Lacunary series in QK type spaces

Under mild conditions on the weight function K we characterize lacunary series in QK(p, q) spaces, where QK(p, q) spaces are QK type spaces of functions analytic in the unit disk.

We assume throughout the paper that (1.1) Otherwise, Q K (p, q) only contains constant functions (cf.[7]).It is easy to see that The space F (p, q, s) was introduced and investigated by R. Zhao in [10].
In order to state the main theorem in the paper, we need an auxiliary function ϕ K defined by The following condition has played a crucial role in the study of Q K type spaces during the last few years: The characterization of the lacunary series in Q K spaces was given in [8].In this paper we obtain a characterization of the lacunary series in general Q K type spaces, that is the following result: has Hadamard gaps, that is, n k+1 /n k ≥ λ > 1 for all k ∈ N, then the following statements are equivalent:

Preliminaries on weight function
In what follows we say f g (for two functions f and g) if there is a constant C such that f ≤ Cg .We say f ≈ g (that is, f is comparable with g ) whenever g f g .
Before proving the Theorem 1.1, we need some lemmas.
Lemma 2.1.If K satisfies condition (1.2), then K has the following properties: The above results can be found in Proposition 4 of [8]. Proof.Let By a change of variables we have Denote and By Lemma 2.1, we have (2.1) Since K(t) is non-decreasing on (0, ∞), we have By (2.1) and (2.2), we have where C(β) is a constant which only depends on β .
On the other hand, recall that K(t) is non-decreasing on (0, ∞).Then We get the desired result and complete the proof.
The above result can be found in [5].
By Jensen's formula, we can directly obtain the above result. where The last inequality holds because of log(1/r) ≥ 1 − r .

Lacunary series in
Since f ∈ Q K (p, q), by Theorem 2.5 we have where The Taylor series of f has at most [log λ 2] + 1 terms a j z nj when n j ∈ I k for k ≥ 1.By Hölder's inequality, we note that Next suppose that condition (3.1) holds.Since K satisfies condition (1.2), K is concave by Lemma 2.1.By Jensen's inequality and Lemma 2.4 we obtain that Given a ∈ Δ, by Theorem 2.5 we have where Proof.In fact, sufficiency is obvious because of Q K,0 (p, q) ⊂ Q K (p, q).Now we will prove necessity of Theorem 3.2.Suppose that the lacunary series f belongs to Q K (p, q).We must show that I(a) → 0 as |a| → 1, where From the proof of Theorem 3.1, we know that f ∈ Q K (p, q) implies that For any > 0, there is a δ ∈ (0, 1) such that We may as well assume that K(0 + 0) = 0 if K satisfies condition (1.2).Then we choose a such that 1 > |a| > δ .By Lemma 2.4 and Theorem 3.1 we have Since is arbitrary, we conclude that I(a) → 0 as |a| → 1.So f ∈ Q K,0 (p, q) and the proof is complete.
We have obtained Theorem 1.1 from Theorem 3.1 and 3.2.Two special cases are worth mentioning.If p = 2, q = 0 , we get a characterization of lacunary series in Q K spaces which was given in [8].