Boundedness of Faber operators
Journal of Inequalities and Applications volume 2013, Article number: 257 (2013)
In this work, we prove the boundedness of the Faber operators that transform the Hardy-Orlicz class into the Smirnov-Orlicz class .
1 Introduction and main results
Let a bounded simply connected domain G with the boundary Γ in the complex plane be given, such that the complement of the closed domain is a simply connected domain , i.e., and . Without loss of generality, we may assume . Let , and .
By the Riemann theorem on a conformal mapping, there exists a unique function meromorphic in which maps the domain conformally and univalently onto the domain and satisfies the conditions
Let the function be the inverse function for . This function maps the domain conformally and univalently onto the domain .
Condition (1) implies that the function , being analytic in the domain without the point , has a simple pole at the point . Therefore its Laurent expansion in some neighborhood of the point ∞ has the form
For a non-negative integer n, we can write
where is a polynomial of order n and is the sum of the infinite number of terms with negative powers.
The polynomial is called the Faber polynomial of order n for the domain G.
Let h be a continuous function on . Its modulus of continuity is defined by
The function h is called Dini-continuous if
The curve Γ is called Dini-smooth if it has a parametrization
such that is Dini-continuous and ≠0 .
If Γ is Dini-smooth, then
for some constants and independent of z.
A continuous and convex function which satisfies the conditions , for ,
is called an N-function.
The complementary N-function to M is defined by
Let M be an N-function and N be its complementary function. By we denote the linear space of Lebesgue measurable functions satisfying the condition, for some ,
The space becomes a Banach space with the norm
The norm is called Orlicz norm and the Banach space is called Orlicz space. Every function in is integrable on Γ (see [, p.50]), i.e., .
Let D be a unit disk and be the image of the circle under some conformal mapping of D onto G, and let M be an N-function.
The class of functions which are analytic in G and satisfy the condition
uniformly in r is called the Smirnov-Orlicz class and denoted by .
The Smirnov-Orlicz class is a generalization of the familiar Smirnov class . In particular, if , , then Smirnov-Orlicz class determined by M coincides with the Smirnov class .
Since (see ) , every function in the class has the nontangential boundary values a.e. on Γ and the boundary value function belongs to . Hence norm can be defined as
Let be an N-function. The class of functions which are analytic in D and satisfy the condition
uniformly in r is called the Hardy-Orlicz class and denoted by .
Since , every function in the class has the nontangential boundary values a.e. on T and the boundary value function belongs to . Hence norm can be defined as
The spaces and are Banach spaces respectively with the norm and .
holds for every and [, p.80].
Let Γ be a Dini-smooth curve, G be a finite domain bounded by Γ and .
The Cauchy-type integral
is called Faber operator for the domain G from into .
The inverse Faber operator from into is defined as
Let Γ be a Dini-smooth curve and G be a finite domain bounded by Γ. Then the boundedness of the Faber operators from into () was proved in [, p.125].
In this paper, we obtain the following results about the boundedness of the Faber operator from into and about the boundedness of the inverse Faber operator from into .
Theorem 1 Let G be a finite domain bounded by a Dini-smooth curve Γ. Then the Faber operator has a finite norm and
Theorem 2 Let G be a finite domain bounded by a Dini-smooth curve Γ. Then the inverse Faber operator has a finite norm and
Corollary 1 Let G be a finite domain bounded by a Dini-smooth curve Γ and be the image of the polynomial defined in the unit disk under the Faber operator. Then
Corollary 2 Let G be a finite domain bounded by a Dini-smooth curve Γ and be the image of the polynomial defined in G under the inverse Faber operator. Then
With the help of these two corollaries, one can carry over the direct and inverse theorems on the order of the best approximations in mean, from the unit disk to the case of a domain with a sufficiently smooth boundary.
2 Proof of the main results
Proof of Theorem 1 For the Faber operator , the equality
holds [, p.123], where
From (2) we obtain
Using the definition of the Orlicz norm, Hölder’s inequality and (3), we get
Therefore we obtain that
Proof of Theorem 2 For the inverse Faber operator , the equality
holds [, p.127]. With the help of this equality, Theorem 2 is proved by the similar method of the proof of Theorem 1. □
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Krasnoselskii MA, Rutickii YB: Convex Functions and Orlicz Spaces. Noordhoff, Groningen; 1961.
Suetin PK: Series of Faber Polynomials. Gordon & Breach, New York; 1988.
The authors declare that they have no competing interests.
The author YEY determined the problem after making the literature research and organized the proofs of the theorems. The author RÇ helped to the proofs of the theorems and wrote the manuscript in the latex.