ON THE MAXIMUM VALUE FOR ZYGMUND CLASS ON AN INTERVAL

We prove that if f ( z ) is a continuous real-valued function on 
 ℝ with the properties f ( 0 ) = f ( 1 ) = 0 and that ‖ f ‖ z = inf x , t | f ( x + t ) − 2 f ( x ) + f ( x − t ) / t | is finite for all x , t ∈ ℝ , which is called Zygmund function on ℝ , then max x ∈ [ 0 , 1 ] | f ( x ) | ≤ ( 11 / 32 ) ‖ f ‖ z . As an 
application, we obtain a better estimate for Skedwed Zygmund 
bound in Zygmund class.


Introduction and the main results. A continuous real-valued function f (x) on
R is said to belong to the class Λ * (R) if there exists a constant C such that for all x, t ∈ R.This class introduced by Zygmund [8] is called Zygmund class, and we denote the infimum of the values C in (1.1) by f z .Gardiner and Sullivan [6] proved that by applying the Beurling-Ahlfors extension formula [1]  has bounded ∂-derivative in the upper half plane H = {(x, y) | y > 0}, where ∂ = ∂ x + i∂ y .On the other hand, from Ahlfors and Bers [2], for any L ∞ complex-valued function µ(z) defined for z in C there is a curve of quasi-conformal homeomorphisms f tµ of Ĉ defined for |t| < µ −1 ∞ such that f tµ is holomorphic as a function of t and its derivative in t is given by the following formula: where the constant in O(t 2 ) is uniform for z in compact sets.If f tµ is normalized to fix 0, 1, and ∞, then F(x) is in the function space Λ * (R).The necessary and sufficient condition for a real-valued function f (x) on R to have an extension F(z) on H with bounded ∂-derivative is f (x) ∈ Λ * (R), which is proved by Gardiner and Sullivan in [6] also by Reich and Chen in [7].It is hoped that the knowledge of the special properties of such functions may be applied to the study of quasi-conformal theory.
In order to prove that the Beurling-Ahlfors extension F BA has bounded ∂-derivative, Gardiner and Sullivan [6] applied these Zygmund function properties by solving the es-timate of max{|f (x)| : 0 ≤ x ≤ 1} when f ∈ Λ * (R) is normalized by f (0) = f (1) = 0, in fact, they proved the following theorem.
More recently, in their joint paper, Baladi et al. [3] also used the Skewed Zygmund bound property to estimate the upper and lower bound for some transfer operators.They introduced the Zygmund space Z on I, where I denotes a compact interval as the complex vector space of continuous functions ϕ : where And they proved the following useful result.
Theorem 1.3 (Skewed Zygmund bound).For all ϕ ∈ Z, x, y ∈ I, where I denotes a compact interval, 0 < t < 1, In this paper, first we will point out that the proof of the theorem has error, so that Theorem 1.2 is not proved.And then we will prove the following theorem.
As an application, we will use our result to obtain a better estimate for the above Skewed Zygmund bound in Section 3.

Preliminary results and the proof of Theorem 1.4.
We assume f ∈ Λ * (R) and f (0) = f (1) = 0, and we need the following results due to Chen and Wei [4].
First, we will point out that there is an error in the proof of Theorem 1.2.Chen and Wei set in [4] that and, by deduction, they set As Chen and Wei used in [4], they denoted by Λ n the Zygmund class on the interval Contradicting to Chen and Wei [4], we say that the method used to obtain (2.5) does not generally hold, and the formulas of (2.5) can only be true in the following For if were true, by the same method used by Chen and Wei [4], we set that (2.10) The following example is used in [4].
We choose that f (x) equals zero when x < 0 and x > 1, the dividing points in [0, 1] and the values of f * at the dividing points are listed as follows: (2.11) By the above example and (2.10), we have max , we obtain that max (2.23) The results we obtain hold for any B ≥ f z , hence we have proved that max (2.24) The proof of Theorem 1.4 is finished.
3. Application to estimate Skewed Zygmund bound.We will use our Theorem 1.4 to improve Skewed Zygmund bound due to Baladi et al. [3].Suppose I is a compact interval and the Zygmund space Z on I as the complex vector space of continuous functions ϕ : I → C such that where Z(ϕ, x, t) = (ϕ(x + t) + ϕ(x − t) − 2ϕ(x))/t.
The vector space Z becomes a Banach space when it is endowed with the norm ϕ = max(sup I |ϕ|, Z(ϕ)), and it has close relation with the Banach space Λ α of α-Hölder functions, that is, functions ϕ : I → C satisfying the norm ϕ α = max(sup I |ϕ|, |ϕ| α ).We know that Z Λ α for 0 < α < 1 and Λ 1 Z. (See [3,5].)Our next result will be stated in the following theorem.Theorem 3.1 (Skewed Zygmund bound).For all ϕ ∈ Z, x, y ∈ I, where I denotes a compact interval, 0 < t < 1, Proof of Theorem 3.1.Suppose a given function ϕ(x) satisfying the conditions in Theorem 3.1, for any x, y ∈ I we define a function in [0, 1] as follows:  (3.9) The proof of Theorem 3.1 is finished.
Remark 3.2.There are also some useful applications of Theorems 1.4 and 3.1, for example, they can be used for the estimate of the upper and lower bound for some transfer operators introduced in [4], we omit it here.