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# Boundedness of Faber operators

*Journal of Inequalities and Applications*
**volume 2013**, Article number: 257 (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.

## 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}.

By the Riemann theorem on a conformal mapping, there exists a unique function w=\mathrm{\Phi}(z) meromorphic in {G}^{-} which maps the domain {G}^{-} conformally and univalently onto the domain {\mathbf{D}}^{-} and satisfies the conditions

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}^{-}.

Condition (1) implies that the function w=\mathrm{\Phi}(z), being analytic in the domain {G}^{-} without the point z=\mathrm{\infty}, has a simple pole at the point z=\mathrm{\infty}. Therefore its Laurent expansion in some neighborhood of the point ∞ has the form

For a non-negative integer *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*.

Let *h* be a continuous function on [0,2\pi ]. 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 {\phi}_{0}^{\mathrm{\prime}}(\tau ) is Dini-continuous and ≠0 [1].

If Γ is *Dini-smooth*, then

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

A continuous and convex function M:[0,\mathrm{\infty})\to [0,\mathrm{\infty}) which satisfies the conditions M(0)=0, M(x)>0 for x>0,

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 {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,

The space {L}_{M}(\mathrm{\Gamma}) becomes a Banach space with the norm

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.

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 {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).

Since (see [3]) {E}_{M}(G)\subset {E}_{1}(G), every function in the class {E}_{M}(G) has the nontangential boundary values a.e. on Γ and the boundary value function belongs to {L}_{M}(\mathrm{\Gamma}). Hence {E}_{M}(G) norm can be defined as

Let M:[0,\mathrm{\infty})\to [0,\mathrm{\infty}) 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 {H}_{M}(\mathbf{D}).

Since {H}_{M}(\mathbf{D})\subset {H}_{1}(\mathbf{D}), every function in the class {H}_{M}(\mathbf{D}) has the nontangential boundary values a.e. on **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})}.

Hölder’s inequality

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}).

The Cauchy-type integral

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

The inverse Faber operator from {E}_{M}(G) into {H}_{M}(\mathbf{D}) 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 {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

holds [[5], p.123], where

From (2) we obtain

Using the definition of the Orlicz norm, Hölder’s inequality and (3), we get

where

Therefore we obtain that

□

*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. □

## References

Israfilov DM, Oktay B, Akgün R: Approximation in Smirnov-Orlicz classes.

*Glas. Mat.*2005, 40(60):87–102.Rao MM, Ren ZD:

*Theory of Orlicz Spaces*. Dekker, New York; 1991.Kokilashvili V: On analytic functions of Smirnov-Orlicz classes.

*Stud. Math.*1968, 31: 43–59.Krasnoselskii MA, Rutickii YB:

*Convex Functions and Orlicz Spaces*. Noordhoff, Groningen; 1961.Suetin PK:

*Series of Faber Polynomials*. Gordon & Breach, New York; 1988.

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The authors declare that they have no competing interests.

### Authors’ contributions

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.

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Yıldırır, Y.E., Çetintaş, R. Boundedness of Faber operators.
*J Inequal Appl* **2013**, 257 (2013). https://doi.org/10.1186/1029-242X-2013-257

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