On composition operators in QK type spaces

Let p ≥ 1, q > −2 and let K : [0,∞) → [0,∞) be nondecreasing. With a different choice of p , q and K , the Banach space QK(p, q) coincides with many well-known analytic function spaces. Boundedness and compactness of the composition operator Cφ from α -Bloch space Bα into QK(p, q) are characterized by a condition depending only on analytic mapping φ : D → D . The same properties are also studied in the case Cφ : QK(p, q) → Bα .


Introduction
The space Q K (p, q) was introduced in [17].To recall the definition, let D denote the unit disc and let σ a (z) = (a − z)/(1 − āz) be the automorphism in D, which interchanges points 0 and a.Then g(z, a) = − log |σ a (z)| is the Green function.Let p > 0, q > −2 and K : [0, ∞) → [0, ∞) be right-continuous and nondecreasing.An analytic function Here and elsewhere dA stands for the Euclidean area element.The space Q K (p, q) is a Banach space with the norm f = |f (0)| + f QK(p,q) for p ≥ 1 .Setting K(x) = x s , s ≥ 0, the space Q K (p, q) equals to F (p, q, s), the function family introduced in [22].Hence, with different parameters Q K (p, q) coincides with many classical function spaces.This list contains spaces like BM OA, Q s , the Bloch space B , and the Hardy space H 2 (see [1], [3], [18] and [22]).On the other hand, Q K (p, q) generalizes the space Q K = Q K (2, 0) (see [4] and [15]).
One important property of Q K (p, q) spaces is the inclusion relationship with α -Bloch spaces.Let α > 0 and let B α denote the α -Bloch space consisting of functions f analytic in D such that It is shown in [17] that if and only if (1.1) Throughout this article it is assumed that ( (1 − r 2 ) q K(− log r)r dr < ∞, since otherwise Q K (p, q) consists only of constant functions.Every analytic function ϕ : D → D induces a composition operator defined by C ϕ f = f • ϕ for f analytic in D. Thus, boundedness and compactness of the operator C ϕ : X → Y can be considered if X and Y are some normed spaces of functions analytic in the unit disc.In the literature there exist many results for the cases where either X or Y equals B α .For instance, bounded and compact operators C ϕ : B → Q s were characterized in [10] and the case C ϕ : B α → F (p, q, s) was solved in [6].Operators C ϕ : Q s → B α were originally studied in [20] and generalized to C ϕ : F (p, q, s) → B α in [9].Bounded and compact operators C ϕ : B → Q K were considered in [16] and [14], respectively.
In this paper, boundedness of the composition operator between two Q K (p, q) spaces is considered in Section 2. Boundedness of the operator characterized by a condition depending only on the parameters and the analytic self-map ϕ : D → D. In the study of the inverse case C ϕ : Q K (p, q) → B α an additional assumption on K is needed to ensure that the sequence of test functions is bounded in Q K (p, q).This assumption is automatically satisfied in the special case C ϕ : F (p, q, s) → B α .In Section 3 compact composition operators between two Q K (p, q) spaces are studied.After considering basic facts in the general case, compactness is characterized in cases C ϕ : B α → Q K (p, q) and C ϕ : Q K (p, q) → B α .Again, the latter case is studied under the additional assumption on K.

Bounded composition operators
In this section bounded composition operators between B α and Q K (p, q) are studied.This is one step on the way to characterize these properties in the more general case )/p at that when (1.1) holds.Therefore it is first convenient to establish some general facts about composition operators from one space of Q K type to another.

Boundedness and seminorm. When studying bounded operators
if and only if there exists a constant C > 0 such that the condition holds for every f ∈ Q K1 (p 1 , q 1 ).
Proof.Assume C ϕ is bounded, let f ∈ Q K1 (p 1 , q 1 ) and set h = f − f (0).Then there exists a constant C > 0 such that Conversely, assume (2.1) is true.By subharmonicity of |f | p , for any r ∈ (0, 1) there exists a constant M (q), depending on parameter q , such that for every f ∈ Q K (p, q), see [22,Lemma 2.9] or [17,Theorem 2.1] for details.This implies first that there exists a constant C 1 > 0 such that for every f ∈ Q K (p, q).Furthermore, , which shows that for every z ∈ D there exists a constant C 2 > 0 such that for every f ∈ Q K (p, q).Then, applying (2.3) and (2.4) for Q K1 (p 1 , q 1 ) and combining this with (2.1), for every f ∈ Q K1 (p 1 , q 1 ) and thus C ϕ is bounded.

Bounded operators from
The next theorem is based on studies in [6] and [10].The lemma used in the proof is first recalled here for reader's convenience, see [23, p. 215]: Lemma 2.2 ([23]).Let p > 0 .If {n k } is a non-decreasing sequence of natural numbers and n k+1 /n k ≥ λ > 1 for all k ∈ N, then there exists a constant A = A(p, λ) > 0 such that for all sequences {a k } ⊂ C.
Theorem 2.3 is related to [10,Theorem 1.5] and [20, Theorem 2.2.1] as the next corollary shows.See [7] for further results on this subject.
Corollary 2.4.Let ϕ be an analytic self-map of D, α > 0 and s ≥ 0 .Then the next conditions are equivalent: Theorem 2.3 also generalizes the result in [16]: Corollary 2.5.Let ϕ be an analytic self-map of D, α > 0 and let K be nonnegative and nondecreasing in [0, ∞).Then the next conditions are equivalent: The original result of [6] is essentially the following: Corollary 2.6.Let ϕ be an analytic self-map of D, α > 0 , p > 0 , q > −2 , and s > 0 .Then the next conditions are equivalent: The idea of using Lemma 2.2 as a solution to remove assumption p ≥ 2 in (b) was first pointed out in [9].

Bounded operators from
an extra assumption on K is needed.This guarantees that the sequence of test functions is bounded in Q K (p, q) and a suitable assumption is found in the next lemma.Lemma 2.7.Let q > −2 and let K be nonnegative and nondecreasing in [0, ∞).Then if and only if one of the following is true: where K[g(re iθ , a)] can be replaced by K[g(re i(θ−γ) , |a|)] choosing γ = arg(a).For fixed r ∈ (0, 1) and a ∈ D the functions are finite almost everywhere in R, decreasing in [0, π) and increasing in (π, 2π], 2π -periodic and even.Let φ be the rearrangement of φ in [0, 2π] (see [5, p. 276]) and denote In other words, Let t ∈ [0, 1).By the change of variable and by elementary estimates (2.9) sup The right hand side of (2.9) is bounded for r ≤ 1/2 and otherwise (see e.g.[13, p. 226]) which completes the proof.
Remark 2.8.By [2, Theorem 6.1] it is easy to see that log(1−z) belongs to Q s with every s > 0 , although log(1 − z) represents an extremal growth in B and Q s B with s < 1 .Similarly a function h with derivative |h (z)| p = |1 − z| −q−2 represents an extremal growth in B (q+2)/p and by Lemma 2.7 there exists parameters p, q and K such that h ∈ Q K (p, q) B (q+2)/p .For example, set K(t) = t s , 0 < s < 1 and q > −s − 1.Then Q K (p, q) coincides with the non-trivial space F (p, q, s).As a by-product Lemma 2.7 generalizes Theorem 2.9 in [4], which states that log(1 − z) belongs to Q K (2, 0) if and only if for the characteristic function of the set E ⊂ R.This notation is first used to combine the conditions (i) and (ii) of Lemma 2.7 in the next theorem.

Compact composition operators
The study of compact operators from one Q K type space to another is divided in three parts.First some general facts are considered, then the compactness of C ϕ : B α → Q K (p, q) is characterized, and in the end composition operators from Q K (p, q) to B α are studied.

Compactness and convergence of sequences.
The next lemma is proved in [11, p. 4690] and [12, p. 31], but it needs to be recalled with assumptions of slightly different shape.Note that both sources have a minor misprint in condition (1), but the proof is correct.Lemma 3.1 ([11]).Let X and Y be two Banach spaces of analytic funtions on D and let T be a linear mapping from X to Y .Assume that the following conditions are true: (1) The point evaluation functionals on Y are bounded; (2) For every bounded sequence in X , there is a subsequence, which converges uniformly to an element of X on compact subsets of D; (3) If {f n } ⊂ X converges uniformly to zero on compact subsets of D, then {T f n } converges uniformly to zero on compact subsets of D. Then T : X → Y is compact if and only if given a bounded sequence {f n } in X such that f n → 0 uniformly on compact sets, then the sequence {T f n } converges to zero in the norm of Y .Lemma 3.2.Let ϕ be an analytic self-map of D, p 1 , p 2 ≥ 1 , q 1 , q 2 > −2 and let K 1 , K 2 be nonnegative and nondecreasing in [0, ∞).Then ϕ induces a compact composition operator from Q K1 (p 1 , q 1 ) to Q K2 (p 2 , q 2 ) if and only if for every bounded sequence {f n } ⊂ Q K1 (p 1 , q 1 ) such that f n → 0 uniformly on compact sets of D.
Proof.Here Lemma 3.1 is applied with T = C ϕ , X = Q K1 (p 1 , q 1 ), and 1) is true, because for every a ∈ D there exists a constant C > 0 such that the mapping λ a :

by (2.3) and (2.4).
To show (2) in Lemma 3.1 holds, let {f n } be a sequence in the unit ball of Q K1 (p 1 , q 1 ).Then {f n } is uniformly bounded on compact sets of D by (2.4).By Montel's theorem there exist a subsequence {f n k } and an analytic function f 0 such that f n k → f 0 uniformly on compact subsets of D. Fatou's lemma implies Finally, it is enough to use the seminorm in (3.1), since f n (z) tends uniformly to zero on compact sets.Lemma 3.1 completes the proof.
see [17].Lemma 3.2 is not valid for operator from Q K1,0 (p 1 , q 1 ) into Q K2 (p 2 , q 2 ), because the limit function f 0 in (3.2) doesn't necessarily belong to Q K1,0 (p 1 , q 1 ) when every f n does.An easy example of this is the sequence which tends to f 0 (z) = log(1 − z) uniformly on compact subsets of D. In fact, f n ∈ B 0 for every n ∈ N, but f 0 / ∈ B 0 .Luckily, proving that the condition in Lemma 3.1 is necessary for T to be compact needs only the assumptions (1) and (3) to hold (see [12] or [11]).Hence, the following lemma is proved as a by-product of Lemma 3.2.
for every bounded sequence {f n } ⊂ Q K1,0 (p 1 , q 1 ) such that f n → 0 uniformly on compact sets.
Note that Lemmas 3.2 and 3.3 also hold if one of the spaces Q K (p, q) or Q K,0 (p, q) is replaced by B α or B α 0 , respectively.This is obvious, since the norms of Q K (p, q) and B (q+2)p −1 are equivalent when the spaces coincide.

Compact operators C
The next lemma and the next theorem are based on [6], where the authors proved the similar result for F (p, q, s) by using norm of Carleson measure type.Once again, D δ = {z ∈ D : |ϕ(z)| > δ} and 1 D δ denotes the characteristic function of D δ .Lemma 3.4.Let ϕ be an analytic self-map of D, α > 0 , p ≥ 1 , q > −2 and let K be nonnegative and nondecreasing in [0, ∞).
It is easy to see that {h n } is bounded in B α 0 .Furthermore, {h n (z)} tends uniformly to zero on compact subsets of D. By Lemma 3.3, as n → ∞.In other words, for every ε > 0 there exists N ∈ N such that n ≥ N implies for every a ∈ D. Then Without a loss of generality, one is free to assume N ≥ 2. Then (3.3) is true by choosing δ such that as n → ∞.In other words, for every ε > 0, there exists N ∈ N such that (3.5) By (3.3) there exists δ ∈ (0, 1) such that Hence, (3.5) and (3.6) imply and the proof is completed.
Theorem 3.5.Let ϕ be an analytic self-map of D, α > 0 , p ≥ 1 , q > −2 and let K be nonnegative and nondecreasing in [0, ∞).Then the following conditions are equivalent: Proof.Since (a) implies (b), assume first that (b) holds.To show (c) is true, let {t n } ⊂ (0, 1) be a sequence converging to 1 .Set f n (z) = f (t n z) for an arbitrary function f in B, the unit ball of B α .Then the sequence {t n } induces a function family For every ε > 0 , there exist finitely many functions g 1 , . . ., g l ∈ B α 0 such that for every g ∈ F there exists an index k ∈ By Lemma 3.4 there exists δ k = δ(k, ε) for every g k such that sup for every g ∈ F .Choose h 1 , h 2 ∈ B (cf. [20]) such that for some constant C > 0 .Setting (h j ) n = h j (t n z) one obtains (h j ) n ∈ F for j = 1, 2 .Applying the inequality (3.8) with these functions it is easy to find a constant C > 0 (depending only on p and C ) such that sup Hence, condition (3.7) follows by Fatou's lemma and ϕ ∈ Q K (p, q) since identical mapping belongs to B α 0 .Assume (c) is true and {f n } is bounded in B α such that f n → 0 uniformly on compact sets.For simplicity, let f n B α ≤ 1.It is easy to get from (c) that for every ε > 0 there exists δ ∈ (0, 1) such that (3.9) sup By (3.9) and (3.10) as n → ∞.Condition (c) implies (a) and the proof is completed.
Theorem 3.5 generalizes the Theorem 1 in [14]: Corollary 3.6.Let ϕ be an analytic self-map of D and let K be nonnegative and nondecreasing in [0, ∞).Then the following statements are equivalent: The next corollary is analogous to Theorem 1.2. in [6]: Corollary 3.7.Let ϕ be an analytic self-map of D, p ≥ 1 , q > −2 and s > 0 such that q + s > −1 .Then the following conditions are equivalent: As in Theorem 2.9, an extra assumption is needed to study compact operators C ϕ : Q K (p, q) → B α using Lemma 3.2.The condition (3.12) below is assumed to be true to ensure that the sequence of test functions is bounded in Q K (p, q).Theorem 3.8.Let ϕ : D → D be analytic, α > 0 , p ≥ 1 and q > −2 .Assume also that K : [0, ∞) → [0, ∞) is nondecreasing and Then ϕ induces a compact operator C ϕ : Q K (p, q) → B α if and only if ϕ ∈ B α and for every n, which is a contradiction.Thus, (3.13) holds when C ϕ : Q K (p, q) → B α is compact.Conversely, let ϕ ∈ B α and let (3.13) be true.Take any bounded sequence {f n } , which converges to zero on compact subsets of D. Then (3.13) implies that for every ε > 0 there exists δ ∈ (0, 1) such that The technique of the proof above is similar to Theorem 2.2.1 in [20], where the special case of Theorem 3.8 is proved, see Corollary 3.9 below.Note also that (3.12) is satisfied with K(t) = t s , s > max{0, −q − 1} .Hence, Theorem 3.8 generalizes one result presented in [9], see Corollary 3.10 below.

Corollary 2 . 11 .
Let α, s > 0 .Then an analytic mapping ϕ : D → D induces a bounded operator C ϕ : Q s → B α if and only if