Singular Integral Operators in Generalized Morrey Spaces on Curves in the Complex Plane

We study the boundedness of the Cauchy singular integral operators on curves in complex plane in generalized Morrey spaces. We also consider the weighted case with radial weights. We apply these results to the study of Fredholm properties of singular integral operators in weighted generalized Morrey spaces.


Introduction
In this paper, we deal with singular integral operators in generalized Morrey spaces. The well-known classical Morrey spaces were widely investigated during last decades; see for instance books [1,21], survey paper [22], and references therein. We study the boundedness of a singular integral operator S Γ in the space L p,ϕ (Γ, ), where Γ is a composite curve which is a union of a finite number of non-intersecting curves without self-intersection, satisfying arc-chord condition. The boundedness of singular integral operators in classical Morrey spaces on a single curve was studied in [23]. We also refer to the paper [20], where conditions for the weighted boundedness of a general class of multidimensional singular integral operators in generalized Morrey spaces are found. In [17] weighted results for singular integral operators were obtained in classical Morrey spaces, but with more general weights. To prove the boundedness of the singular integral operator S Γ in the weighted generalized Morrey space L p,ϕ (Γ, ), first we have to prove the non-weighted boundedness of the maximal operator along such a curve in L p,ϕ (Γ). Then we derive the non-weighted boundedness of S Γ via the Alvarez-Pérez-type point-wise estimate For two-weight estimates for the maximal operator in local Morrey spaces we refer to [26]. We apply the obtained results to the study of Fredholm properties of singular integral operators in weighted generalized Morrey spaces. The theory of the Riemann boundary value problem and singular integral equations on curves in the complex plane, including Fredholm properties, is well known, see the books [5,15,16]. In particular, this theory was extensively developed in such spaces as Lebesgue, Orlicz and recently in variable exponent Lebesgue spaces and their weighted versions; see [7,10]. For the case of composite curves we refer to [7].
We study the Fredholmness of the following singular integral operator: in weighted generalized Morrey space L p,ϕ (Γ, ), where Γ is a set of nonintersecting oriented closed curves without self-intersection, satisfying arcchord condition. Γ may be such a single curve or union of such curves. Fredholmness of such operators in classical weighted Morrey spaces was studied in [24]. The case of generalized Morrey spaces on an interval was studied in [13]. We apply the methods from these papers to extend the results obtained there to the case of generalized weighted Morrey spaces on composite curves.
The paper is organized as follows: in Sect. 2, we provide necessary definitions on generalized Morrey spaces, Zygmund classes of functions and Matuszewska-Orlicz indices. In Sect. 3, we describe some known facts we use. In Sect. 4, we present our new results on the boundedness and Fredholmness of singular integral operators in weighted generalized Morrey spaces on composite curves. First we prove the Fefferman-Stein inequality Mf L p,ϕ (X) ≤ C M # f L p,ϕ (X) for a metric space X to derive the non-weighted boundedness of S Γ via (1.1). Then we prove the boundedness of S Γ and Fredholmness of A in the weighted case.

Generalized Morrey Spaces on Homogeneous Underlying Spaces
Let (X, d, μ) be a homogeneous metric measure space with quasi-distance d and measure μ. We restrict ourselves to the case where X has constant dimension: there exists a number N > 0 (not necessarily integer) such that where the constants C 1 and C 2 do not depend on x ∈ X and r > 0. In this case, the generalized Morrey space L p,ϕ (X) may be defined by the norm: where 1 ≤ p < ∞ and 0 ≤ M (ϕ) < N and the standard notation B(x, r) = {y ∈ X : d(x, y) < r} is used. Everywhere in the sequel it is assumed that ϕ : R + → R + is a measurable function satisfying the following assumptions: 1. ϕ(r) is continuous in a neighborhood of the origin; 2. ϕ(0) = 0; 3. inf r>δ ϕ(r) > 0 for every δ > 0 and ϕ(r) ≥ cr n (2.3) for 0 < r ≤ l, if l < ∞, and 0 < r ≤ N with an arbitrary N > 0, if l = ∞, the constant c depends on N in the latter case. Condition (2.3) makes the space L p,ϕ (X) non-trivial (see [18,Corollary 3.4]).

Curves Satisfying Arc-Chord Condition
Let Γ be a bounded curve in the complex plane C. We denote τ = t(σ), t = t(s), where σ and s stand for the arc abscissas of the points τ and t, and dμ(τ ) = dσ will stand for the arc-measure on Γ. We also use the notation so that Γ * (t, r) ⊆ Γ(t, r), and denote = μΓ = length of Γ.
Finally, a curve Γ is said to satisfy the (uniform) arc-chord condition, if In the sequel Γ is always assumed to be a curve satisfying the arc-chord condition.
The generalized Morrey space L p,ϕ (Γ) is defined by the norm (2.6) For a non-negative weight function (t) the weighted generalized Morrey space is introduced as

On Admissible Weight Functions
In the sequel, when studying the singular operator S Γ along a curve Γ in weighted generalized Morrey space, we deal with radial type weights of the form We introduce below the class of weight functions w k (x), x ∈ [0, ], admitted for our goals. Although the functions w k should be defined only on [0, d], . By V ± we denote the classes of functions w ∈ W defined by the following conditions: For the proof of this lemma we refer to [18].

Definition 2.5.
We say that a function ϕ ∈ W belongs to the Zygmund class (2.12) and to the Zygmund class It is known that the property of a function to be almost increasing or almost decreasing after the multiplication (division) by a power function is closely related to the notion of the so-called Matuszewska-Orlicz indices. We refer for instance to [9,19] for the properties of the indices of such a type.
For a function ϕ ∈ W(R + ) := W W such indices at the origin are defined as follows: ln r .
(2.14) The indices m(ϕ) are finite numbers when ϕ ∈ W (R + ) and M (ϕ) are finite numbers when ϕ ∈ W (R + ). Besides this, and We will also use the following known properties: (2.18) For other properties of the indices of functions ϕ ∈ W(R + ), we refer for instance to the paper [25, Section 6] and references therein.

Maximal Operator
To prove the boundedness of the singular integral operator in generalized Morrey space, we need to consider the maximal operator in this space. The boundedness of the maximal operator in the space L p(·),ϕ(·) (X) is known, see [8], in the setting of quasi-metric measure spaces. In particular, we can use this result for curves with arcchord condition.
In the proof of Lemma 4.2 for maximal operator, we will use the following lemma.

Point-Wise Estimate for the Weighted Singular Integral Operator on a Curve Which Satisfies the Arc-Chord Condition
Let Γ be a curve satisfying the arc-chord condition. We consider radial type weights of the form (t) = w(|t − t 0 |), t 0 ∈ Γ.
We define the operator K in the following way: In [23, Theorem 6.1] the following point-wise estimate was proved: when w ∈ V + , and It is easy to see that in the right-hand sides of (3.2)-(3.3) we have Hardy-type operators. Thus, according to the point-wise estimate (3.2)-(3.3) the boundedness of the weighted singular integral operator is reduced to the boundedness of the corresponding Hardy-type operators.

Hardy-Type Operators
We study the following Hardy-type operators: For Hardy-type operators in various function spaces, we refer for instance to [3,14], and the recent book [12], see also references therein. The boundedness of the operators (3.4) in generalized Morrey spaces was proved in [13]. We present here this result in a slightly modified form so as we need it here.

Main Results
In this section, we study singular integral operators in generalized Morrey spaces on composite curves, which satisfy the arc-chord condition. The boundedness of singular integral operators on the single such a curve in classical Morrey spaces was proved in [23].

Singular Integral Operator on Composite Curves
By a composite curve we mean a union Γ = m k=1 Γ k of a finite number of nonintersecting curves without self-intersection, satisfying arc-chord condition. Then the singular integral operator S Γ f (t) on such curves can be defined in the following way: It is convenient to treat the function f (t), t ∈ Γ, defined on Γ, as follows: Then we treat f (t) as . Then We define, for k = 1, 2, . . . , m: Hence, where T k is an operator with bounded kernel. We define the norm in the Morrey space on composite curves in a natural way, namely as where ϕ(r) = (ϕ 1 (r), ϕ 2 (r), . . . , ϕ m (r)) . From (4.3) there follows the obvious statement: if the singular integral operator is bounded in the generalized Morrey spaces on every separate curve, then it is also bounded in this space on the composite curve.

Boundedness of the Singular Integral Operator on Composite Curves: Non-weighted Case
The main result in this case reads. To prove this theorem we need the following new lemma, which is a generalization of the Lemma 5.3, proved in [23] for classical Morrey spaces.
Let M # be defined as follows:

Lemma 4.2.
Let X be a metric measure space with μ(X) = ∞. Under the condition (2.1), i.e., C 1 r N ≤ μB(x, r) ≤ C 2 r N , the following estimate holds: Proof. To prove this statement, we use the following point-wise estimate from [23, Lemma 5.3]: we need to prove the boundedness of S Γ k and T k .
Using the property f p,ϕ k = f s p s ,ϕ k , 0 < s < 1, of the norm we have To prove the boundedness of S Γ k we apply Lemma 4.2 and the inequality (1.1), and obtain To this end, we need the boundedness of the maximal operator in L p,ϕ k (Γ k ). According to [8] such a boundedness holds under the condition: where C does not depend on τ and r. It is easy to check that the conditions of our theorem imply the validity of the condition (4.5). Thus, the boundedness of S Γ k is proved under the conditions of our theorem. Boundedness of T k is evident, since T k is the operator with bounded kernel. Indeed, 1 τ −t k =: K j (τ, t k ) is bounded, since τ ∈ Γ j and t k ∈ Γ k , k = j. Therefore, The proof is complete.

Weighted Case
For simplicity we fix the weight at a single point t k 0 on each curve Γ k . Therefore, and ϕ k (r) r be an almost decreasing function. (4.8)
Proof. According to the representation (4.3) to prove the boundedness of the weighted singular integral operator on composite curves we have to prove the weighted boundedness of S Γ k and T k . To prove the weighted boundedness of S Γ k in generalized Morrey space we use the point-wise estimates (3.2)-(3.3). According to these estimates, we need the boundedness of the non-weighted S Γ k and weighted boundedness of the corresponding Hardy-type operators. The non-weighted boundedness of S Γ k f k is proved under the right-hand side condition in (4.7) in Theorem 4.1. The boundedness of the Hardy-type operators H w and H w was proved in Theorem 3.2 under Conditions (4.7), (4.8) and (4.9).
The boundedness of T k f in the weighted space L p,ϕ (Γ, ) can be proved by using the same arguments as in the non-weighted case, proved in Theorem 4.1. The right-hand side condition in (4.9) is sufficient for such boundedness. The proof is complete.

On Some Basics Related to Fredholmness
In the sequel Γ = is to the left of Γ , and D − = C\D + = m k=1 D − k , where D − k , k ∈ 1, m, are to the right of the contours Γ k , k ∈ 1, m. For simplicity we assume that the origin is in D + .
Let X = X(Γ) be any Banach space of functions on Γ. We recall that a linear operator A in a Banach space X is called Fredholm if its kernel ker A := {u ∈ X : Au = 0} has a finite dimension α := dim(ker A) < ∞, the range R(A) := {f ∈ X : f = Au, u ∈ X} is closed in X and has a finite dimension β := dim(R(A)) < ∞. Following [6], the ordered pair (α, β) will be referred to as the d-characteristic of the operator A. The difference Ind X A := α − β is called the index of A.
In this section we consider the operator It is well known that the following conditions: (1) the singular integral operator S Γ in L p,ϕ (Γ, ) is bounded; (2) the commutator gS Γ − S Γ g in L p,ϕ (Γ, ), where g ∈ C(Γ) is compact; (3) S 2 Γ = I in this setting guarantee the Fredholmness of the singular integral operator aI + bS Γ with continuous coefficients a and b, under the condition a(t) = ±b(t), t ∈ Γ. See for instance [11,Theorem A], where Fredholmness of singular integral operators in the setting of an abstract Banach space of functions on curves was proved.
Proof. From (4.3) it is easy to see that to prove compactness of commutator S Γ g−gS Γ , we have to prove the compactness of the commutators S Γ k g−gS Γ k and T k g − gT k , k = 1, 2, . . . , m. We start with the commutator S Γ k g − gS Γ k . From the famous Mergelyan's result, see for instance [4], p. 169, it is known that the continuous functions may be uniformly approximated by rational functions for an arbitrary Jordan curve Γ. Since the compactness of such commutator for rational functions is known, and the boundedness of our singular integral operators is proved in Theorem 4.3, the statement of this theorem for this commutator is proved. Now we consider the commutator T k g − gT k . Since the function g is continuous and a product of a compact operator with a continuous function is compact, we need to prove the compactness of the operator We denote Thus, Since, by the Stone-Weierstrass theorem, any continuous function on a compact set Π in R n can be uniformly approximated by a polynomial, we have that for any ε > 0, ∃ p n (s, σ) = 0≤μ+ν≤n C μ,ν s μ σ ν , such that 11) uniformly in both variables.
We denote (4.12) Then Since P n is a finite dimensional operator, it is enough to prove its boundedness by the norm, to claim that P n is compact.
Hence, for the operator P n , defined by (4.12), we have , since σ and s are bounded.
It is not hard to show that, under the assumptions of our theorem, Therefore, by (4.13) we obtain that P n f j L p,ϕ j (Γj , j ) ≤ C f j L p,ϕ j (Γj , j ) .
Thus, the operator P n is compact in the space L p,ϕj (Γ j , j ). Now we have to estimate |R n f j | = |T kj f j − P n f j | (4.14) by the norm of our space. Passing to the norm in (4.14), by (4.11) and (4.13) we obtain R n f j L p,ϕ j (Γj , j ) = T kj f j − P n f j L p,ϕ j (Γj , j ) ≤ Cε f j L 1 (Γj ) ≤ ε f j L p,ϕ j (Γj , j ) → 0.
Thus, we can conclude that the operator T k f is compact in L p,ϕ (Γ, ), since T k f is the limit of the sequence of compact operators. The proof is complete.
Now we can formulate the theorem on Fredholmness of the operator A, which indeed is proved by the above statements.