Boundedness of Faber operators
© Yıldırır and Çetintaş; licensee Springer. 2013
Received: 14 December 2012
Accepted: 4 May 2013
Published: 21 May 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 .
Let the function be the inverse function for . This function maps the domain conformally and univalently onto the domain .
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.
such that is Dini-continuous and ≠0 .
for some constants and independent of z.
is called an N-function.
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.
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 .
uniformly in r is called the Hardy-Orlicz class and denoted by .
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 .
is called Faber operator for the domain G from into .
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 .
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
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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