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# Boundedness of Faber operators

- Yunus Emre Yıldırır
^{1}Email author and - Ramazan Çetintaş
^{1}

**2013**:257

https://doi.org/10.1186/1029-242X-2013-257

© Yıldırır and Çetintaş; licensee Springer. 2013

**Received:**14 December 2012**Accepted:**4 May 2013**Published:**21 May 2013

## Abstract

In this work, we prove the boundedness of the Faber operators that transform the Hardy-Orlicz class ${H}_{M}(\mathbf{D})$ into the Smirnov-Orlicz class ${E}_{M}(G)$.

**MSC:**41A10, 42A10.

## Keywords

- Faber operator
- Hardy-Orlicz class
- 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 $G\cup \mathrm{\Gamma}$ is a simply connected domain ${G}^{-}$, *i.e.*, $G:=int\mathrm{\Gamma}$ and ${G}^{-}=ext\mathrm{\Gamma}$. Without loss of generality, we may assume $0\in G$. Let $\mathbf{T}=\{w\in \mathbb{C}:|w|=1\}$, $\mathbf{D}=int\mathbf{T}$ and ${\mathbf{D}}^{-}=ext\mathbf{T}$.

Let the function $z=\mathrm{\Psi}(w)$ be the inverse function for $w=\mathrm{\Phi}(z)$. This function maps the domain ${\mathbf{D}}^{-}$ conformally and univalently onto the domain ${G}^{-}$.

*n*, we can write

where ${F}_{n}(z)$ is a polynomial of order *n* and ${E}_{n}(z)$ is the sum of the infinite number of terms with negative powers.

The polynomial ${F}_{n}(z)$ is called the *Faber polynomial* of order *n* for the domain *G*.

*h*be a continuous function on $[0,2\pi ]$. Its modulus of continuity is defined by

*h*is called

*Dini-continuous*if

*Dini-smooth*if it has a parametrization

such that ${\phi}_{0}^{\mathrm{\prime}}(\tau )$ is Dini-continuous and ≠0 [1].

*Dini-smooth*, then

for some constants ${c}_{1}$ and ${c}_{2}$ independent of *z*.

is called an *N*-*function*.

*The complementary*

*N*-

*function*to

*M*is defined by

*M*be an

*N*-function and

*N*be its complementary function. By ${L}_{M}(\mathrm{\Gamma})$ we denote

*the linear space of Lebesgue measurable functions*$f:\mathrm{\Gamma}\to \mathbb{C}$ satisfying the condition, for some $\alpha >0$,

where $\rho (g;N):={\int}_{\mathrm{\Gamma}}N[|g(z)|]|dz|$.

The norm ${\parallel \cdot \parallel}_{{L}_{M}(\mathrm{\Gamma})}$ is called *Orlicz norm* and the Banach space ${L}_{M}(\mathrm{\Gamma})$ is called *Orlicz space*. Every function in ${L}_{M}(\mathrm{\Gamma})$ is integrable on Γ (see [[2], p.50]), *i.e.*, ${L}_{M}(\mathrm{\Gamma})\subset {L}_{1}(\mathrm{\Gamma})$.

Let **D** be a unit disk and ${\mathrm{\Gamma}}_{r}$ be the image of the circle $\{w\in \mathbb{C}:|w|=r,0<r<1\}$ under some conformal mapping of **D** onto *G*, and let *M* be an *N*-function.

*G*and satisfy the condition

uniformly in *r* is called the *Smirnov-Orlicz class* and denoted by ${E}_{M}(G)$.

The Smirnov-Orlicz class is a generalization of the familiar Smirnov class ${E}_{p}(G)$. In particular, if $M(x):={x}^{p}$, $1<p<\mathrm{\infty}$, then Smirnov-Orlicz class ${E}_{M}(G)$ determined by *M* coincides with the Smirnov class ${E}_{p}(G)$.

*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 ${H}_{M}(\mathbf{D})$.

**T**and the boundary value function belongs to ${L}_{M}(\mathbf{T})$. Hence ${H}_{M}(\mathbf{D})$ norm can be defined as

The spaces ${H}_{M}(\mathbf{D})$ and ${E}_{M}(G)$ are Banach spaces respectively with the norm ${\parallel f\parallel}_{{L}_{M}(\mathbf{T})}$ and ${\parallel f\parallel}_{{L}_{M}(\mathrm{\Gamma})}$.

holds for every $f\in {L}_{M}(\mathrm{\Gamma})$ and $g\in {L}_{N}(\mathrm{\Gamma})$ [[4], p.80].

Let Γ be a Dini-smooth curve, *G* be a finite domain bounded by Γ and $\phi \in {H}_{M}(\mathbf{D})$.

is called *Faber operator* for the domain *G* from ${H}_{M}(\mathbf{D})$ into ${E}_{M}(G)$.

Let Γ be a Dini-smooth curve and *G* be a finite domain bounded by Γ. Then the boundedness of the Faber operators from ${H}_{p}(\mathbf{D})$ into ${E}_{p}(G)$ ($p\ge 1$) was proved in [[5], p.125].

In this paper, we obtain the following results about the boundedness of the Faber operator from ${H}_{M}(\mathbf{D})$ into ${E}_{M}(G)$ and about the boundedness of the inverse Faber operator from ${E}_{M}(G)$ into ${H}_{M}(\mathbf{D})$.

**Theorem 1**

*Let*

*G*

*be a finite domain bounded by a Dini*-

*smooth curve*Γ.

*Then the Faber operator*${F}_{0}:{H}_{M}(\mathbf{D})\to {E}_{M}(G)$

*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*${F}_{1}:{E}_{M}(G)\to {H}_{M}$

*has a finite norm and*

**Corollary 1**

*Let*

*G*

*be a finite domain bounded by a Dini*-

*smooth curve*Γ

*and*${P}_{n}$

*be the image of the polynomial*${\phi}_{n}$

*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*${\phi}_{n}$

*be the image of the polynomial*${P}_{n}$

*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 $({F}_{0}\phi )(z)$, the equality

□

*Proof of Theorem 2*For the inverse Faber operator $({F}_{1}f)(w)$, the equality

holds [[5], p.127]. With the help of this equality, Theorem 2 is proved by the similar method of the proof of Theorem 1. □

## Declarations

## Authors’ Affiliations

## References

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## Copyright

This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.