Matrix Riemann–Hilbert problems with jumps across Carleson contours

We develop a theory of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n \times n$$\end{document}n×n-matrix Riemann–Hilbert problems for a class of jump contours and jump matrices of low regularity. Our basic assumption is that the contour \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Gamma $$\end{document}Γ is a finite union of simple closed Carleson curves in the Riemann sphere. In particular, unbounded contours with cusps, corners, and nontransversal intersections are allowed. We introduce a notion of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^p$$\end{document}Lp-Riemann–Hilbert problem and establish basic uniqueness results and Fredholm properties. We also investigate the implications of Fredholmness for the unique solvability and prove a theorem on contour deformation.

values m + and m − from the left and right sides of exist, are continuous, and satisfy the jump condition m + = m − v on . Uniqueness is ensured by requiring that m approaches the identity matrix at infinity.
The theory of scalar RH problems is well-developed in the classical set-up in the complex plane [26] as well as for problems on Riemann surfaces [28,30]. Constructive existence and uniqueness results are available, at least within classes of Hölder continuous functions [1,26,30]. We refer to the monograph [22] for more recent developments and further references in the case of less regular solutions.
The theory of matrix RH problems is substantially more complicated than the scalar theory. Only very special classes of problems (such as problems with a rational jump matrix, see Chapter I of [6]) can be solved explicitly. Uniqueness can often be established by means of Liouville's theorem, but existence results are rare and usually rely on the presence of some special symmetry, see [1,8].
Matrix RH problems are essential in the analysis of integrable systems, orthogonal polynomials, and random matrices. The RH approach is particularly powerful when it comes to determining asymptotics. Indeed, the asymptotic behavior of solutions of many RH problems can be efficiently determined by means of the nonlinear steepest descent method introduced by Deift and Zhou [10], building on earlier work of Its [19] and Manakov [25]. This method and generalizations thereof have been instrumental in several recent advances in random matrix theory and in the analysis of large-time asymptotics of solutions of integrable PDE's [8,9,15,16,20,21].
The classical formulation of a RH problem, which involves a piecewise smooth contour and a smooth (or at least Hölder continuous) jump matrix v, is sufficient for many applications. However, in order to obtain a more convenient setting for the application of functional analytic techniques, it is essential to extend the formulation of a RH problem to the L p -setting [5,24]. Deift and Zhou and others [8,11,12,16,29] have extended the definition of a RH problem to the case where the jump matrix v and its inverse v −1 belong to appropriate Lebesgue spaces, and the contour is a finite union of closed simple smooth curves in the Riemann sphere with a finite number of transversal intersection points. In particular, the relationship between the unique solvability of a RH-problem and the Fredholmness of a certain associated singular operator was explained in [29].
Our goal in this paper is to lay the foundation for a theory of matrix RH problems for a class of possibly unbounded jump contours of very low regularity. Our basic assumption is that the contour is a finite union of closed Carleson curves in the Riemann sphere. The contours are allowed to pass through infinity and to have cusps, corners, and nontransversal intersections. We introduce a notion of L p -Riemann-Hilbert problem for this class of contours and establish basic uniqueness results and Fredholm properties. We also investigate the implications of Fredholmness for the unique solvability and prove a theorem on contour deformation. We mainly develop those parts of the theory which seem most relevant for applications to integrable equations. For example, at several places in Sect. 5 we assume that the jump matrix has unit determinant and we do not consider possible generalizations of partial indices.
The matrix RH problems considered here are different from the vector RH problems studied, for example, in [4] and [24]. However, the Fredholm theories of these two problems are closely related, so in this regard our main contribution is to extend results known for bounded curves to unbounded curves. Such an extension is important for applications to integrable equations where most contours naturally pass through infinity.
The formulation of a successful theory of RH problems is intricately linked to the boundedness of the Cauchy singular operator S defined in Eq. (2.3) below. Indeed, this operator is the key ingredient in the Sokhotski-Plemelj formulas for the boundary values of an analytic function. Since it has been proved in recent years that S is a bounded operator on L p ( ) if and only if is Carleson (cf. [4]), it is natural to expect that the class of Carleson contours is the most general class of contours for which a clean RH theory exists. This is the reason we choose to consider Carleson jump contours.
We emphasize that RH problems with contours involving nontransversal intersections are important in applications to integrable evolution equations. For example, the analysis of the Degasperis-Procesi equation on the half-line naturally leads to a RH problem with the jump contour displayed in Fig. 1, see [23]. The results of the present paper can be used to rigorously derive the long-time asymptotics of the solutions of this equation via the nonlinear steepest descent method [3].
An additional reason for writing this paper is to make accessible detailed and rigorous proofs of several basic results on matrix RH problem. Many of these results are well-known to the experts (at least if the contour is sufficiently well-behaved), but their proofs are scattered or absent in the literature. It turns out that the basic results can be proved in the more general setting of Carleson jump contours with little extra effort.
In Sect. 2, we summarize several properties of Smirnoff classes and Cauchy integrals over rectifiable Jordan curves. In Sect. 3, we introduce the notion of a Carleson jump contour as well as a number of function spaces which turn out to be convenient when dealing with contours passing through infinity. In Sect. 4, we establish several properties of Cauchy integrals over general Carleson jump contours. In Sect. 5, we introduce a notion of L p -Riemann-Hilbert problem for a general Carleson jump contour and develop the basics of a theory for these problems.

Preliminaries
A subset ⊂ C is an arc if it is homeomorphic to a connected subset I of the real line which contains at least two distinct points. If ϕ : I → is a homeomorphism onto an arc and (a, b) ⊂ I is the interior of I with a ∈ R ∪ {−∞} and b ∈ R ∪ {∞}, then lim t→a + ϕ(t) and lim t→b − ϕ(t) are referred to as endpoints of whenever the limits exist and are finite. An arc may have two, one, or no endpoints. An arc that does not contain its endpoints is an open arc. If I = [a, b] is a closed interval, the length | | of is defined by If I is not closed, the length of is defined as the supremum of |ϕ([a, b])| as [a, b] ranges over all closed subintervals of I . The arc is rectifiable if its length is finite. A subset ⊂ C is a composed curve if it is connected and may be represented as the union of finitely many arcs each pair of which have at most endpoints in common. A composed curve is oriented if it can be represented as the union of finitely many oriented arcs each pair of which have at most endpoints in common. A subset ⊂ C is a Jordan curve if it is homeomorphic to the unit circle S 1 .
Let ⊂ C be a composed curve. If z ∈ C, r ∈ (0, ∞), and D(z, r ) denotes the open disk of radius r centered at z, then ∩ D(z, r ) is an at most countable union of arcs. If all of these arcs are rectifiable and the sum of their lengths is finite, we say that is rectifiable for every z ∈ and every r ∈ (0, ∞). A composed curve is locally rectifiable if and only if ∩ D(0, r ) is rectifiable for every r ∈ (0, ∞).

Carleson curves
Let ⊂ C be a locally rectifiable composed curve. We equip with Lebesgue length measure and denote the measure of a measurable subset γ ⊂ by |γ |; see e.
Moreover, is Carleson if and only if each of its finite number of arcs is Carleson. We refer to Chapter 1 of [4] for more information on Carleson curves.

Cauchy singular operator
Let be a composed locally rectifiable curve. Let C ∞ 0 ( ) denote the set of all restrictions of smooth functions f : whenever the limit exists. If h ∈ C ∞ 0 ( ), then (S h)(z) exists for almost all z ∈ (see Theorem 4.14 of [4]). If w is a weight on , we say that the Cauchy singular operator S generates a bounded operator on L p ( , w) if C ∞ 0 ( ) is dense in L p ( , w) and It was realized in the early eighties that the Carleson condition is the essential condition for ascertaining boundedness of S in L p -spaces [7]. More precisely, if 1 < p < ∞ and is a composed locally rectifiable curve, then S generates a bounded operator on L p ( ) if and only if is Carleson. In dealing with unbounded contours, we will need a more general version of this result valid for weighted L pspaces.
Let p ∈ (1, ∞). We define A p ( ) as the set of weights w ∈ L p loc ( ) such that 1/w ∈ L q loc ( ) and

Smirnoff classes
Let ⊂ C be a rectifiable Jordan curve oriented counterclockwise. LetĈ = C ∪ ∞ denote the Riemann sphere and let D + and D − be the two components ofĈ\ .
Assuming that ∞ ∈ D − , we refer to D + and D − as the interior and exterior components respectively. Let 1 ≤ p < ∞. A function f analytic in D + belongs to the Smirnoff class E p (D + ) if there exists a sequence of rectifiable Jordan curves {C n } ∞ 1 in D + , tending to the boundary in the sense that C n eventually surrounds each compact subdomain of D + , such that (2.5) A function f analytic in D − is said to be of class E p (D − ) if there exists a sequence of rectifiable Jordan curves {C n } ∞ 1 in D − , tending to the boundary in the sense that every compact subset of D − eventually lies outside n , such that (2.5) holds. We leṫ E p (D − ) denote the subspace of E p (D − ) consisting of all functions f ∈ E p (D − ) that vanish at infinity.

Basic results on Cauchy integrals
Given a locally rectifiable composed contour ⊂ C and a measurable function h defined on , we define the Cauchy integral (Ch)(z) for z ∈ C\ by whenever the integral converges. To avoid confusion, we will sometimes indicate the dependence of C on by writing C for C.
In the next two propositions, we collect a number of properties of the Cauchy integral and its relation to the Smirnoff classes; we refer to Chapter 10 of [14] and Chapter 6 of [4] for proofs. Given a Jordan curve ⊂C, we let D + and D − denote the interior and exterior components ofĈ\ . (2.6).

Theorem 2.2 Let C denote the Cauchy integral operator defined in
then the nontangential limits of f (z) as z approaches the boundary exist a.e. on ; if f + (z) denotes the boundary function, then f + ∈ L p ( ) and If f ∈ E p (D − ), then the nontangential limits of f (z) as z approaches the boundary exist a.e. on . If f − (z) denotes the boundary function, then f − ∈ L p ( ) and Suppose is a Carleson Jordan curve. Then the Cauchy singular operator S : Theorem 2.2 implies that if is a Carleson Jordan curve and h ∈ L p ( ) for some 1 < p < ∞, then the left and right nontangential boundary values of Ch, which we denote by C + h and C − h, lie in L p ( ). This allows us to view C ± as linear operators C ± : h → C ± h on L p ( ). Theorem 2.3 Let 1 < p < ∞ and let ⊂ C be a Carleson Jordan curve. Then C ± are bounded operators on L p ( ) with the following properties: • The Sokhotski-Plemelj formulas are valid. • C ± are complementary projections on L p ( ) in the sense that

Carleson jump contours
RH problems are conveniently formulated on the Riemann sphereĈ = C∪∞. In order to allow for jump contours passing through infinity, we introduce a class of curves J , which in addition to the rectifiable Jordan curves considered in the previous section also includes unbounded contours. Recall that ⊂ C is referred to as a Carleson curve if is a locally rectifiable composed curve satisfying (2.1). We extend this notion to the Riemann sphere by calling a subset ⊂Ĉ a Carleson curve if and only if is connected and ∩ C is a Carleson curve.

The class J
Let J denote the collection of all subsets of the Riemann sphereĈ such that is homeomorphic to the unit circle and is a Carleson curve. If ∞ / ∈ , then ∈ J if and only if ⊂ C is a Carleson Jordan curve. However, J also includes curves passing through infinity. In fact, the next proposition shows that J is invariant under the action of the group of linear fractional transformations. This shows that J is a natural extension of the family of Carleson Jordan curves in that it puts ∞ on an equal footing with the other points in the Riemann sphere.  Proof See "Appendix 1".

Remark 3.2
If γ : S 1 →Ĉ is an injective C 1 map such that |γ (s)| = 0 for all s ∈ S 1 , then := γ (S 1 ) belongs to J . Indeed, being a continuous bijection from a compact space onto a Hausdorff space, γ is a homeomorphism S 1 → . In view of Proposition 3.1, we may assume that ∞ / ∈ . Then, since S 1 is compact, we may cover S 1 with a finite number of open sets {U j } n 1 such that the restriction of γ to each U j is a C 1 graph; Proposition 1.1 of [4] now implies that is Carleson.

Remark 3.3
The Carleson condition is essential in Proposition 3.1. In fact, the family of composed locally rectifiable (but not necessarily Carleson) curves is not invariant under the action of the group of linear fractional transformations. Indeed, let = {te −it 2 |1 < t < ∞} and ψ(z) = z −1 . Then is locally rectifiable, but ψ( ) is not locally rectifiable because ψ( ) = {t −1 e it 2 |1 < t < ∞} ⊂ D(0, 1) has infinite length: This example does not contradict Proposition 3.1. Indeed, the estimate implies that | ∩D(0,r )| r is unbounded as r → ∞; hence is not Carleson.

Carleson jump contours
We call a connected subset ⊂Ĉ of the Riemann sphere a Carleson jump contour if it has the following properties: (a) ∩ C is an oriented composed curve.
is a Carleson jump contour. Indeed, using that any logarithmic spiral {re −iδ ln r | 0 < r < 1} for δ ∈ R is a Carleson arc (see Example 1.  Our goal is to establish generalizations of Theorems 2.2 and 2.3 which are valid in the case of a general Carleson jump contour . These generalizations will be stated and proved in Sect. 4; in the remainder of this section, we introduce a number of function spaces which will be needed for the formulation of these theorems. and z 0 is any point in C\ . It is easy to see that E p (D) does not depend on the choice consists of the functions in E p (D) that vanish at infinity, so that the present definition ofĖ p (D) is consistent with the definition given in Sect. 2.3. If D = D 1 ∪ · · · ∪ D n is the union of a finite number of disjoint subsets ofĈ each of which is bounded by a curve in J , we define E p (D) andĖ p (D) as the set of functions f analytic in D such that f | D j ∈ E p (D j ) and f | D j ∈Ė p (D j ) for each j, respectively.

Properties of E p ( D) andĖ p ( D)
Our definitions of the generalized Smirnoff classes E p (D) andĖ p (D) are designed in such a way that these classes possess convenient transformation properties under the action of the group of linear fractional transformations.

Proposition 3.6 Let 1 ≤ p < ∞. Let be a Carleson jump contour and let D be the union of any number of components ofĈ\ . Let ψ(z) be an arbitrary linear fractional transformation of the form
Proof Without loss of generality, we may assume that D is one of the components of C\ where ∈ J .
Since the linear fractional transformations form a group, this will prove (a). in D tending to the boundary such that (2.5) holds. It follows that {ψ(C n )} ∞ 1 is a sequence of rectifiable Jordan curves in ψ(D) tending to ψ( ) and If c = 0, then ψ (z) is a finite constant. If c = 0, then our assumption that ψ(D) is bounded implies that the point z = −d/c does not belong toD; hence the function is bounded on D. It follows that the right-hand side of (3.4) is finite.
). This completes the proof of (a). (b) This part is a consequence of (a) and the definitions. Indeed, suppose first that c = 0. By definition ofĖ Using (a) and the fact that cz + d = bc−ad cw−a when w = ψ(z), the latter condition is It is possible to characterize the spaces E p (D) andĖ p (D) in terms of conditions analogous to (2.5) also when ∞ ∈ ∂ D.
tending to in the sense that C n eventually surrounds each compact subset of D, such that It remains to prove the opposite implications. Let u → ψ(u) be a conformal isomorphism from the open unit disk onto ϕ(D) and let r be the image under ψ of the circle |u| = r . Suppose f ∈ E p (D). Then Proposition 3.6 (a) implies that f • ϕ −1 ∈ E p (ϕ(D)). Hence, by Theorem 10.1 in [14], Applying the change of variables w = ϕ(z), we find Hence, by the change of variables w = ϕ(z) and Theorem 10.1 in [14], This completes the proof of (b).

then the functions z f (z)g(z) and f g belong toĖ r (D).
Proof (a) The result is immediate from the definitions if ∞ / ∈ . Thus suppose ∞ ∈ . Let z 0 ∈ C\ and define ϕ as in (3.3).
another application of Proposition 3.6 shows that f ∈Ė r (D). This proves (a).
Without loss of generality, we may assume that A n = B n = C n for all n ≥ 1 where {C n } ∞ 1 are level curves of an arbitrary conformal map of the unit disk onto D (see Theorem 10.1 in [14]). Then, by Hölder's inequality, Using Proposition 3.6 again, we conclude that f g/ϕ, f g ∈Ė r (D). Part (b) follows.

The spacesL p ( )
Let be a Carleson curve. We defineL p ( ) as the set of all If h ∈L p ( ), then the value of the Cauchy integral (Ch)(z) is well-defined for all z ∈ C\ . Indeed, if 1/ p + 1/q = 1, then the Carleson property of implies | · −z| − 2 q L q ( ) < ∞; hence, by Hölder's inequality, If is bounded, thenL p ( ) = L p ( ). Lemma 3.9 Let 1 ≤ p < ∞ and let be a Carleson curve. Let z 0 ∈ C\ and let ϕ be given by (3.3).
is a bijectionL p ( ) → L p (ϕ( )) and where acts on a function f : We infer that is a bijectionL p ( ) → L p (ϕ( )) with inverse given by For each z 0 ∈ C\ , we define a norm onL p ( ) by The spaceL p ( ) with the norm · L p ( ),z 0 is nothing but the weighted space p . Different choices of z 0 ∈ C\ induce different norms onL p ( ), but these norms are all equivalent. We say that an operator T onL p ( ) is bounded if it is bounded with respect to one (and hence all) of these norms.
then the functions z f (z)g(z) and f g belong toL r ( ).

The Cauchy singular operator
Theorem 2.1 can be used to establish boundedness of the Cauchy singular operator S onL p ( ) if 1 < p < ∞ and is Carleson.
The result follows from Theorem 2.1 if we can show that w ∈ A p ( ). If p = 2, this is an immediate consequence of being Carleson. Thus suppose p = 2.
Define q by 1/ p + 1/q = 1 and let z ∈ . If 0 < r ≤ |z−z 0 | 2 and z ∈ D(z, r ), then Using the Carleson condition (2.1) on the disk D(z, r ), we find that there exists a constant C α > 0 depending only on α such that with C p > 0 depending only on p.
On the other hand, if R = |z − z 0 | + r , then the Carleson condition on the disk D(z 0 , 2 1−n R) yields It follows from (3.14) and (3.17) that w(z) satisfies the Muckenhoupt condition (2.4).
Our next objective is to determine how S transforms under the change of variables w = 1/(z − z 0 ). We need the following lemma.
A similar argument shows that Consequently, for all small enough r > 0, where the function Given > 0 small enough, there exists a unique r = r ( ) > 0 such that δ(r )/2 = .
It follows that → for every h ∈L p ( ). In other words, the following diagram commutes: Proof We will show that S h = −1 S ϕ( ) h a.e. on whenever h ∈ C ∞ 0 ( ). Since C ∞ 0 ( ) is dense inL p ( ) and the operators S and −1 S ϕ( ) are bounded onL p ( ) by Lemma 3.9 and Proposition 3.11, this will prove (3.24).
Let h ∈ C ∞ 0 ( ). A change of variables shows that for all z ∈ ∩ C and > 0. As → 0, the left-hand side of (3.25) tends to (S h)(z) for a.e. z ∈ . It remains to prove that the right-hand side of (3.25) tends to for a.e. z ∈ as → 0. The proof of this fact is complicated by the fact that, in general, the disk ϕ (D(z, )) is not centered at ϕ(z).
Let z ∈ and let 0 < < |z − z 0 |. Then Noting that where The function ( h)(w) = w −1 h(w −1 + z 0 ) is the restriction to ϕ( ) of a smooth function which approaches zero as w → ∞. Hence there exists an M > 0 such that |( h)(w)| ≤ M for all w ∈ ϕ( ). We estimate

Cauchy integrals over Carleson jump contours
The following two theorems generalize Theorems 2.2 and 2.3 to the case where is a general Carleson jump contour.
If f ∈Ė p (D − ), then the nontangential limits of f (z) as z approaches the boundary exist a.e. on . If f − (z) denotes the boundary function, then f − ∈L p ( ) and In particular, Theorem 4.1 implies that if is a Carleson jump contour and h ∈L p ( ) for some 1 < p < ∞, then the left and right nontangential boundary values of Ch, which we denote by C + h and C − h, lie inL p ( ). This allows us to define two linear operators C ± : h → C ± h onL p ( ). Theorem 4.2 Let 1 < p < ∞ and let ⊂Ĉ be a Carleson jump contour. Then C ± are bounded operators onL p ( ) with the following properties: • The Sokhotski-Plemelj formulas

4)
are valid. • C ± are orthogonal projections onL p ( ) in the sense thaṫ In the special case of a jump contour consisting of a single rectifiable Jordan curve, Theorems 4.1 and 4.2 reduce to Theorems 2.2 and 2.3, respectively.

Proof of (a)
Suppose first that ∞ / ∈ , so that ⊂ C is bounded. Let f ∈Ė p (D + ). Represent as the union of finitely many arcs each pair of which have at most endpoints in common. If z ∈ is not one of these finitely many endpoints, then z belongs to ∂ D + j for exactly one component D + j of D + . Since Theorem 2.2 implies that f | D + j has nontangential limits a.e. on ∂ D + j , it follows that f has nontangential limits a.e. on . Another application of Theorem 2.2 shows that f Now suppose z ∈ D + k for some 1 ≤ k ≤ n. Since z lies in the region exterior to ∂ D + j for each j = k, Theorem 2.2 yields If z ∈ D − , then z lies in the region exterior to ∂ D + j for every j, so a similar computation implies (C f + )(z) = 0. This proves (4.1). Similar arguments apply when f ∈Ė p (D − ). This proves (a) in the case when is bounded.

A convergence lemma
For the proof of (b), we need the following lemma. Proof Part (i) is a consequence of Theorem 17.2 in Chapter III of [27]. In order to prove (ii), let { f n } ∞ 1 be a sequence of functions in E p (D − ) such that f n− → h in L p ( ). Let z 0 ∈ D + and let ϕ(z) = 1/(z − z 0 ). Then h • ϕ −1 ∈ L p (ϕ( )) and Proposition 3.6 implies that f n • ϕ −1 ∈ E p (ϕ(D − )) for each n. Assuming for simplicity that both and ϕ( ) are oriented counterclockwise, we have ( f n • ϕ −1 ) + = f n− • ϕ −1 , and so Hence, by (i), there exists a function g ∈ E p (ϕ(D − )) such that f n • ϕ −1 → g uniformly on compact subsets of ϕ(D − ) and g , then each f n vanishes at ∞. Hence f vanishes at ∞, and so f ∈Ė p (D − ).

Proof of (b)
Suppose 1 < p < ∞ and h ∈L p ( ). We first assume that ∞ / ∈ . Switching the orientation of if necessary, we may suppose that ∞ ∈ D − . Let R( ) be the set of all rational functions with no poles on . Every function r ∈ R( ) can be written as r = r + + r − , where r + is analytic in D + , r − is analytic in D − , and r − vanishes at infinity. That is, r + ∈Ė p (D + ) and r − ∈Ė p (D − ). We claim that Indeed, if consists of a single Carleson Jordan curve, then (4.7) follows from Lemma 6.5 of [4]. If is the union of multiple Carleson Jordan curves {∂ D + j } n 1 , then we write r − = n j=1 r − j where r − j is analytic outside D + j and r − j (∞) = 0 for each j. Let χ j be the characteristic function of ∂ D + j . Decomposing r + and r − j into partial fractions and using that (4.7) is valid in the case when is a Carleson Jordan curve, we find and a.e. on . Equation (4.9) implies Thus S r − = −r − . Similarly, equation (4.8) implies S r + = r + . This proves (4.7). Equation (4.7) implies that S 2 r = r for every r ∈ R( ). Since R( ) is dense in L p ( ) (see Lemma 9.14 in [4]), it follows that S 2 h = h for every h ∈ L p ( ).
Let h ∈ L p ( ). Let r n be a sequence in R( ) converging to h in L p ( ). The boundedness of S on L p ( ) implies ∈ . It also follows that showing that the Sokhotski-Plemelj formulas (4.4) are valid for all h ∈ L p ( ).

Definition
Let n ≥ 1 be an integer. Given an n × n-matrix valued function v : → G L(n, C), we define a solution of the L p -RH problem determined by ( , v) to be an n ×n-matrix valued function m ∈ I +Ė p (D) such that the nontangential boundary values m ± satisfy m + = m − v a.e. on .

Properties of m ±
In order to make contact with earlier works on L p -RH problems on smooth contours, we show that m is a solution of the It follows that m satisfies the L p -RH problem determined by ( , v).

Remark 5.2
In most earlier references on L p -RH problems [11,12,16,29], a solution of an L p -RH problem is defined as a pair of functions m ± ∈ I + L p ( ) satisfying (RH1)-(RH2) (or properties very similar to (RH1)-(RH2)); the associated function m(z) is then referred to as the 'extension of m ± '. Here, in an effort to mimic the classical formulation of a RH problem as closely as possible, we have chosen to define a solution directly in terms of m. Proposition 5.1 shows that in the set-up provided by the spacesL p ( ) andĖ p (D), the definitions in terms of m and m ± are equivalent. (RH1) is equivalent to the condition that m ± ∈ I + C ±L p ( ).

Uniqueness results
We will show that the solution of the L p -RH problem determined by ( , v) is unique provided that det v = 1 and n ≤ p.
Proof Suppose m,m ∈ I +Ė p (D) are two solutions of the L p -RH problem determined by ( , v). Suppose m −1 ∈ I +Ė q (D). By Lemma 3.8, Using Theorem 4.1 and the fact that It follows that m =m on D. Remark 5.5 The assumption in Lemma 5.4 that m −1 ∈ I +Ė q (D) implies that m ± deliver a so-called L p -canonical factorization of v; the uniqueness of the latter is known, see [18,24]. Recall that the adjugate adj A of an n × n matrix A is defined by where m i j (A) denotes the (i j)th minor of A. By Cramer's rule, the inverse of A is given by A −1 = adj(A)/ det(A) whenever det(A) = 0. We continue to assume that 1 < p < ∞. Hence, by (a), so that (b) follows from Lemma 5.4.

Remark 5.7
For a sufficiently smooth contour, the special case n = p = 2 of Theorem 5.6 was proved in [8,13]. Theorem 5.6 generalizes this result to the case of a Carleson contour and any 1 ≤ n ≤ p. As an application, we note that the case n = 3 is relevant for the 3 × 3-matrix RH problem associated with the Degasperis-Procesi equation, see Fig. 1.

A singular integral equation
Given a Banach space X , we let B(X ) denote the space of bounded linear operators on X . Given two functions w ± ∈L p ( ) ∩ L ∞ ( ), we define the operator C w : We fix a point z 0 ∈ C\ and let · L p ( ) denote the associated norm onL p ( ) defined in (3.12). The estimate 2) The next proposition shows that if v = (v − ) −1 v + and w ± = ±v ± ∓ I then the L p -RH problem determined by ( , v) is equivalent to the following singular integral equation for μ ∈ I +L p ( ) cf. [2]: Conversely, suppose μ ∈ I +L p ( ) satisfies (5.3).

Fredholm properties
Given v : → G L(n, C), we define a solution of the homogeneous L p -RH problem determined by ( , v) to be an n × n-matrix valued function m ∈Ė p (D) such that m + = m − v a.e. on .
Hence μ = (m − +m − )(v − ) −1 also satisfies Eq. (5.3). By Proposition 5.8 and uniqueness of m, we conclude that  Let C( ) denote the set of restrictions to of continuous functionsĈ → C. If ⊂Ĉ is given the subspace topology, Tietze's extension theorem implies that C( ) coincides with the set of continuous functions → C. We will show that if w ± ∈ C( ) then the operator I − C w is Fredholm. If, in addition, w ± are nilpotent, the Fredholm index of this operator is zero, so that all four statements where the right multiplication operator R g is defined for functions g(z) and h(z) by Proof Since ⊂Ĉ is compact, there exists a c such that | det v ± | ≥ c > 0 on .
Assume first that ∞ / ∈ . Step 1. We will show that T w and Tw defined by (5.5) are compact operators on L p ( ). By Mergelyan's rational approximation theorem (see p. 119 of [17] the operators R w + n S − S R w + n are integral operators with continuous kernels. A standard argument based on Ascoli's theorem implies that they are compact L p ( ) → C( ); hence they are also compact L p ( ) → L p ( ). Since as n → ∞, it follows that is compact. Since the compact operators form a two-sided ideal, we find that is a compact operator on L p ( ). Similar arguments apply to the other terms in (5.5). This shows that T w and Tw are compact on L p ( ).
Step 2. We will show that I − C w is Fredholm onL p ( ). Let h ∈ L p ( ). Then In view of the identities w +w+ = −w + −w + and w −w− = w − +w − , the right-hand side equals T w h + C w h + Cwh. Hence Interchanging w andw in the above argument, we find It follows that I −C w is invertible modulo compact operators; hence I −C w is Fredholm on L p ( ). Since the norms of L p ( ) andL p ( ) are equivalent when is bounded, this proves (a) in the case of ∞ / ∈ .
If w ± are nilpotent, then tw ± ∈ C( ) and det(tw + + I ) = det(I −tw − ) = 1, thus the operator I − C tw is Fredholm on L p ( ) for t ∈ [0, 1] by Step 2. Since the Fredholm index is constant on connected components, this proves (b) in the case of ∞ / ∈ .

Reversal of subcontours
It is sometimes convenient to consider RH problems with jumps across contours which are not Carleson jump contours but which can be turned into Carleson jump contours by reorienting an appropriate subcontour. We make the following definition: If˜ denotes the Carleson jump contour with the orientation reversed on a subset 0 ⊂ andṽ is defined byṽ then we say that m ∈ I +Ė p (D) satisfies the L p -RH problem determined by (˜ ,ṽ) if and only if m satisfies the L p -RH problem determined by ( , v).

Contour deformations
Many applications of RH problems rely on arguments involving contour deformations. For example, in the nonlinear steepest descent method of [10], the jump contour is deformed in such a way that w = v − I is exponentially small away from a finite number of critical points. Theorem 5.12 below gives conditions under which the deformed RH problem is equivalent to the original one. Proof The result is immediate when ∞ / ∈ . The case of ∞ ∈ can be reduced to the case of ∞ / ∈ by means of Proposition 3.6.
Letˆ = ∪γ denote the union of the Carleson jump contour and a curve γ ∈ J , see Figs. 5 and 6. Suppose that, reversing the orientation on a subcontour if necessary, is a Carleson jump contour. To be definite, we henceforth fix an orientation on the contourˆ which turns it into a Carleson jump contour, and we endow the contours The contours γ and Fig. 6 The contoursˆ = ∪ γ and γ ± and γ with the orientations they inherit as subsets ofˆ . Then is a Carleson jump contour up to reorientation; we define a solution of the L p -RH problem determined by ( , v) as in Sect. 5.6. Let B + and B − denote the two components ofĈ\γ . Without loss of generality, we may assume that ∞ ∈B − . LetD ± be the open sets such thatĈ\ˆ =D + ∪D − and ∂D + = −∂D − =ˆ . Let U ± =D ± ∩ B + . LetD =D + ∪D − and U = U + ∪U − . Let γ + and γ − be the parts of γ that belong to the boundary of U + and U − , respectively. The orientations of γ ± are such that B + lies to the left of γ + , whereas B + lies to the right of γ − .
Then the L p -RH problems determined by ( , v) and (ˆ ,v) are equivalent in the following sense: If m ∈ I +Ė p (D) satisfies the L p -RH problem determined by ( , v), then the functionm(z) defined for z ∈D bŷ satisfies the L p -RH problem determined by (ˆ ,v). Conversely, ifm ∈ I +Ė p (D) satisfies the L p -RH problem determined by (ˆ ,v), then the function m(z) defined for z ∈D by Conversely, supposem ∈ I +Ė p (D) satisfies the L p -RH problem determined by (ˆ ,v) and define m(z) for z ∈ C\ˆ by (5.8). By Lemma 5.11, m ∈ I +Ė p (D). The nontangential boundary values m ± ∈ I +L p (ˆ ) satisfy Moreover, m + =m + m 0+ and m − =m − on γ + , while m + =m + and m − =m − m 0− on γ − . It follows that m + = m − v a.e. on and that m + = m − a.e. on γ . Using becomes valid for all z ∈ D. It follows that m satisfies the L p -RH problem determined by ( , v).

Conclusions
We have taken a first few steps toward developing a theory of L p -Riemann-Hilbert problems for a class of jump contours of very low regularity. More precisely, we have considered jump contours which are the union of a finite number of possibly unbounded simple Carleson curves. Several results well-known from the case of smooth contours have been shown to generalize to this more general setting. Our definition of a solution of the L p -RH problem has been novel in that it has been given directly in terms of m(z) using appropriate Smirnoff classes (and not in terms of m ± as in [11,12,16,29]). Moreover, we have established uniqueness of the L p -RH problem for n × n matrices for any 1 ≤ n ≤ p (see Theorem 5.6; for n = p = 2 this result was proved in [8,13] for sufficiently smooth contours). Overall it has been demonstrated that the theory of L p -RH problems extends virtually unimpeded to the setting of Carleson jump contours.
On the other hand, it is natural to expect the class of Carleson contours to be the largest class of contours for which a clean RH theory exists. Indeed, the Cauchy singular operator S , which is essential in the RH formalism, is known to be bounded on L p ( ), 1 < p < ∞, if and only if is a Carleson curve [4].
The presented results can be used to determine rigorously the long-time asymptotics of solutions of integrable evolution equations via the method of nonlinear steepest descent. We mention in this regard that RH problems with complicated contours that do not fit into the traditional framework arise in the analysis of initial-boundary value problems for integrable PDEs. For example, the analysis of the Degasperis-Procesi equation on the half-line leads to a RH problem with an unbounded jump contour involving nontransversal intersections, see Fig. 1.
Taking the supremum over all partitions and all closed subintervals [c, d] ⊂ I , we find (6.1).