Remark on the Boundedness of the Cauchy Singular Integral Operator on Variable Lebesgue Spaces with Radial Oscillating Weights

Recently V. Kokilashvili, N. Samko, and S. Samko have proved a sufficient condition for the boundedness of the Cauchy singular integral operator on variable Lebesgue spaces with radial oscillating weights over Carleson curves. This condition is formulated in terms of Matuszewska-Orlicz indices of weights. We prove a partial converse of their result.


Introduction and main result
Let Γ be a rectifiable curve in the complex plane. We equip Γ with Lebesgue length measure |dτ |. We say that a curve Γ is simple if it does not have selfintersections. In other words, Γ is said to be simple if it is homeomorphic either to a line segment or to to a circle. In the latter situation we will say that Γ is a Jordan curve. The Cauchy singular integral of f ∈ L 1 (Γ) is defined by This integral is understood in the principal value sense, that is, where Γ(t, R) := {τ ∈ Γ : |τ − t| < R} for R > 0. David [4] (see also [3,Theorem 4.17]) proved that the Cauchy singular integral generates the bounded operator S on the Lebesgue space L p (Γ), 1 < p < ∞, if and only if Γ is a Carleson (Ahlfors-David regular ) curve, that is, where for any measurable set Ω ⊂ Γ the symbol |Ω| denotes its measure. To have a better idea about Carleson curves, consider the following example. Let α > 0 and Γ := {0} ∪ τ ∈ C : τ = x + ix α sin(1/x), 0 < x ≤ 1 .
One can show (see [3,Example 1.3]) that Γ is not rectifiable for 0 < α ≤ 1, Γ is rectifiable but not Carleson for 1 < α < 2, and Γ is a Carleson curve for α ≥ 2. A measurable function w : Γ → [0, ∞] is referred to as a weight function or simply a weight if 0 < w(τ ) < ∞ for almost all τ ∈ Γ. Suppose p : Γ → [1, ∞] is a measurable a.e. finite function. Denote by L p(·) (Γ, w) the set of all measurable complex-valued functions f on Γ such that Γ |f (τ )w(τ )/λ| p(τ ) |dτ | < ∞ for some λ = λ(f ) > 0. This set becomes a Banach space when equipped with the Luxemburg-Nakano norm If p is constant, then L p(·) (Γ, w) is nothing else but the weighted Lebesgue space. Therefore, it is natural to refer to L p(·) (Γ, w) as a weighted generalized Lebesgue space with variable exponent or simply as a weighted variable Lebesgue space. This is a special case of Musielak-Orlicz spaces [19] (see also [13]). Nakano [20] considered these spaces (without weights) as examples of so-called modular spaces, and sometimes the spaces L p(·) (Γ, w) are referred to as weighted Nakano spaces.
where A Γ is a positive constant depending only on Γ. Let w 1 , . . . , w n ∈ W and the weight w be given by (1.1). If then the Cauchy singular integral operator S is bounded on L p(·) (Γ, w).
This condition is also necessary for the boundedness of S on the variable Lebesgue space L p(·) (Γ, w) with the Khvedelidze weight w (see [10]).
The author have proved in [8] that for Jordan curves condition (1.3) is necessary for the boundedness of the operator S. The proof of this result given in [8] essentially uses that Γ is closed. In this paper we embark on the situation of non-closed curves. Our main result is a partial converse of Theorem 1.1. It follows from our results [6,8]   . Suppose w 1 , . . . , w n ∈ W and the weight w is given by (1.1). If the Cauchy singular integral operator S is bounded on L p(·) (Γ, w), then Γ is a Carleson curve and Moreover, if there exists an ε 0 > 0 such that the Cauchy singular integral operator S is bounded on L p(·) (Γ, w 1+ε ) for all ε ∈ (−ε 0 , ε 0 ), then For standard Lebesgue spaces, the boundedness of the operator S on L p (Γ, w), 1 < p < ∞, implies that S is also bounded on L p (Γ, w 1+ε ) for all ε in a sufficiently small neighborhood of zero (see [3,Theorems 2.31 and 4.15]). Hence if 1 < p < ∞, Γ is a simple Carleson curve, w 1 , . . . , w n ∈ W, and the weight w is given by (1.1), then S is bounded on the standard Lebesgue space L p (Γ, w), 1 < p < ∞, if and only if We believe that all weighted variable Lebesgue spaces have this stability property. Conjecture 1.4. Let Γ be a simple rectifiable curve, p : Γ → [1, ∞] be a measurable a.e. finite function, and w : Γ → [0, ∞] be a weight such that the Cauchy singular integral operator S is bounded on L p(·) (Γ, w). Then there is a number ε 0 > 0 such that S is bounded on L p(·) (Γ, w 1+ε ) for all ε ∈ (−ε 0 , ε 0 ).
If this conjecture would be true, we were able to prove the complete converse of Theorem 1.1 for non-closed curves, too.

Proof
In this section we formulate several results from [3,6,8] and show that Theorem 1.3 easily follows from them.
2.1. Muckenhoupt type condition. Suppose Γ is a simple rectifiable curve and p : Γ → (1, ∞) is a continuous function. Since Γ is compact, one has is well defined and also bounded and bounded away from zero. We say that a weight  [6] for Jordan curves, however its proof remains the same for curves homeomorphic to line segments, see also [7, Theorem 3.2]).
Consider the function Combining Lemmas 4.8-4.9 and Theorem 5.9 of [6] with Theorem 3.4, Lemma 3.5 of [3], we arrive at the following.

Matuszewska-Orlicz indices as indices of powerlikeness.
If ̺ ∈ W, then Φ 0 ̺ is a regular and submultiplicative function and its indices are nothing else but the Matuszewska-Orlicz indices m(̺) and M (̺). The next result shows that for radial oscillating weights indices of powerlikeness and Matuszewska-Orlicz indices coincide.
Theorem 2.4 (see [8,Theorem 2.8]). Suppose Γ is a simple Carleson curve. If w 1 , . . . , w n ∈ W and w(τ ) = n k=1 w k (|τ − t k |), then for every t ∈ Γ the function V 0 t w is regular and submultiplicative and Note that in [8], Theorem 2.4 is proved for Jordan curves. But the proof does not use the assumption that Γ is closed. It works also for non-closed curves considered in this paper.

Proof of Theorem 1.3.
Proof. Suppose S is bounded on L p(·) (Γ, w). From Theorem 2.1 it follows that w ∈ A p(·) (Γ). By Hölder's inequality this implies that Γ is a Carleson curve. Fix an arbitrary t ∈ Γ. Then, in view of Theorems 2.2 and 2.3 the function V 0 t w is regular and submultiplicative, so its indices are well defined and satisfy 0 ≤ 1/p(t)+α(V 0 t w) and 1/p(t) + β(V t w) ≤ 1. From these inequalities and Theorem 2.4 it follows that If S is bounded on all spaces L p(·) (Γ, w 1+ε ) for all ε in a neighborhood of zero, then as before for all ε in a neighborhood of zero and for all k ∈ {1, . . . , n}. These inequalities immediately imply that 0 < 1/p(t k )+m(w k ) and 1/p(t k )+M (w k ) < 1 for all k.
Remark 2.5. The presented proof involves the notion of indices of powerlikeness, which were invented to treat general Muckenhoupt weights (see [3]). Weights considered in the present paper are continuous except for a finite number of points. So, it would be rather interesting to find a direct proof of the fact that w ∈ A p(·) (Γ) implies (2.1), which does not involve the indices of powerlikeness α(V 0 t w) and β(V 0 t w). 2.6. Final remarks. In connection with Conjecture 1.4, we would like to note that for standard Lebesgue spaces L p (Γ, w) there are two different proofs of the stability of the boundedness of S on L p (Γ, w 1+ε ) for small ε. Simonenko's proof [22] is based on the stability of the Fredholm property of some singular integral operators related to the Riemann boundary value problem. Another proof is based on the self-improving property of Muckenhoupt weights (see e.g. [3, Theorem 2.31]). One may ask whether does w ∈ A p(·) (Γ) imply w 1+ε ∈ A p(·) (Γ) for all ε ∈ (−ε 0 , ε 0 ) with some fixed ε > 0? The positive answer would give a proof of the complete converse of (1.3). The author does not know any stability result for the boundedness of S or a self-improving property for w ∈ A p(·) (Γ).
After this paper had been submitted, P. Hästö and L. Diening [5] have found a necessary and sufficient condition for the boundedness of the classical Hardy-Littlewood maximal function on weighted variable Lebesgue spaces in the setting of R n . Note that they write a weight as a measure (outside of | · | p(τ ) ). Their condition is another generalization of the classical Muckenhoupt condition. In the setting of Carleson curves (and the weight written inside of | · | p(τ ) ), the Hästö-Diening condition takes the form (2.2) sup t∈Γ sup R>0 1 R p Γ(t,R) Γ(t,R) w(τ ) p(τ ) |dτ | w(·) −p(·) χ Γ(t,R) (·) q(·)/p(·) < ∞,