The Maximal Operator in Weighted Variable Spaces L p ( · )

We study the boundedness of the maximal operator in the weighted spaces Lp(·)(ρ) over a bounded open set Ω in the Euclidean space Rn or a Carleson curve Γ in a complex plane. The weight function may belong to a certain version of a general Muckenhoupt-type condition, which is narrower than the expected Muckenhoupt condition for variable exponent, but coincides with the usual Muckenhoupt class Ap in the case of constant p. In the case of Carleson curves there is also considered another class of weights of radial type of the form ρ(t) = ∏m k=1 wk(|t− tk|), tk ∈ Γ, wherewk has the property that r 1 p(tk) wk(r) ∈ Φ1, whereΦ1 is a certain Zygmund-Bari-Stechkin-type class. It is assumed that the exponent p(t) satisfies the Dini–Lipschitz condition. For such radial type weights the final statement on the boundedness is given in terms of the index numbers of the functions wk (similar in a sense to the Boyd indices for the Young functions defining Orlich spaces).

1 , where Φ 0 1 is a certain Zygmund-Bari-Stechkin-type class.It is assumed that the exponent p(t) satisfies the Dini-Lipschitz condition.For such radial type weights the final statement on the boundedness is given in terms of the index numbers of the functions w k (similar in a sense to the Boyd indices for the Young functions defining Orlich spaces).

Introduction
Within the frameworks of variable exponent spaces L p(•) (Ω), the boundedness of maximal operators was proved in L. Diening [7] for bounded domains in R n and in D.Cruz-Uribe, A.Fiorenza and C.J. Neugebauer [6] and A.Nekvinda [22], [21] for unbounded domains.The weighted boundedness with power weights was proved in V.Kokilashvili and S.Samko [11] in the case of bounded domains.We refer also to L.Diening [8] and D.Cruz-Uribe, A.Fiorenza, J.M.Martell, and C.Perez [5] for problems of boundedness of maximal operators in variable exponent spaces.
In [11] the power weights |x − x 0 | γ were considered and one of the main points in the result obtained in [11] was that in condition on γ only the values of p(x) at the point x 0 are of importance: − n p(x 0 ) < γ < n q(x 0 ) (under the usual log-condition on p(x)).However, an explicit description in terms of Muckenhoupt-type condition of general weights for which the maximal operator is bounded in the spaces L p (•) still remains an open problem.
A certain subclass of general weights was considered in [10], where for the case of bounded domains Ω in the Euclidean space, the boundedness of the maximal operator in the spaces L p(•) (Ω, ρ) was proved.This subclass may be characterized as a class of radial type weights which satisfy the Zygmund-Bari-Stechkin condition.Radial weights w in this class are almost increasing or almost decreasing and may oscillate between two power functions with different exponents and have non-coinciding upper and lower indices m w and M w (of the type of Boyd indices).In comparison with the approach in [11], the main problems arising are related to the situation when the indices m w and M w do not coincide, in particular when m w is negative while M w is positive.
In this paper, because of applications to weighted boundedness of singular integral operator along Carleson curves, we prove similar results for the maximal operator along Carleson curves.This extension from the Euclidean space to the case of Carleson curves required an essential modification of certain means used in [10].To obtain this result, we first prove a certain general theorem with a certain version of the Muckenhoupt-type condition.
The weighted results obtained for the maximal operator pave the way to the study of Fredholmness of singular integral equations on Carleson curves in case of more general weights.In fact the main Theorems A, A and B may be rewritten in terms on function spaces defined on metric spaces.However, because of application to the theory of singular integral equations, we prefer to present the results in the context of Carleson curves.
The paper is organized as follows.In Section 2 we formulate the main results -Theorems A,A and B -on the weighted boundedness of the maximal operator.In Section 4 we recall the notion of the upper and lower indices of almost increasing non-negative functions and develop some properties of weights in the Zygmund-Bari-Stechkin class, which we need to prove the main result.In Sections 5 and 6 we give the proof of Theorems A, A and B.

Notation
where f is a non-negative function on R 1 + ; Γ is an arbitrary bounded Carleson curve on the complex plane, closed or open; γ denotes an arbitrary portion of Γ; γ r (t) = {τ ∈ Γ : |τ − t| < r}; dν(t) = ds denotes the arc measure on Γ; In what follows, X will always denote either a bounded open set Ω in R n , or a bounded Carleson curve Γ.The variable exponent p(•) defined on X is supposed to satisfy the conditions By L p(•) (X, ρ), where ρ(t) ≥ 0, we denote the weighted Banach space of measurable functions f : X → C such that where dµ(t) stands for the arc-length measure dν(t) in case X = Γ and dµ(t) = dt in case X = Ω.

Statement of the Main Results
We use the notation M ρ both for and The boundedness of the operator M ρ was proved in the case of the power weight ρ(x) = |x − x 0 | β , x 0 ∈ Ω in [11] and ρ(t) = |t − t 0 | β , t 0 ∈ Γ in [12] under the following (necessary and sufficient) condition − n p(x 0 ) < β < n q(x 0 ) or respectively.We prove two main results given in Theorems A and B. In Theorem A stated below we consider some general Muckenhoupt type weights, the proof being the same both for Carleson curves and domains in R n .In Theorem B, in the case of Carleson curves we deal with a special class of radial type weights in the Zygmund-Bari-Stechkin class.Such a result for the Euclidean case was earlier obtained in [10].The proof for the case of Carleson curves required an essential modification of the technique used.
The class of weights in Theorem A is narrower than the naturally expected Muckenhoupt class A p(•) should be.However, it coincides with the Muckenhoupt class A p in case p is constant.Theorem B is proved by means of Theorem A, but it is not contained in Theorem A, being more general in its range of applicability.
We introduce the following "ersatz"of the Muckenhoupt condition which coincides with the Muckenhoupt condition in the case p(x) ≡ p * is constant, as well as its version for Carleson curves in the complex plane.
Observe that the class of weights satisfying condition (2.4)-(2.5) is evidently narrower that what we expect from the "real" Muckenhoupt class A p(•) .Thus, in the case of power weights , where the boundedness of the maximal operator holds [11].Obviously conditions (2.4)-(2.5)are sharp on those power functions which are "fixed" to a point at which the minimum of p(•) is reached.

Theorem A . Let the exponent p(t) satisfy conditions (1.1), (1.2) and the weight ρ fulfill condition (2.5). Then the operator
In the next theorem, we deal with weights of the form where w k (x) may oscillate as x → 0+ between two power functions (radial Zygmund-Bari-Stechkin type weights).The Zygmund-Bari-Stechkin class Φ 0 1 of weights and the upper and lower indices of weights (of the type of the Boyd indices) used in the theorem below are defined in Section 4. Note that various non-trivial examples of functions in Zygmund-Bari-Stechkini-type classes with coinciding indices may be found in [23], Section II; [24], Section 2.1, and with non-coinciding indices in [26].

Theorem B. Let Γ be a bounded Carleson curve and p(t) satisfy conditions (1.1), (1.2) on Γ.
The operator M is bounded in L p(•) (Γ, ρ) with the weight (2.6), where w k (r) are such functions that r (2.7) A similar statement for bounded domains in R n was proved in [10].

Some basics for variable exponent spaces
The weighted space L p(•) (Γ, ρ) was introduced in (1.3).We write We recall some basic facts for the variable exponent spaces L p(•) (Γ) and refer e.g. to [14] for details.
The Hölder inequality holds in the form and the norm f p(•) are simultaneously greater than one and simultaneously less than 1: and Let Γ be a bounded Carleson curve, the exponent p satisfy condition (1.2) and let w be any function such that there exist exponents a, b ∈ R 1 and the constants c 1 > 0 and where C > 1 does not depend on t, t 0 ∈ Γ.
which is bounded by the condition on w. 24 Preliminaries on Zygmund-Bari-Stechkin classes.
4.1 Index numbers m w and M w of non-negative a. i. functions The numbers (see [23], [26], [25]), will be referred to as the lower and upper indices of the function w(x) (compare these indices with the Matuszewska-Orlicz indices, see [18], p. 20; they are of the type of the Boyd indices, see [15], p. 75; [16], or [3], p. 149 about the Boyd indices).We have The indices m ω and M ω may be also well defined for functions w(x) positive for x > 0 which do not necessarily belong to W , for example, if there exists an a ∈ R 1 such that w a (x) := x a w(x) is in W . Obviously, So we also introduce the class
, where Z 0 is the class of functions w ∈ W satisfying the condition and Z δ is the class of functions w ∈ W satisfying the condition where c = c(w) > 0 does not depend on h ∈ (0, ]. In the sequel we refer to the above conditions as (Z 0 )-and (Z δ )-conditions.
Theorem 4.2.Let w ∈ W . Then w ∈ Z 0 if and only if m w > 0, and w ∈ Z δ , δ > 0, if and only if M w < δ, so that Besides this, for w ∈ Φ 0 δ and any ε > 0 there exist constants The following properties are also valid ) Statements (4.3)-(4.5)remain valid for the case when M w or m w may be non-positive.Namely, the following corollary from Theorem 4.2 is valid.Corollary 4.3.Let w(x), 0 < x ≤ , be such a function that t a w(t) ∈ Z 0 for some a ∈ R 1 .Then formula (4.4) remains valid and for any ε > 0 there exists c 1 > 0 such that Similarly, if x a w(x) ∈ Z δ , then (4.5) is valid and for any ε > 0 there exists c 2 > 0 such that Indeed, let a ∈ R 1 be such that w a (x) = x a w(x) ∈ W . Then according to (4.4) the function x mw a −ε is a.i.for every ε > 0. But m w a = m w + a, so that w(x) x m w −ε is a.i.for every ε > 0. Since m w > 0, then the function w itself is a.i., which means that it is in W . which follows from the fact that the function w(r) r ν with µ > M w is a.d.according to (4.5).

On examples of functions in
Power and power-logarithmic functions w(x) = x µ , x µ ln 1 Examples of non-equilibrated characteristics are much less trivial.An example of such a function w with different indices m(w) and M (w) was given in [1]; in the context of submultiplicative convex functions another example of functions with non-coinciding Matuszewska-Orlicz indices was given in [17], the latter example been also exposed in [18], p.93.In [26] there was explicitly constructed a family of functions with different indices belonging to the class Φ 0 γ .

Auxiliary lemmas
where the constant C > 0 does not depend on r ∈ [0, ]; it does not depend also on 0 Proof.The function w 1 (x) = x δ [w(x)] λ is almost increasing, because the function w(x) x δ/λ is almost decreasing when δ λ > M ω , according to formula (4.5), the validity of which follows from Corollary 4.3.Therefore, w 1 ∈ W .By the definition of the lower index, we easily obtain that m w 1 = δ − λM w .Hence m w 1 > 0 and consequently w 1 ∈ Z 0 by Theorem 4.2.
In order to show that the constant C in (4.10) does not depend on the appropriate choice of λ, we proceed as follows The function w(x) x M w +ε is almost decreasing for every ε > 0 by Corollary 4.3.Therefore, the expression in the brackets is bounded from above.Since λ ≥ 0, we get under the choice of ε sufficiently small: where C > 0 does not depend on t ∈ Γ and r ∈ [0, ].
Proof.Let µ < m w and w µ (x) = w(x) x µ so that w µ (x) is an a.i.function according to (4.4) and Corollary 4.3.We proceed as follows: where The inequality is valid, which follows from the direct estimation similar to the above arguments: By (4.14) from (4.13) we get where in the last inequality we used Lemma 4.6 with δ = 1. 2 5 Proof of Theorems A and A .

Proof of Theorem A.
To Prove Theorem A, we have to show that Following the idea in [7], we represent I p (M ρ f ) as where We make use of the known estimate (see [7], valid for for all for all f ∈ L p(•) with f p ≤ c.Estimate (5.3) is obtained by means of the usual Hölder inequality with the constant exponents p * and q * = p * p * −1 , taking into account that ∞, the latter following from condition (2.4).In view of (5.3), we may apply estimate (5.2).Then (5.1) implies Since Ω [ρ(x)] p(x) dx < ∞ by (2.4), we obtain As is known [29], p. 201, the weighted maximal operator M ρ 1 is bounded in L p * with a constant p * > 1, if the weight [ρ(x)] p 1 (x) is in A p * , which is nothing else but condition (2.4).
Therefore, by the boundedness of the weighted operator M ρ 1 in L p * , from (5.4) we get Since M is sublinear, this yields its boundedness in the space L p(•) (Ω, ρ).

Proof of Theorem A .
The proof of Theorem A is essentially the same.We only mention that an analogue of the pointwise estimate (5.2) for Carleson curves is also known, see Theorem 3.3 in [13] and Subsection 4.2 in [12], and the boundedness of the maximal operator along Carleson curves with Muckenhoupt weights satisfying the A p -condition (p ≡ const) is also known, see [4], p.149.

Proof itself of Theorem B.
1 0 The case (6.7).This case is covered by Theorem A , because in the case (6.7) the weight w(|t − t 0 |) satisfies condition (2.5) by Theorem 6.3.
By the continuity of p(t) we can choose δ so that a(δ) < 1 q(t 0 ) − M w .Then 1 q δ > M w and condition (6.13) is fulfilled.Then the operator M w is bounded in the space L p(•) (γ δ ) which completes the proof.