Yıldırır and Çetinta¸s Journal of Inequalities and Applications 2013, 2013:257 http://www.journalofinequalitiesandapplications.com/content/2013/1/257
RESEARCH
Open Access
Boundedness of Faber operators Yunus Emre Yıldırır* and Ramazan Çetinta¸s *
Correspondence:
[email protected] 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 class HM (D) into the Smirnov-Orlicz class EM (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 in the complex plane be given, such that the complement of the closed domain G ∪ 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 = int T and D– = ext T. By the Riemann theorem on a conformal mapping, there exists a unique function w = (z) meromorphic in G– which maps the domain G– conformally and univalently onto the domain D– and satisfies the conditions (∞) = ∞
and
lim
z→∞
(z) = γ > . z
()
Let the function z = (w) be the inverse function for w = (z). This function maps the domain D– conformally and univalently onto the domain G– . Condition () implies that the function w = (z), being analytic in the domain G– without the point z = ∞, has a simple pole at the point z = ∞. Therefore its Laurent expansion in some neighborhood of the point ∞ has the form (z) = γ z + γ +
γk γ + ··· + k + ··· . z z
For a non-negative integer n, we can write n (z) = Fn (z) + En (z),
z ∈ G– ,
where Fn (z) is a polynomial of order n and En (z) is the sum of the infinite number of terms with negative powers. The polynomial Fn (z) is called the Faber polynomial of order n for the domain G. Let h be a continuous function on [, π]. Its modulus of continuity is defined by w(t, h) := sup h(t ) – h(t ) : t , t ∈ [, π], |t – t | ≤ t ,
t ≥ .
© 2013 Yıldırır and Çetinta¸s; licensee Springer. 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.
Yıldırır and Çetinta¸s Journal of Inequalities and Applications 2013, 2013:257 http://www.journalofinequalitiesandapplications.com/content/2013/1/257
The function h is called Dini-continuous if
π
t – w(t, h) dt < ∞.
The curve is called Dini-smooth if it has a parametrization : ϕ (τ ),
≤ τ ≤ π
such that ϕ (τ ) is Dini-continuous and = []. If is Dini-smooth, then < c ≤ (z) ≤ c < ∞,
z∈
for some constants c and c independent of z. A continuous and convex function M : [, ∞) → [, ∞) which satisfies the conditions M() = , M(x) > for x > , lim
x→
M(x) = , x
lim
x→∞
M(x) =∞ x
is called an N -function. The complementary N -function to M is defined by N(y) := max xy – M(x) , x≥
y ≥ .
Let M be an N -function and N be its complementary function. By LM () we denote the linear space of Lebesgue measurable functions f : → C satisfying the condition, for some α > ,
M α f (z) |dz| < ∞.
The space LM () becomes a Banach space with the norm
f (z)g(z)|dz| : g ∈ LN (), ρ(g; N) ≤ ,
f LM () := sup
where ρ(g; N) := N[|g(z)|]|dz|. The norm · LM () is called Orlicz norm and the Banach space LM () is called Orlicz space. Every function in LM () is integrable on (see [, p.]), i.e., LM () ⊂ L (). Let D be a unit disk and r be the image of the circle {w ∈ C : |w| = r, < r < } 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 r
M f (z) |dz| < ∞
uniformly in r is called the Smirnov-Orlicz class and denoted by EM (G).
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Yıldırır and Çetinta¸s Journal of Inequalities and Applications 2013, 2013:257 http://www.journalofinequalitiesandapplications.com/content/2013/1/257
The Smirnov-Orlicz class is a generalization of the familiar Smirnov class Ep (G). In particular, if M(x) := xp , < p < ∞, then Smirnov-Orlicz class EM (G) determined by M coincides with the Smirnov class Ep (G). Since (see []) EM (G) ⊂ E (G), every function in the class EM (G) has the nontangential boundary values a.e. on and the boundary value function belongs to LM (). Hence EM (G) norm can be defined as
f EM (G) := f LM () ,
f ∈ EM (G).
Let M : [, ∞) → [, ∞) be an N -function. The class of functions which are analytic in D and satisfy the condition
π
M f reit dt < ∞
uniformly in r is called the Hardy-Orlicz class and denoted by HM (D). Since HM (D) ⊂ H (D), every function in the class HM (D) has the nontangential boundary values a.e. on T and the boundary value function belongs to LM (T). Hence HM (D) norm can be defined as
f HM (D) := f LM (T) ,
f ∈ HM (D).
The spaces HM (D) and EM (G) are Banach spaces respectively with the norm f LM (T) and
f LM () . Hölder’s inequality
f (z)g(z)|dz| ≤ f L () g L () M N
holds for every f ∈ LM () and g ∈ LN () [, p.]. Let be a Dini-smooth curve, G be a finite domain bounded by and ϕ ∈ HM (D). The Cauchy-type integral (F ϕ)(z) = πi
ϕ((ζ )) dζ , ζ –z
z ∈ G,
is called Faber operator for the domain G from HM (D) into EM (G). The inverse Faber operator from EM (G) into HM (D) is defined as (F f )(w) =
πi
f [(t)] dt, |t|= t – w
|w| < .
Let be a Dini-smooth curve and G be a finite domain bounded by . Then the boundedness of the Faber operators from Hp (D) into Ep (G) (p ≥ ) was proved in [, p.]. In this paper, we obtain the following results about the boundedness of the Faber operator from HM (D) into EM (G) and about the boundedness of the inverse Faber operator from EM (G) into HM (D).
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Yıldırır and Çetinta¸s Journal of Inequalities and Applications 2013, 2013:257 http://www.journalofinequalitiesandapplications.com/content/2013/1/257
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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
ϕ HM (D) .
Theorem Let G be a finite domain bounded by a Dini-smooth curve . Then the inverse Faber operator F : EM (G) → HM has a finite norm and
(F f ) H
M (D)
≤ F
f EM (G) .
Corollary Let G be a finite domain bounded by a Dini-smooth curve and Pn be the image of the polynomial ϕn defined in the unit disk under the Faber operator. Then
(F ϕ) – Pn E
M (G)
≤ F
ϕ – ϕn HM (D) .
Corollary Let G be a finite domain bounded by a Dini-smooth curve and ϕn be the image of the polynomial Pn defined in G under the inverse Faber operator. Then
(F f ) – ϕn H
M (D)
≤ F
f – Pn EM (G) .
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 For the Faber operator (F ϕ)(z), the equality (F ϕ)(z) = ϕ (z) + πi
|t|=
ϕ(t)F t, (z) dt
()
holds [, p.], where (t) – , (t) – (w) t – w
F(t, w) =
|t| ≥ , |w| ≥ .
From () we obtain
(F ϕ)(z)g(z)|dz|
≤ ϕ (z) g(z)|dz| + πi
|t|=
ϕ(t)g(z)F t, (z) |dt| |dz|.
Using the definition of the Orlicz norm, Hölder’s inequality and (), we get
(F ϕ) E
M (G)
≤ m () + m () ϕ HM (D) ,
where m () = sup (w), |w|=
m () =
πi
g(z) F H (D) |dz|. N
()
Yıldırır and Çetinta¸s Journal of Inequalities and Applications 2013, 2013:257 http://www.journalofinequalitiesandapplications.com/content/2013/1/257
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Therefore we obtain that
(F ϕ) ≤ m () + m () and
(F ϕ) ≤ F
ϕ HM (D) . E (G) M
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 Applications 2013 2013:257.
