Boundedness of Faber operators

(1)

R E S E A R C H

Open Access

Boundedness of Faber operators

Yunus Emre Yıldırır

*

_{and Ramazan Çetinta¸s}

*_{Correspondence:}

yildirir@balikesir.edu.tr Department of Mathematics, Necatibey Faculty of Education, Balikesir University, Balikesir, 10100, Turkey

Abstract

In this work, we prove the boundedness of the Faber operators that transform the Hardy-Orlicz classHM(D) into the Smirnov-Orlicz classEM(G).

MSC: 41A10; 42A10

Keywords: Faber operator; Hardy-Orlicz class; Smirnov-Orlicz class

1 Introduction and main results

Let a bounded simply connected domainGwith the boundaryin the complex plane be given, such that the complement of the closed domainG∪is a simply connected domain

G–_,_i.e._,_G_:=_int _and_G–₌_ext_{. Without loss of generality, we may assume }_∈_G_{. Let}

T={w∈C:|w|= },D=intTandD–₌_ext_T_.

By the Riemann theorem on a conformal mapping, there exists a unique functionw=

(z) meromorphic inG–which maps the domainG–conformally and univalently onto the domainD–_{and satisﬁes the conditions}

(∞) =∞ and lim z→∞

(z)

z =γ > . ()

Let the functionz=(w) be the inverse function forw=(z). This function maps the domainD–conformally and univalently onto the domainG–_.

Condition () implies that the functionw=(z), being analytic in the domainG–

with-out the pointz=∞, has a simple pole at the pointz=∞. Therefore its Laurent expansion in some neighborhood of the point∞has the form

(z) =γz+γ+

γ

z +· · ·+ γk

zk +· · ·. For a non-negative integern, we can write

n(z) =Fn(z) +En(z), z∈G–,

whereFn(z) is a polynomial of ordernandEn(z) is the sum of the inﬁnite number of terms with negative powers.

The polynomialFn(z) is called theFaber polynomialof ordernfor the domainG. Lethbe a continuous function on [, π]. Its modulus of continuity is deﬁned by

w(t,h) :=suph(t) –h(t):t,t∈[, π],|t–t| ≤t

, t≥.

(2)

The functionhis calledDini-continuousif π



t–w(t,h)dt<∞.

The curveis calledDini-smoothif it has a parametrization

:ϕ(τ), ≤τ≤π

such thatϕ_(τ) is Dini-continuous and=  []. IfisDini-smooth, then

 <c≤(z)≤c<∞, z∈

for some constantscandcindependent ofz.

A continuous and convex functionM: [,∞)→[,∞) which satisﬁes the conditions

M() = ,M(x) >  forx> ,

lim x→

M(x)

x = , x→∞lim

M(x)

x =∞

is called anN-function.

The complementary N-functiontoMis deﬁned by

N(y) :=max x≥

xy–M(x), y≥.

LetMbe anN-function andNbe its complementary function. ByLM() we denotethe linear space of Lebesgue measurable functions f :→Csatisfying the condition, for some

α> ,

Mαf(z) |dz|<∞.

The spaceLM() becomes a Banach space with the norm

fLM():=sup

f(z)g(z)|dz|:g∈LN(),ρ(g;N)≤

,

whereρ(g;N) :=N[|g(z)|]|dz|.

The norm · LM()is calledOrlicz normand the Banach spaceLM() is calledOrlicz space. Every function inLM() is integrable on(see [, p.]),i.e.,LM()⊂L().

LetDbe a unit disk andrbe the image of the circle{w∈C:|w|=r,  <r< }under some conformal mapping ofDontoG, and letMbe anN-function.

The class of functions which are analytic inGand satisfy the condition

r

Mf(z) |dz|<∞

(3)

The Smirnov-Orlicz class is a generalization of the familiar Smirnov classEp(G). In par-ticular, ifM(x) :=xp_{,  <}_p_<_∞_{, then Smirnov-Orlicz class}_E

M(G) determined byM coin-cides with the Smirnov classEp(G).

Since (see [])EM(G)⊂E(G), every function in the classEM(G) has the nontangen-tial boundary values a.e. onand the boundary value function belongs toLM(). Hence

EM(G) norm can be deﬁned as

fEM(G):=fLM(), f ∈EM(G).

LetM: [,∞)→[,∞) be anN-function. The class of functions which are analytic in Dand satisfy the condition

π



Mfreit _dt_<_∞

uniformly inris called theHardy-Orlicz classand denoted byHM(D).

SinceHM(D)⊂H(D), every function in the classHM(D) has the nontangential bound-ary values a.e. onTand the boundary value function belongs toLM(T). Hence HM(D) norm can be deﬁned as

fHM(D):=fLM(T), f ∈HM(D).

The spacesHM(D) andEM(G) are Banach spaces respectively with the normfLM(T)and

fLM().

Hölder’s inequality

f(z)g(z)|dz| ≤ fLM()gLN()

holds for everyf ∈LM() andg∈LN() [, p.].

Letbe a Dini-smooth curve,Gbe a ﬁnite domain bounded byandϕ∈HM(D). The Cauchy-type integral

(Fϕ)(z) =

 πi

ϕ((ζ))

ζ–z dζ, z∈G,

is calledFaber operatorfor the domainGfromHM(D) intoEM(G). The inverse Faber operator fromEM(G) intoHM(D) is deﬁned as

(Ff)(w) =

 πi

|t|=

f[(t)]

t–w dt, |w|< .

Letbe a Dini-smooth curve andGbe a ﬁnite domain bounded by. Then the bound-edness of the Faber operators fromHp(D) intoEp(G) (p≥) was proved in [, p.].

(4)

Theorem  Let G be a ﬁnite domain bounded by a Dini-smooth curve.Then the Faber operator F:HM(D)→EM(G)has a ﬁnite norm and

(Fϕ)_E_M₍_G₎≤ FϕHM(D).

Theorem  Let G be a ﬁnite domain bounded by a Dini-smooth curve.Then the inverse Faber operator F:EM(G)→HMhas a ﬁnite norm and

(Ff)_H

M(D)≤ FfEM(G).

Corollary  Let G be a ﬁnite domain bounded by a Dini-smooth curve and Pnbe the

image of the polynomialϕndeﬁned in the unit disk under the Faber operator.Then

(Fϕ) –Pn_E

M(G)≤ Fϕ–ϕnHM(D).

Corollary  Let G be a ﬁnite domain bounded by a Dini-smooth curveandϕnbe the

image of the polynomial Pndeﬁned in G under the inverse Faber operator.Then

(Ff) –ϕn_H

M(D)≤ Ff–PnEM(G).

With the help of these two corollaries, one can carry over the direct and inverse theo-rems 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 For the Faber operator (Fϕ)(z), the equality

(Fϕ)(z) =ϕ

(z)+  πi

|t|=

ϕ(t)Ft,(z)dt ()

holds [, p.], where

F(t,w) = ₍_t₎

(t) –(w)– 

t–w, |t| ≥,|w| ≥.

From () we obtain

(Fϕ)(z)g(z)|dz|

≤

ϕ(z)g(z)|dz|+  πi

|t|=

ϕ(t)g(z)Ft,(z)|dt|

|dz|. ()

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

(Fϕ)_E

M(G)≤

m() +m()

ϕHM(D),

where

m() =sup

|w|=

(w), m() =

 πi

(5)

Therefore we obtain that

(Fϕ)≤

m() +m()

and

(Fϕ)_E_M₍_G₎≤ FϕHM(D).

Proof of Theorem For the inverse Faber operator (Ff)(w), the equality

(Ff)(w) =f

(z)–  πi

|t|=

f(t)F(t,w)dt

holds [, p.]. With the help of this equality, Theorem  is proved by the similar method

of the proof of Theorem .

Competing interests

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.

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

References

1. Israﬁlov, DM, Oktay, B, Akgün, R: Approximation in Smirnov-Orlicz classes. Glas. Mat.40(60), 87-102 (2005) 2. Rao, MM, Ren, ZD: Theory of Orlicz Spaces. Dekker, New York (1991)

3. Kokilashvili, V: On analytic functions of Smirnov-Orlicz classes. Stud. Math.31, 43-59 (1968) 4. Krasnoselskii, MA, Rutickii, YB: Convex Functions and Orlicz Spaces. Noordhoﬀ, Groningen (1961) 5. Suetin, PK: Series of Faber Polynomials. Gordon & Breach, New York (1988)

doi:10.1186/1029-242X-2013-257

Cite this article as:Yıldırır and Çetinta¸s:Boundedness of Faber operators.Journal of Inequalities and Applications2013