒟_{s_{1}} → Q_{s_{2},0} is bounded (respectively compact) if and only if C_{φ}: 𝒟_{s_{1}} → Q_{s_{2}} is bounded (respectively compact) and C_{φ}(𝒟_{s_{1}}) ⊂ Q_{s_{2},0} , the operator C_{φ}: 𝒟_{s_{1}}^{∗} → Q_{s_{2},0}^{∗} is said to be bounded, if C_{φ}: 𝒟_{s_{1}}^{∗} → Q_{s_{2}}^{∗} is bounded and C_{φ}(𝒟_{s_{1}}^{∗}) ⊂ Q_{s_{2},0}^{∗}.

Theorem 1.8. Let −1 < s_{1} < ∞, 0 < s_{2} < ∞ and φ ∈ B(D). Then C_{φ}: 𝒟_{s_{1}} → Q_{s_{2},0} is bounded if and only if C_{φ}: 𝒟_{s_{1}} → Q_{s_{2}} is bounded and the following two conditions are satisfied:

(1) φ ∈ Q_{s_{2},0} ;

(2) lim_{|a|,|b|,t→1} ∫_{|φ(z)|≥t}|φ′_{a}(φ(z))|^{2+s_{1}}|φ′(z)|^{2}g^{s_{2}}(z, b) dA(z) = 0.

Similarly, if s_{1} ≤ 0 and s_{2} ≤ 1, then C_{φ}: 𝒟_{s_{1}}^{∗} → Q_{s_{2},0}^{∗} is bounded if and only if C_{φ}: 𝒟_{s_{1}}^{∗} → Q_{s_{2}}^{∗} is bounded, φ ∈ Q_{s_{2},0}^{∗} and (2) is satisfied.

It is easy to show that the conditions (1) and (2) in Theorem 1.8 together are equivalent to

(3) lim_{|b|→1} sup_{a∈D} ∫_{D}|φ′_{a}(φ(z))|^{2+s_{1}}|φ′(z)|^{2}g^{s_{2}}(z, b) dA(z) = 0,

and hence the first part of Theorem 1.8 implies the following result.

Theorem 1.9. Let −1 < s_{1} < ∞, 0 < s_{2} < ∞ and φ ∈ B(D). Then C_{φ}: 𝒟_{s_{1}} → Q_{s_{2},0} is compact if and only if C_{φ}: 𝒟_{s_{1}} → Q_{s_{2}} is compact and the condition (3) above is satisfied.

The remaining part of the paper is organized as follows. In Section 2, some background material and auxiliary results needed later on are recalled, and Section 3 contains the proofs of the results presented in this section.

Acknowledgments. The research reported in this paper was supported in part by the grants of MEC, Spain, BFM2002-02098 and MTM2004-21420-E. The authors would like to thank Dragan Vukotić for some useful comments and discussions.

2. Background material

A positive Borel measure μ on D is a bounded s-Carleson measure, if

sup_{I} \frac{μ(S(I))}{|I|^{s}} < ∞, 0 < s < ∞,

where |I| denotes the arc length of a subarc I of the boundary of

S(I) = \{z ∈ D : \frac{z}{|z|} ∈ I, 1 − |I| ≤ |z|\}

is the Carleson box based on I, and the supremum is taken over all subarcs I such that |I| ≤ 1. Moreover, if

lim_{|I|→0} \frac{μ(S(I))}{|I|^{s}} = 0, 0 < s < ∞,

then μ is a compact s-Carleson measure. If s = 1, then a bounded (respectively compact) 1-Carleson measure is just a standard bounded (respectively compact) Carleson measure.

Some well-known and useful characterizations of bounded s-Carleson measures are gathered in the following lemma. For the proof, see [1, Theorem 13], [3, Lemma 2.1], [10, pp. 89–90], and [11, Proposition 2.1].

Lemma B. Let μ be a positive Borel measure on D, 1 < s < ∞, 0 < r < 1 and 0 < τ < ∞. Then the following statements are equivalent:

(1) K_{1} = sup_{I} \frac{μ(S(I))}{|I|^{s}} < ∞;

(2) K_{2} = sup_{z∈D} \frac{μ(D(z, r))}{(1 − |z|^{2})^{s}} < ∞;

(3) K_{3} = sup_{z∈D} ∫_{D}(\frac{(1 − |z|^{2})^{τ}}{|1 − z̄w|^{1+τ}} )^{s} dμ(w) < ∞.

Moreover, K_{1} , K_{2} and K_{3} are comparable.

Here D(a, r) = {z ∈ D : |φ_{a}(z)| < r} is the pseudo-hyperbolic disc of center a ∈ D and radius 0 < r < 1. The pseudo-hyperbolic disc D(a, r) is an Euclidean disc centered at (1 − r^{2})a/(1 − |a|^{2}r^{2}) with radius (1 − |a|^{2})r/(1 − |a|^{2}r^{2}), see [8, p. 3].

The following change of variables formula by C. S. Stanton, [6] and [15], was apparently first used by J. H. Shapiro [12] in the study of composition operators, and it also plays a key role in some of the proofs in this paper.

Lemma C. Let g and u be positive measurable functions on D, and let φ ∈ B(D). Then

∫_{D}(g ∘ φ)(z)|φ′(z)|^{2}u(z) dA(z) = ∫_{D} g(w)U(φ, w) dA(w),

where

U(φ, w) = ∑_{z∈φ^{−1}{w}} u(z), w ∈ D \ {φ(0)}.

If u(z) = (− log |z|)^{s}, then U(φ, w) is the generalized Nevanlinna counting function

N_{φ,s}(w) = ∑_{z∈φ^{−1}\{w\}}(log \frac{1}{|z|})^{s}.

For the study of compactness, the following well-known result is needed. See [4, Proposition 3.11] for a similar result.

Lemma D. Let −1 < s_{1} < ∞, 0 ≤ s_{2} < ∞ and φ ∈ B(D). Then C_{φ}: 𝒟_{s_{1}} → Q_{s_{2}} is compact if and only if for any bounded sequence {f_{n}} in 𝒟_{s_{1}} with f_{n} → 0 uniformly on compact subsets of D as n → ∞, ∥f_{n} ∘φ∥_{Q_{s_{2}}} → 0 as n → ∞.

3. Proofs

Proof of Theorem 1.1. It is enough to prove the implications (1) ⇒ (2) and (3) ⇒ (1) since (2) ⇒ (3) is clearly true.

Suppose C_{φ}: ℬ_{α} → Q_{s} is bounded, that is, (1) is satisfied. If h ∈ ℬ_{α}^{∗}, then

∫_{D}((h ∘ φ)^{∗}(z))^{2}g^{s}(z, a) dA(z) ≤ ∥h∥_{ℬ_{α}^{∗}}^{2} ∫_{D} \frac{|φ′(z)|^{2}}{(1 − |φ(z)|^{2})^{2α}} g^{s}(z, a) dA(z),

where the last integral is uniformly bounded for all a ∈ D by Theorem A, and therefore C_{φ}: ℬ_{α}^{∗} → Q_{s}^{∗} is bounded. Thus (1) implies (2) is proved.

To prove (3) ⇒ (1), let first α = 1, and suppose that C_{φ}: ℬ_{0}^{∗} → Q_{s}^{∗} is bounded. If h_{b}(z) = bz, b ∈ D, then h^{∗}(z) = |b|(1 − |bz|^{2})^{−1} and h_{b} ∈ ℬ_{0}^{∗} for all b ∈ D, and

sup_{a∈D} ∫_{D} \frac{|b|^{2}|φ′(z)|^{2}}{(1 − |bφ(z)|^{2})^{2}} g^{s}(z, a) dA(z) ≤ C∥h∥_{ℬ^{∗}}^{2} ≤ C|b|^{2}

for some positive C by the assumption. Taking limit as |b| → 1, b ∈ D, Fatou’s lemma with Theorem A implies that C_{φ}: ℬ → Q_{s} is bounded, that is, (1) with α = 1 holds.

If 0 < α < 1, functions with Hadamard gaps may be used. Define

g_{n}(z) = ∑_{k=0}^{∞} 2^{k(α−1)}(b_{n}z)^{2^{k}},

where {b_{n}} ⊂ D and |b_{n}| → 1, as n → ∞. Then g_{n} ∈ ℬ_{α,0} by [18, Theorem 1]. Since |g_{n}(z)| ≤ ∑_{k=0}^{∞} 2^{k(α−1)} , there is a positive constant C, depending only on α, such that h_{n} = C^{−1}g_{n} satisfies |h_{n}(z)| ≤ \frac{1}{2} for all z ∈ D and n ∈ N, and therefore h_{n}^{∗}(z) ≃ |h′_{n}(z)| in D. Now one may argue as in [1, p. 133] and use Fatou’s lemma to conclude that the condition (3) with 0 < α < 1 in Theorem A is satisfied, and hence (1) with 0 < α < 1 holds.

Remark. To characterize bounded composition operators from Q_{s_{1}}^{∗} to Q_{s_{2}}^{∗} when 0 < s_{1}, s_{2} < 1 appears to be more complicated. However, Example 1.2 shows that there is a function φ for which C_{φ}: ℬ^{∗} → Q_{s_{2}}^{∗} is bounded if and only C_{φ}: Q_{s_{1}}^{∗} → Q_{s_{2}}^{∗} is bounded, provided 0 < s_{1} ≤ 1.

Proof of Theorem 1.3. It suffices to prove the implications (3) ⇒ (4) and (4) ⇒ (2) since (2) ⇒ (3) is clearly true and (1) is equivalent to (4) by [17, Theorem 2.2.1(iii)].

Suppose C_{φ}: Q_{s}^{∗} → ℬ_{α}^{∗} is bounded, that is, (3) holds, and let first 0 < s ≤ 1. If ϕ_{β,a}(z) = 1 − (1 − āz)^{β}, where 0 < β < 1 and a ∈ D, then

(3.1)  ϕ_{β,a}^{∗}(z) = \frac{β|a| |1 − āz|}{1 − |1 − (1 − āz)^{β}|} ≃ \frac{β|a|}{|1 − āz|}, z ∈ D.

By [7, Lemma 2.5], there is a positive constant C_{1} such that

(3.2)  \begin{aligned}∫_{D}(ϕ_{β,a}^{∗}(z))^{2}(1 − |φ_{b}(z)|^{2})^{s} dA(z) & ≃ β^{2}|a|^{2}(1 − |b|^{2})^{s} \\ & · ∫_{D} \frac{(1 − |z|^{2})^{s}}{|1 − b̄z|^{2s}|1 − āz|^{2}} dA(z) \\ & ≤ C_{1}^{2}β^{2}|a|^{2} \frac{(1 − |b|^{2})^{s}}{|1 − āb|^{s}} , \end{aligned}

and it follows that ∥ϕ_{β,a}∥_{Q_{s}^{∗}} ≤ C_{1}2^{s/2}β|a| for all a ∈ D, 0 < β < 1 and 0 < s ≤ 1. By the assumption there exists a positive constant C_{2} such that

\frac{β|a|}{2} \frac{|φ′(z)|}{|1 − āφ(z)|}(1−|z|^{2})^{α} ≤ (ϕ_{β,a}∘φ)^{∗}(z)(1−|z|^{2})^{α} ≤ C_{2}∥ϕ_{β,a}∥_{Q_{s}^{∗}} ≤ C_{2}C_{1}2^{s/2}β|a|,

and the assertion φ ∈ ℬ_{α}^{∗} follows by choosing a = φ(z). The case s = 0 can be proved in a similar manner by choosing the test function φ_{a}(z)/2.

If h ∈ ℬ^{∗} = B(D), then

(h ∘ φ)^{∗}(z)(1 − |z|^{2})^{α} ≤ ∥h∥_{ℬ^{∗}} \frac{|φ′(z)|}{1 − |φ(z)|^{2}} (1 − |z|^{2})^{α} ≤ ∥h∥_{ℬ^{∗}} ∥φ∥_{ℬ_{α}^{∗}} ,

and (4) ⇒ (2) follows.

Proof of Theorem 1.4. To prove that (2) implies (1), the reasoning in the proof of [10, Theorem 2.2] is followed. By Lemma C and the subharmonicity of |f′(z)|^{2} , there is a positive constant C_{1} such that

\begin{aligned}_{∫D} & |(f ∘ φ)′(z)|^{2}g^{s_{2}}(z, a) dA(z) = ∫_{D} |f′(w)|^{2}dμ_{a,s_{2}}(w) \\ & ≤ C_{1} ∫_{D}(\frac{1}{(1 − |w|^{2})^{2}} ∫_{D(w,1/2)} |f′(z)|^{2}dA(z))dμ_{a,s_{2}}(w), \end{aligned}

where dμ_{a,s_{2}}(w) = N_{φ∘φ_{a},s_{2}}(w) dA(w). Then the symmetry

χ_{D(z,r)}(w) = χ_{D(w,r)}(z)

of the characteristic functions of pseudohyperbolic discs and Fubini’s theorem yield

(3.3) ∫_{D}|(f ∘ φ)′(z)|^{2}g^{s_{2}}(z, a) dA(z) ≤ C_{1}∫_{D}|f′(z)|^{2}(∫_{D(z,1/2)}\frac{dμ_{a,s_{2}}(w)}{(1 − |w|^{2})^{2}} ) dA(z).

By Lemmas B and C, the assumption (2) is equivalent to

(3.4)  sup_{a∈D} ∫_{D(z,1/2)} dμ_{a,s_{2}}(w) ≤ C_{2}(1 − |z|^{2})^{2+s_{1}}, z ∈ D,

for some positive constant C_{2} . Since 1−|w| ≃ 1−|z| for w ∈ D(z, 1/2), it follows by (3.3) and (3.4) that C_{φ}: 𝒟_{s_{1}} → Q_{s_{2}} is bounded.

Suppose then that C_{φ}: 𝒟_{s_{1}} → Q_{s_{2}} is bounded. For a ∈ D, define f_{a}(z) = ∫_{0}^{z}(φ′_{a}(w))^{1+s_{1}/2} dw. Then, by Forelli–Rudin estimates [19, Lemma 4.2.2] there is a positive constant C_{1} , depending only on s_{1} , such that

∥f_{a}∥_{𝒟_{s_{1}}}^{2} = (1 − |a|^{2})^{1+s_{1}/2} ∫_{D} \frac{(1 − |z|^{2})^{s_{1}}}{|1 − āz|^{2+s_{1}}} dσ(z) ≤ C_{1}

for all a ∈ D, and thus the family {f_{a} : a ∈ D} is norm bounded uniformly in 𝒟_{s_{1}} . Since C_{φ}: 𝒟_{s_{1}} → Q_{s_{2}} is bounded, there is a positive constant C_{2} such that

\begin{aligned}sup_{b∈D} ∫_{D} |φ′_{a}(φ(z))|^{2+s_{1}}|φ′(z)|^{2}(1 − |φ_{b}(z)|^{2})^{s_{2}} dσ(z) & = ∥f_{a} ∘ φ∥_{Q_{s_{2}}}^{2} \\ & ≤ C_{2}∥f_{a}∥_{𝒟_{s_{1}}}^{2} ≤ C_{1}C_{2} \end{aligned}

for all a ∈ D, and the condition (2) follows.

Since (2) and (3) are clearly equivalent, it is now proceeded to consider the hyperbolic case. Suppose that s_{1} ≤ 0 and s_{2} ≤ 1. If (2) is satisfied, then the same reasoning as in the first part of the proof shows that C_{φ}: 𝒟_{s_{1}}^{∗} → Q_{s_{2}}^{∗} is bounded since also (h^{∗})^{2} is a subharmonic function in D.

Suppose then that C_{φ}: 𝒟_{s_{1}}^{∗} → Q_{s_{2}}^{∗} is bounded. For a ∈ D and \frac{1}{2} < γ ≤ 1 let

(3.5) f_{a,γ}(z) = ∫_{0}^{z}(φ′_{a}(w))^{γ} dw =\begin{cases}\frac{(1 − |a|^{2})^{γ}}{ā(1 − 2γ)} ((1 − āz)^{1−2γ} − 1), & a ∈ D \ \{0\}, \\ z, & a = 0, \end{cases}

and

(3.6)  h_{a,γ}(z) = \{ \begin{matrix}\frac{(2γ − 1)ā}{6}f_{a,γ}(z), & a ∈ D \ \{0\}, \\ \frac{z}{2}, & a = 0. \end{matrix}

Then ∥h_{a,γ}∥_{∞} ≤ \frac{1}{2} for all a ∈ D, and therefore h_{a,γ}^{∗}(z) ≃ |h′_{a,γ}(z)| in D. The reasoning in the proof of the implication (1) ⇒ (2) with the functions

h_{a,1+s_{1}/2}(z) = \frac{(1 + s_{1})ā}{6} ∫_{0}^{z}(φ′_{a}(w))^{1+s_{1}/2} dw

yields the assertion (2).

Proof of Theorem 1.7. Suppose first that C_{φ}: 𝒟_{s_{1}} → Q_{s_{2}} is compact and consider the functions f_{a}(z) = ∫_{0}^{z}(φ′_{a}(w))^{1+s_{1}/2} dw. Since, by the proof of Theorem 1.4, there is a positive constant C such that ∥f_{a}∥_{𝒟_{s_{1}}} ≤ C for all a ∈ D, and further f_{a} → 0 uniformly on compact subsets as |a| → 1, Lemma D gives (2).

Suppose now that (2) holds. Let {f_{n}} ⊂ 𝒟_{s_{1}} such that ∥f_{n}∥_{𝒟_{s_{1}}} ≤ C_{1} < ∞ for all n ∈ N, and f_{n} → 0 uniformly on compact subsets of D. By Lemma D, it suffices to show that ∥f_{n}∘φ∥_{Q_{s_{2}}} → 0 as n → ∞. For 0 < δ < 1, let Δ(0, δ) denote the Euclidean disc centered at the origin and of radius δ. A similar reasoning as in the proof of Theorem 1.4 with the fact 1−|w|^{2} ≃ 1−|z|^{2} ≃ |1−z̄w|, w ∈ D(z, 1/2), yields

∥f_{n} ∘ φ∥_{Q_{s_{2}}}^{2} = sup_{a∈D} ∫_{D} |f′_{n}(w)|^{2} dμ_{a,s_{2}}(w)

≤ C_{2} ∫_{D} |f′_{n}(z)|^{2}(1 − |z|^{2})^{s_{1}}(sup_{a∈D} ∫_{D(z,1/2)} |φ′_{z}(w)|^{2+s_{1}} dμ_{a,s_{2}}(w)) dA(z)

≤ C_{2} ∫_{D\Δ(0,δ)} |f′_{n}(z)|^{2}(1 − |z|^{2})^{s_{1}}(sup_{a∈D} ∫_{D} |φ′_{z}(w)|^{2+s_{1}} dμ_{a,s_{2}}(w)) dA(z)

+ \frac{C_{2}4^{1+s_{1}}}{(1 − δ)^{2}} ∥φ∥_{Q_{s_{2}}}^{2} ∫_{Δ(0,δ)} |f′_{n}(z)|^{2} dA(z)

= I_{1}(δ) + I_{2}(δ), where C_{2} is a positive constant. For a given ε > 0, by the assumption (2) and Lemma C, there exists a δ_{0} ∈ (0, 1) such that, for all |z| > δ_{0} ,

sup_{a∈D} ∫_{D} |φ′_{z}(w)|^{2+s_{1}} dμ_{a,s_{2}}(w) ≤ \frac{ε^{2}}{2C_{1}^{2}C_{2}} ,

and it follows that I_{1}(δ_{0}) < ε^{2}/2. In view of Theorem 1.4, the assumption (2) implies that C_{φ}: 𝒟_{s_{1}} → Q_{s_{2}} is bounded, and hence φ ∈ Q_{s_{2}} . Since f_{n} → 0 uniformly on compact subsets of D, in particular, in Δ(0, δ_{0}), there exists an N ∈ N such that, for n ≥ N,

∫_{Δ(0,δ_{0})} |f′_{n}(z)|^{2} dA(z) < \frac{ε^{2}(1 − δ_{0})^{2}}{2C_{2}4^{1+s_{1}}∥φ∥_{Q_{s_{2}}}^{2}} ,

and therefore I_{2}(δ_{0}) < ε^{2}/2. Thus, as n ≥ N, ∥f_{n} ∘ φ∥_{Q_{s_{2}}} < ε, and C_{φ}: 𝒟_{s_{1}} → Q_{s_{2}} is compact by Lemma D.

Since (2) and (3) are clearly equivalent, the proof is complete.

Proof of Theorem 1.8. Suppose first that C_{φ}: 𝒟_{s_{1}} → Q_{s_{2},0} is bounded, that is, C_{φ}: 𝒟_{s_{1}} → Q_{s_{2}} is bounded and C_{φ}(𝒟_{s_{1}}) ⊂ Q_{s_{2},0} . Then, by using the functions f(z) = z ∈ 𝒟_{s_{1}} and f_{a}(z) = ∫_{0}^{z}(φ′_{a}(w))^{1+s_{1}/2} dw ∈ 𝒟_{s_{1}} , the inclusion C_{φ}(𝒟_{s_{1}}) ⊂ Q_{s_{2},0} implies that (1) and (2) are satisfied.

Suppose now that C_{φ}: 𝒟_{s_{1}} → Q_{s_{2}} is bounded, and the conditions (1) and (2) are satisfied. It suffices to show that f ∘ φ ∈ Q_{s_{2},0} for f ∈ 𝒟_{s_{1}} . A similar reasoning as in the proof of Theorem 1.7 yields

(3.7)  \begin{aligned}_{∫D} & |(f ∘ φ)′(z)|^{2}g^{s_{2}}(z, a) dA(z) ≤ C ∫_{D\Δ(0,t)} |f′(z)|^{2}(1 − |z|^{2})^{s_{1}} \\ & · (∫_{D} |φ′_{z}(w)|^{2+s_{1}}dμ_{a,s_{2}}(w)) dA(z) \\ & + C ∫_{Δ(0,t)} |f′(z)|^{2}(∫_{D(z,1/2)} \frac{dμ_{a,s_{2}}(w)}{(1 − |z|^{2})^{2}} ) dA(z) \\ & = I_{1}(t) + I_{2}(t), \end{aligned}

where C is a positive constant and 0 < t < 1. To deal with I_{1}(t), write

∫_{D} |φ′_{z}(w)|^{2+s_{1}} dμ_{a,s_{2}}(w) = ∫_{Δ(0,t)} |φ′_{z}(w)|^{2+s_{1}} dμ_{a,s_{2}}(w)

+ ∫_{D\Δ(0,t)} |φ′_{z}(w)|^{2+s_{1}} dμ_{a,s_{2}}(w)

≤ \frac{2^{2+s_{1}}}{(1 − t)^{2+s_{1}}} ∫_{D} |φ′(u)|^{2}g^{s_{2}}(u, a) dσ(u)

+ ∫_{D\Δ(0,t)} |φ′_{z}(w)|^{2+s_{1}} dμ_{a,s_{2}}(w) = I_{3}(t) + I_{4}(t).

By the assumption (2), for a given ε > 0, there exists a δ_{1} ∈ (0, 1) such that

(3.8)  I_{4}(t) < \frac{ε^{2}}{3C∥f∥_{𝒟_{s_{1}}}^{2}}

for all |a|, |z|, t ≥ δ_{1} . Let t ≥ δ_{1} be fixed. Since φ ∈ Q_{s_{2},0} by the assumption (1), there exists a δ_{2} ∈ [δ_{1}, 1) such that

(3.9)  I_{3}(t) < \frac{ε^{2}(1 − t)^{2+s_{1}}}{3 · 2^{2+s_{1}}C∥f∥_{𝒟_{s_{1}}}^{2}}

for all |a| ≥ δ_{2} . Since |z| ≥ t in the term I_{1}(t), it follows by combining (3.7), (3.8) and (3.9) that I_{1}(t) ≤ 2ε^{2}/3 for |a| ≥ δ_{2} . Further, since φ ∈ Q_{s_{2},0} by the assumption (1), there exists a δ_{3} ∈ [δ_{2}, 1) such that

I_{2}(t) ≤ \frac{Cπt^{2}}{(1 − t^{2})^{2}} sup_{|z|=t} |f′(z)|^{2} ∫_{D} |φ′(z)|^{2}g^{s_{2}}(z, a) dA(z) < \frac{ε^{2}}{3}

for all |a| ≥ δ_{3} . Therefore one finally concludes

∫_{D} |(f ∘ φ)′(z)|^{2}g^{s_{2}}(z, a) dA(z) ≤ I_{1}(t) + I_{2}(t) < \frac{2ε^{2}}{3} + \frac{ε^{2}}{3} = ε^{2}

for all |a| ≥ δ_{3} , that is, f ∘ φ ∈ Q_{s_{2},0} as one wished to prove.

Proof of Theorem 1.9. As it was pointed out in Section 1, Theorem 1.9 follows by Theorem 1.8.

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Received 18 May 2005