On asymptotic behavior of fractional Cauchy transform

We study non-regularity of growth of the fractional Cauchy transform $$\begin{aligned} f(z)=\int _{-\pi }^{\pi } \frac{d\psi (t)}{(1-ze^{-it})^\alpha }, \quad \alpha >0, \psi \in BV[-\pi ,\pi ], \end{aligned}$$f(z)=∫-ππdψ(t)(1-ze-it)α,α>0,ψ∈BV[-π,π],in terms of the modulus of continuity of the function $$\psi $$ψ. Sharp estimates of the lower logarithmic order of f are found. In the case $$\alpha \in (0,1)$$α∈(0,1) the estimates are of different form than that for the logarithmic order.

Our main focus is to study of the following Cauchy type integral B Igor Chyzhykov chyzhykov@matman.uwm.edu.pl Galyna Beregova gberegova@yahoo.com 1 where ψ is a complex-valued function of bounded variation, i.e., ψ ∈ BV [−π, π], or a fractional Cauchy transform of the measure ψ * on ∂D associated with ψ. The family of all functions of the form (1) is denoted by F α . A study of F α can be found in the book [11], here we mention only few properties. F α is a Banach space with respect to the norm f F α = inf ψ * ψ * , where ψ * denotes the total variation of ψ * . A straightforward consequence of the definition is that for f ∈ F α one has where M(r , f ) = max{| f (re iθ )| : |z| = r }.
Many relations with Hardy, Besov, and Dirichlet-type spaces are obtained in the mentioned monograph.
Theorem A If 0 < α ≤ 1, then F α ⊂ H p for 0 < p < 1/α. If 0 < p ≤ 1 then H p ⊂ F 1/ p , where H p is the Hardy space of analytic functions in D satisfying where m is the planar Lebesgue measure.
Functions from F α appear frequently in representation theorems for different classes of functions defined on the unit disc D. In particular, Djrbashian introduced [5] classes A α , α > −1, of analytic functions in D. For α > 0, an analytic function f belongs to the class A α , α > 0, if and only if Note that the union of α>0 A α contains the class of analytic functions f in D of finite order of growth, that is log log M(r , f ) = O log 1 1−r (r → 1−). Due to results of Djrbashian [5,Chap. IX] functions f ∈ F α+1 appear in a parametric representation of the class A α , α > −1.
Radial and non-tangential limits of f ∈ F α were investigated in many papers, e.g., Hallenbeck and MacGregor [8,9], and Sheremeta [13], see also [11,Chap. 10;12]. It turns out that the estimate (2) can be improved in terms of the modulus of continuity for ψ. Let be the modulus of continuity of ψ.
The next result in this spirit is a consequence of Theorem 3.4 [4] (see also Theorem 3.1 [1]).
Theorem C Let f be an analytic function in D of the form (1). Let α > γ and 0 < γ ≤ 1. Then ψ ∈ Λ γ , if and only if

Remark 1
More general results using the concept of a proximate order are obtained in [4]. There are counterparts for harmonic functions represented by Poisson-type integrals as well, see [1,3].

Remark 2
Growth of pth means of functions f ∈ F α is described in [2].
A common feature of the above mentioned results is that the growth of f depends on which Hölder class Λ γ the modulus of continuity of ψ belongs to. Let denote the logarithmic order and the lower logarithmic order, respectively. It follows from Theorem C that for an arbitrary f ∈ F α and appropriate α, γ we have p[ f ] ≤ α − γ if and only if ψ ∈ Λ γ . The following question appears naturally. Does a smoothness of the modulus of continuity ω(δ, ψ) on a sequence (δ n ) of values δ tending to zero imply an upper estimate for the lower logarithmic order similar to that for the logarithmic order?
We shall see that the answer is in the positive provided that α > 1, but in general, the situation is more complicated.
, and that f is of the form (1).

Remark 4
The quantity Δ in (4) is continuous at 1 as a function of α. Moreover,

Proof of Theorem 3
Let ϕ be arbitrary on [−π, π]. We extend ψ on R by the formula . Since the kernel of the integral (1) is a 2πperiodic function in t, integrating by parts gives us Therefore, uniformly in ϕ, ω(δ) := ω(δ; ψ) We start with the simple cases (i) and (ii). Suppose that 0 < α ≤ γ . Then ω(δ) = O(δ γ ), and standard estimates yield where C is a positive constant. Hence, the statements (i) and (ii) follow.
Let now α > γ ≥ 0. Consider two cases. First, we suppose that α < 1. We then chooseδ In what follows the inequality g 1 (n) g 2 (n) means that there exists a constant C independent of n such that g 1 (n) ≤ Cg 2 (n) (n → ∞).
We put r n = 1 −δ n . Since a modulus of continuity is nondecreasing, we have Then Taking into account (5), and settingδ n = δ Finally, taking into account thatδ n δ n , from the above estimates we deduce This finishes the proof in the first case. Let now α > 1. We put r n := 1 − δ n in this case. Arguing similarly as in the first case, we obtain Similarly, as in the first case, Thus, using the definition ofδ n and the latter estimates we deduce 1−γ < α − λ is equivalent to α > 1 provided that α > γ and γ < 1.
In the case α = 1, all the estimates for the case α > 1 except (14) remain true. Instead of (14) we have The theorem is proved.

Example of a function of non-regular growth
The following theorem shows the sharpness of the corollary.

Remark 7
In what follows all asymptotic relations as n → ∞ that depend on parameters r , t, etc. hold uniformly in these parameters at their specified ranges. For simplicity, this fact will no longer be emphasized.
Then, using (21) for n ≤ k, and (22), for n ≥ k + 1, we obtain for some k 0 ∈ N (see Remark 7) Further, let 1 − r ∈ [A k t k , t k /A k ]. In this case, using (21) for n < k, (22) for n ≥ k + 1, and (26) for n = k, definitions of (t n ) and (t n ), we deduce (k 1 ∈ N) We show that the second addend in the last correlation dominates, i.e.