Matrix Riemann–Hilbert problems with jumps across Carleson contours
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Abstract
We develop a theory of \(n \times n\)matrix Riemann–Hilbert problems for a class of jump contours and jump matrices of low regularity. Our basic assumption is that the contour \(\Gamma \) 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 \(L^p\)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.
Keywords
Matrix Riemann–Hilbert problem Cauchy integral Carleson contourMathematics Subject Classification
35Q15 30E25 45E051 Introduction
A Riemann–Hilbert (RH) problem consists of finding a sectionally analytic function with prescribed jumps across some given contour in the complex plane. In its simplest formulation, the problem involves a smooth simple closed contour \(\Gamma \) dividing the complex plane into an interior domain \(D_+\) and an exterior domain \(D_\), as well as a smooth ‘jump matrix’ v(z) defined for \(z \in \Gamma \). The problem consists of finding an \(n\times n\)matrixvalued function m(z) which is analytic in \(D_+ \cup D_\) and whose boundary values \(m_+\) and \(m_\) from the left and right sides of \(\Gamma \) exist, are continuous, and satisfy the jump condition \(m_+ = m_ v\) on \(\Gamma \). Uniqueness is ensured by requiring that m approaches the identity matrix at infinity.
The theory of scalar RH problems is welldeveloped in the classical setup 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 largetime 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 \(\Gamma \) 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 \(\Gamma \) 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 RHproblem 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 \(\Gamma \) 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 \(\mathcal {S}_\Gamma \) defined in Eq. (2.3) below. Indeed, this operator is the key ingredient in the SokhotskiPlemelj formulas for the boundary values of an analytic function. Since it has been proved in recent years that \(\mathcal {S}_\Gamma \) is a bounded operator on \(L^p(\Gamma )\) if and only if \(\Gamma \) 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 DegasperisProcesi equation on the halfline 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 longtime 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 wellknown to the experts (at least if the contour \(\Gamma \) is sufficiently wellbehaved), 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.
2 Preliminaries
Let \(\Gamma \subset \mathbb {C}\) be a composed curve. If \(z \in \mathbb {C}\), \(r \in (0, \infty )\), and D(z, r) denotes the open disk of radius r centered at z, then \(\Gamma \cap D(z, r)\) 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 \(\Gamma \cap D(z, r)\) is rectifiable. \(\Gamma \) is locally rectifiable if \(\Gamma \cap D(z, r)\) is rectifiable for every \(z \in \Gamma \) and every \(r \in (0,\infty )\). A composed curve \(\Gamma \) is locally rectifiable if and only if \(\Gamma \cap D(0, r)\) is rectifiable for every \(r \in (0,\infty )\).
2.1 Carleson curves
2.2 Cauchy singular operator
It was realized in the early eighties that the Carleson condition is the essential condition for ascertaining boundedness of \(\mathcal {S}_\Gamma \) in \(L^p\)spaces [7]. More precisely, if \(1< p < \infty \) and \(\Gamma \) is a composed locally rectifiable curve, then \(\mathcal {S}_\Gamma \) generates a bounded operator on \(L^p(\Gamma )\) if and only if \(\Gamma \) is Carleson. In dealing with unbounded contours, we will need a more general version of this result valid for weighted \(L^p\)spaces.
Theorem 2.1
Let \(1< p < \infty \) and let \(\Gamma \) be a composed locally rectifiable curve. Let w be a weight on \(\Gamma \). Then \(\mathcal {S}_\Gamma \) generates a bounded operator \(\tilde{\mathcal {S}}_\Gamma \) on \(L^p(\Gamma , w)\) if and only if \(\Gamma \) is Carleson and \(w \in A_p(\Gamma )\). Moreover, if \(f \in L^p(\Gamma , w)\) and \(\mathcal {S}_\Gamma \) generates a bounded operator on \(L^p(\Gamma , w)\), then the limit in (2.3) exists and \((\mathcal {S}_\Gamma f)(z) = (\tilde{\mathcal {S}}_\Gamma f)(z)\) for a.e. \(z \in \Gamma \).
Proof
See Theorem 4.15 and Remark 5.23 of [4]. \(\square \)
2.3 Smirnoff classes
2.4 Basic results on Cauchy integrals
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 \(\Gamma {\subset }\mathbb {C}\), we let \(D_+\) and \(D_\) denote the interior and exterior components of \(\hat{\mathbb {C}} {\setminus } \Gamma \).
Theorem 2.2
 (a)Let \(1 \le p < \infty \). Suppose \(\Gamma \subset \mathbb {C}\) is a rectifiable Jordan curve. If \(f \in E^p(D_+)\), then the nontangential limits of f(z) as z approaches the boundary exist a.e. on \(\Gamma \); if \(f_+(z)\) denotes the boundary function, then \(f_+ \in L^p(\Gamma )\) andIf \(f \in E^p(D_)\), then the nontangential limits of f(z) as z approaches the boundary exist a.e. on \(\Gamma \). If \(f_(z)\) denotes the boundary function, then \(f_ \in L^p(\Gamma )\) and$$\begin{aligned} (\mathcal {C} f_+)(z) = {\left\{ \begin{array}{ll} f(z), &{}\quad z \in D_+, \\ 0, &{}\quad z \in D_. \end{array}\right. } \end{aligned}$$(2.7)$$\begin{aligned} (\mathcal {C} f_)(z) = {\left\{ \begin{array}{ll} f(\infty ), &{}\quad z \in D_+, \\ f(\infty )  f(z), &{}\quad z \in D_. \end{array}\right. } \end{aligned}$$(2.8)
 (b)Let \(1< p < \infty \). Suppose \(\Gamma \) is a Carleson Jordan curve. Then the Cauchy singular operator \(\mathcal {S}_\Gamma :L^p(\Gamma ) \rightarrow L^p(\Gamma )\) defined in (2.3) satisfies \(\mathcal {S}_\Gamma ^2 = I\). Moreover, if \(h \in L^p(\Gamma )\), then$$\begin{aligned} (\mathcal {C}h)_{D_+} \in E^p(D_+), \qquad (\mathcal {C}h)_{D_} \in \dot{E}^p(D_). \end{aligned}$$
Theorem 2.2 implies that if \(\Gamma \) is a Carleson Jordan curve and \(h \in L^p(\Gamma )\) for some \(1< p < \infty \), then the left and right nontangential boundary values of \(\mathcal {C}h\), which we denote by \(\mathcal {C}_+ h\) and \(\mathcal {C}_ h\), lie in \(L^p(\Gamma )\). This allows us to view \(\mathcal {C}_\pm \) as linear operators \(\mathcal {C}_\pm :h \mapsto \mathcal {C}_\pm h\) on \(L^p(\Gamma )\).
Theorem 2.3
 The SokhotskiPlemelj formulasare valid.$$\begin{aligned} \mathcal {C}_+ = \frac{1}{2}(I + \mathcal {S}_\Gamma ), \qquad \mathcal {C}_ = \frac{1}{2}(I + \mathcal {S}_\Gamma ), \end{aligned}$$
 \(\mathcal {C}_\pm \) are complementary projections on \(L^p(\Gamma )\) in the sense thatand$$\begin{aligned} L^p(\Gamma ) = \mathcal {C}_+L^p(\Gamma ) \oplus \mathcal {C}_L^p(\Gamma ) \end{aligned}$$$$\begin{aligned} \mathcal {C}_+  \mathcal {C}_ = \mathbb {I}, \qquad \mathcal {C}_+^2 = \mathcal {C}_+, \qquad \mathcal {C}_^2 = \mathcal {C}_, \qquad \mathcal {C}_+\mathcal {C}_ = \mathcal {C}_\mathcal {C}_+ = 0. \end{aligned}$$
 If \(h = \mathcal {C}_+h  \mathcal {C}_h \in L^p(\Gamma )\), then$$\begin{aligned} (\mathcal {C}h)_{D_+} = (\mathcal {C}\mathcal {C}_+h)_{D_+} \in E^p(D_+), \qquad (\mathcal {C}h)_{D_} = (\mathcal {C}\mathcal {C}_h)_{D_} \in \dot{E}^p(D_). \end{aligned}$$

The map \(h \mapsto (\mathcal {C}h)_{D_+}\) is a bijection \(\mathcal {C}_+L^p(\Gamma ) \rightarrow E^p(D_+)\) with inverse \(f \mapsto f_+\).

The map \(h \mapsto (\mathcal {C}h)_{D_}\) is a bijection \(\mathcal {C}_L^p(\Gamma ) \rightarrow \dot{E}^p(D_)\) with inverse \(f \mapsto f_\).
3 Carleson jump contours
RH problems are conveniently formulated on the Riemann sphere \(\hat{\mathbb {C}} = \mathbb {C}\cup \infty \). In order to allow for jump contours passing through infinity, we introduce a class of curves \(\mathcal {J}\), which in addition to the rectifiable Jordan curves considered in the previous section also includes unbounded contours. Recall that \(\Gamma \subset \mathbb {C}\) is referred to as a Carleson curve if \(\Gamma \) is a locally rectifiable composed curve satisfying (2.1). We extend this notion to the Riemann sphere by calling a subset \(\Gamma \subset \hat{\mathbb {C}}\) a Carleson curve if and only if \(\Gamma \) is connected and \(\Gamma \cap \mathbb {C}\) is a Carleson curve.
3.1 The class \(\mathcal {J}\)
Let \(\mathcal {J}\) denote the collection of all subsets \(\Gamma \) of the Riemann sphere \(\hat{\mathbb {C}}\) such that \(\Gamma \) is homeomorphic to the unit circle and \(\Gamma \) is a Carleson curve. If \(\infty \notin \Gamma \), then \(\Gamma \in \mathcal {J}\) if and only if \(\Gamma \subset \mathbb {C}\) is a Carleson Jordan curve. However, \(\mathcal {J}\) also includes curves passing through infinity. In fact, the next proposition shows that \(\mathcal {J}\) is invariant under the action of the group of linear fractional transformations. This shows that \(\mathcal {J}\) is a natural extension of the family of Carleson Jordan curves in that it puts \(\infty \) on an equal footing with the other points in the Riemann sphere.
Proposition 3.1
Proof
See “Appendix 1”. \(\square \)
Remark 3.2
If \(\gamma :S^1 \rightarrow \hat{\mathbb {C}}\) is an injective \(C^1\) map such that \(\gamma '(s) \ne 0\) for all \(s \in S^1\), then \(\Gamma := \gamma (S^1)\) belongs to \(\mathcal {J}\). Indeed, being a continuous bijection from a compact space onto a Hausdorff space, \(\gamma \) is a homeomorphism \(S^1 \rightarrow \Gamma \). In view of Proposition 3.1, we may assume that \(\infty \notin \Gamma \). Then, since \(S^1\) is compact, we may cover \(S^1\) with a finite number of open sets \(\{U_j\}_1^n\) such that the restriction of \(\gamma \) to each \(U_j\) is a \(C^1\) graph; Proposition 1.1 of [4] now implies that \(\Gamma \) is Carleson.
Remark 3.3
3.2 Carleson jump contours
 (a)
\(\Gamma \cap \mathbb {C}\) is an oriented composed curve.
 (b)
\(\hat{\mathbb {C}}{\setminus }\Gamma \) is the union of two disjoint open sets \(D_+\) and \(D_\) each of which has a finite number of simply connected components in \(\hat{\mathbb {C}}\).
 (c)
\(\Gamma \) is the positively oriented boundary of \(D_+\) and the negatively oriented boundary of \(D_\), i.e. \(\Gamma = \partial D_+ = \partial D_\).
 (d)
If \(\{D_j^+\}_1^n\) and \(\{D_j^\}_1^m\) are the components of \(D_+\) and \(D_\), then \(\partial D_j^+ \in \mathcal {J}\) for \(j = 1, \ldots , n\), and \(\partial D_j^ \in \mathcal {J}\) for \(j = 1, \ldots , m\).
Example 3.4
Proposition 3.1 implies the following result.
Proposition 3.5
The family of Carleson jump contours is invariant under the action of the group of linear fractional transformations. In other words, if \(\psi : \hat{\mathbb {C}} \rightarrow \hat{\mathbb {C}}\) is given by (3.1) for some constants \(a,b,c,d \in \mathbb {C}\) with \(ad  bc \ne 0\), then \(\Gamma \) is a Carleson jump contour if and only if \(\psi (\Gamma )\) is a Carleson jump contour.
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 \(\Gamma \). 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.
3.3 Generalized Smirnoff classes
In Sect. 2.3, we defined the Smirnoff class \(E^p(D)\) for \(D = D_+\) and \(D = D_\), where \(D_+\) and \(D_\) are the domains interior and exterior to a rectifiable Jordan curve, respectively. We now extend this definition to allow for situations where D is an arbitrary finite disjoint union of domains bounded by curves in \(\mathcal {J}\).
If \(D = D_1 \cup \cdots \cup D_n\) is the union of a finite number of disjoint subsets of \(\hat{\mathbb {C}}\) each of which is bounded by a curve in \(\mathcal {J}\), we define \(E^p(D)\) and \(\dot{E}^p(D)\) as the set of functions f analytic in D such that \(f_{D_j} \in E^p(D_j)\) and \(f_{D_j} \in \dot{E}^p(D_j)\) for each j, respectively.
3.4 Properties of \(E^p(D)\) and \(\dot{E}^p(D)\)
Our definitions of the generalized Smirnoff classes \(E^p(D)\) and \(\dot{E}^p(D)\) 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
 (a)
\(f \in E^p(D)\) if and only if \(f \circ \psi ^{1} \in E^p(\psi (D))\).
 (b)
\(f \in \dot{E}^p(D)\) if and only if \(\Psi f \in \dot{E}^p(\psi (D))\) where \((\Psi f)(w) = (cw  a)^{1} f(\psi ^{1}(w))\).
Proof
 (a)
 We will prove that \(f \in E^p(D)\) if and only if \(f \circ \psi ^{1} \in E^p(\psi (D))\) whenever \(\psi (D)\) is bounded and either (i) \(\infty \in D\), (ii) D is bounded, or (iii) \(\infty \in \Gamma \). Since the linear fractional transformations form a group, this will prove (a). Case (i). Suppose \(\psi (D)\) is bounded, \(\infty \in D\), and \(f \in E^p(D)\). By the definition of \(E^p(D)\) in Sect. 2.3, there exists a sequence of rectifiable Jordan curves \(\{C_n\}_1^\infty \) in D tending to the boundary \(\Gamma \) such that (2.5) holds. It follows that \(\{\psi (C_n)\}_1^\infty \) is a sequence of rectifiable Jordan curves in \(\psi (D)\) tending to \(\psi (\Gamma )\) andIf \(c = 0\), then \(\psi '(z)\) is a finite constant. If \(c \ne 0\), then our assumption that \(\psi (D)\) is bounded implies that the point \(z = d/c\) does not belong to \(\bar{D}\); hence the function$$\begin{aligned} \sup _{n \ge 1} \int _{\psi (C_n)} f(\psi ^{1}(w))^p dw = \sup _{n \ge 1} \int _{C_n} f(z)^p \psi '(z) dz. \end{aligned}$$(3.4)is bounded on D. It follows that the righthand side of (3.4) is finite. Thus \(f \circ \psi ^{1} \in E^p(\psi (D))\). Conversely, if \(\psi (D)\) is bounded, \(\infty \in D\), and \(f \circ \psi ^{1} \in E^p(\psi (D))\), then a similar argument shows that \(f \in E^p(D)\). Case (ii). Suppose both \(\psi (D)\) and D are bounded. An argument similar to that used in Case (i) shows that \(f \in E^p(D)\) if and only if \(f \circ \psi ^{1} \in E^p(\psi (D))\). Case (iii). Suppose \(\psi (D)\) is bounded and \(\infty \in \Gamma \). Let \(z_0 \in \mathbb {C}{\setminus }(D \cup \Gamma )\). By the definition of \(E^p(D)\), \(f \in E^p(D)\) if and only if \(f \circ \varphi ^{1} \in E^p(\varphi (D))\), where \(\varphi (z) = 1/(z  z_0)\). But \(\psi \circ \varphi ^{1}\) is a linear fractional transformation mapping the bounded domain \(\varphi (D)\) onto the bounded domain \(\psi (D)\); hence Case (ii) implies that \(f \circ \varphi ^{1} \in E^p(\varphi (D))\) if and only if \(f \circ \varphi ^{1} \circ (\psi \circ \varphi ^{1})^{1} = f \circ \psi ^{1}\) belongs to \(E^p(\psi (D))\). This completes the proof of (a).$$\begin{aligned} \psi '(z) = \frac{ad  bc}{(cz + d)^2} \end{aligned}$$
 (b)

This part is a consequence of (a) and the definitions. Indeed, suppose first that \(c \ne 0\). By definition of \(\dot{E}^p(D)\), \(f \in \dot{E}^p(D)\) if and only if \(f(z), zf(z) \in E^p(D)\). Since \(E^p(D)\) is a linear space, this is the case if and only if \(f(z), (cz + d)f(z) \in E^p(D)\). Using (a) and the fact that \(cz + d = \frac{bc  ad}{cw  a}\) when \(w = \psi (z)\), the latter condition is equivalent to \(f(\psi ^{1}(w)), \frac{bc  ad}{cw  a}f(\psi ^{1}(w)) \in E^p(\psi (D))\). Using that \(E^p(D)\) is a linear space again, this holds if and only if \((cw  a)^{1} f(\psi ^{1}(w)) \in \dot{E}^p(\psi (D))\). The proof when \(c = 0\) is similar. This proves (b).
It is possible to characterize the spaces \(E^p(D)\) and \(\dot{E}^p(D)\) in terms of conditions analogous to (2.5) also when \(\infty \in \partial D\).
Lemma 3.7
 (a)\(f \in E^p(D)\) if and only if there exist curves \(\{C_n\}_1^\infty \subset \mathcal {J}\) in D, tending to \(\Gamma \) in the sense that \(C_n\) eventually surrounds each compact subset of D, such that$$\begin{aligned} \sup _{n \ge 1} \int _{C_n} z  z_0^{2} f(z)^p dz < \infty . \end{aligned}$$(3.5)
 (b)\(f \in \dot{E}^p(D)\) if and only if there exist curves \(\{C_n\}_1^\infty \subset \mathcal {J}\) in D, tending to \(\Gamma \) in the sense that \(C_n\) eventually surrounds each compact subset of D, such that$$\begin{aligned} \sup _{n \ge 1} \int _{C_n} z  z_0^{p2} f(z)^p dz < \infty . \end{aligned}$$(3.6)
Proof
Lemma 3.8
 (a)
\(\dot{E}^p(D) \subset \dot{E}^r(D)\) whenever \(1 \le r \le p < \infty \).
 (b)
Suppose \(p,q,r \in [1, \infty )\) satisfy \(1/p + 1/q = 1/r\). If \(f \in \dot{E}^p(D)\) and \(g \in \dot{E}^q(D)\), then the functions zf(z)g(z) and fg belong to \(\dot{E}^r(D)\).
Proof
 (a)

The result is immediate from the definitions if \(\infty \notin \Gamma \). Thus suppose \(\infty \in \Gamma \). Let \(z_0 \in \mathbb {C}{\setminus }\Gamma \) and define \(\varphi \) as in (3.3). If \(f \in \dot{E}^p(D)\), then Proposition 3.6 implies that \(w^{1}f(\varphi ^{1}(w)) \in \dot{E}^p(\varphi (D))\); since \(\dot{E}^p(\varphi (D)) \subset \dot{E}^r(\varphi (D))\), another application of Proposition 3.6 shows that \(f \in \dot{E}^r(D)\). This proves (a).
 (b)
 Suppose \(p,q,r \in [1, \infty )\) satisfy \(1/p + 1/q = 1/r\). Let \(f \in \dot{E}^p(D)\) and \(g \in \dot{E}^q(D)\). We first suppose D is bounded. Then there exist sequences of rectifiable Jordan curves \(\{A_n\}_1^\infty \) and \(\{B_n\}_1^\infty \) in D tending to the boundary of D such thatWithout loss of generality, we may assume that \(A_n = B_n = C_n\) for all \(n \ge 1\) where \(\{C_n\}_1^\infty \) 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,$$\begin{aligned} \sup _{n \ge 1} \Vert f\Vert _{L^p(A_n)}< \infty , \qquad \sup _{n \ge 1} \Vert g\Vert _{L^q(B_n)} < \infty . \end{aligned}$$Hence \(fg \in E^r(D) = \dot{E}^r(D)\). This proves (b) when D is bounded. If D is unbounded, then pick \(z_0 \in \mathbb {C}{\setminus }(D \cup \Gamma )\) and let \(\varphi (z) = 1/(z  z_0)\). By Proposition 3.6, \(w^{1} f(\varphi ^{1}(w)) \in \dot{E}^p(\varphi (D))\) and \(w^{1} g(\varphi ^{1}(w)) \in \dot{E}^q(\varphi (D))\). Hence, by the preceding paragraph, \(w^{2}(fg)(\varphi ^{1}(w)) = w^{1}(fg/\varphi )(\varphi ^{1}(w)) \in \dot{E}^r(\varphi (D))\). Since \(\varphi (D)\) is bounded, we also have \(w^{1}(fg)(\varphi ^{1}(w)) \in \dot{E}^r(\varphi (D))\). Using Proposition 3.6 again, we conclude that \(fg/\varphi , fg \in \dot{E}^r(D)\). Part (b) follows.$$\begin{aligned} \sup _{n \ge 1} \Vert fg\Vert _{L^r(C_n)} \le \sup _{n \ge 1} \Vert f\Vert _{L^p(C_n)} \Vert g\Vert _{L^q(C_n)} < \infty . \end{aligned}$$
3.5 The spaces \(\dot{L}^p(\Gamma )\)
Lemma 3.9
 (a)The map \(\Phi \) defined for \(h \in \dot{L}^p(\Gamma )\) byis a bijection \(\dot{L}^p(\Gamma ) \rightarrow L^p(\varphi (\Gamma ))\) and$$\begin{aligned} (\Phi h)(w) = w^{1} h(\varphi ^{1}(w)) \end{aligned}$$(3.9)for all \(h \in \dot{L}^p(\Gamma )\).$$\begin{aligned} \Vert \cdot  z_0^{1\frac{2}{p}} h\Vert _{L^p(\Gamma )} = \Vert \Phi h\Vert _{L^p(\varphi (\Gamma ))} \end{aligned}$$
 (b)If \(h \in \dot{L}^p(\Gamma )\), thenwhere \(\Psi \) acts on a function \(f:\hat{\mathbb {C}}{\setminus } \Gamma \rightarrow \mathbb {C}\) by \((\Psi f)(w) = w^{1} f(\varphi ^{1}(w))\).$$\begin{aligned} (\mathcal {C}_\Gamma h)(z) = (\Psi ^{1} \mathcal {C}_{\varphi (\Gamma )} \Phi h)(z) \quad \text {for all }z \in \mathbb {C}{\setminus } \Gamma , \end{aligned}$$(3.10)
Proof
 (a)
 If \(h \in \dot{L}^p(\Gamma )\), then the change of variables \(w = \varphi (z)\) impliesWe infer that \(\Phi \) is a bijection \(\dot{L}^p(\Gamma ) \rightarrow L^p(\varphi (\Gamma ))\) with inverse given by \((\Phi ^{1}H)(z) = \varphi (z) H(\varphi (z))\). This proves (a).$$\begin{aligned} \Vert zz_0^{1\frac{2}{p}} h(z)\Vert _{L^p(\Gamma )}^p&= \int _\Gamma zz_0^{p2} h(z)^p dz\\&= \int _{\varphi (\Gamma )} w^{1} h(\varphi ^{1}(w))^p dw = \Vert \Phi h\Vert _{L^p(\varphi (\Gamma ))}^p. \end{aligned}$$
 (b)
 If \(h \in \dot{L}^p(\Gamma )\), \(z \in \mathbb {C}{\setminus } \Gamma \) and \(w = \varphi (z)\), thenwhich proves (b).$$\begin{aligned} (\mathcal {C}_{\varphi (\Gamma )} \Phi h)(w)&= \frac{1}{2\pi i} \int _{\varphi (\Gamma )} \frac{(h \circ \varphi ^{1})(w')}{w'  w}\frac{dw'}{w'} \nonumber \\&= \frac{z  z_0}{2\pi i} \int _{\Gamma } \frac{h(z')}{z'  z} dz' = (zz_0) (\mathcal {C}_\Gamma h)(z) = \Psi (\mathcal {C}_\Gamma h)(w), \end{aligned}$$(3.11)
Lemma 3.10
 (a)
\(\dot{L}^p(\Gamma ) \subset \dot{L}^r(\Gamma )\) whenever \(1 \le r \le p < \infty \).
 (b)
Suppose \(p,q,r \in [1, \infty )\) satisfy \(1/p + 1/q = 1/r\). If \(f \in \dot{L}^p(\Gamma )\) and \(g \in \dot{L}^q(\Gamma )\), then the functions zf(z)g(z) and fg belong to \(\dot{L}^r(\Gamma )\).
Proof
 (a)

The result is immediate from the definitions if \(\infty \notin \Gamma \). Thus suppose \(\infty \in \Gamma \). Let \(z_0 \in \mathbb {C}{\setminus } \Gamma \) and define \(\varphi \) as in (3.3). If \(h \in \dot{L}^p(\Gamma )\), then Lemma 3.9 implies that \(w^{1}h(\varphi ^{1}(w)) \in \dot{L}^p(\varphi (\Gamma ))\); since \(\dot{L}^p(\varphi (\Gamma )) \subset \dot{L}^r(\varphi (\Gamma ))\), another application of Lemma 3.9 shows that \(h \in \dot{L}^r(\Gamma )\). This proves (a).
 (b)
 Suppose \(p,q,r \in [1, \infty )\) satisfy \(1/p + 1/q = 1/r\). Let \(f \in \dot{L}^p(\Gamma )\) and \(g \in \dot{L}^q(\Gamma )\). Suppose \(\infty \notin \Gamma \). Then, by Hölder’s inequality,Hence \(fg \in L^r(\Gamma ) = \dot{L}^r(\Gamma )\). This gives (b) when \(\infty \notin \Gamma \). If \(\infty \in \Gamma \), then pick \(z_0 \in \mathbb {C}{\setminus } \Gamma \) and define \(\varphi \) as in (3.3). By Lemma 3.9, \(w^{1} f(\varphi ^{1}(w)) \in L^p(\varphi (\Gamma ))\) and \(w^{1} g(\varphi ^{1}(w)) \in L^q(\varphi (\Gamma ))\). Hence, by the preceding paragraph, \(w^{2}(fg)(\varphi ^{1}(w)) = w^{1}(fg/\varphi )(\varphi ^{1}(w)) \in L^r(\varphi (\Gamma ))\). Since \(\varphi (\Gamma )\) is bounded, we also have \(w^{1}(fg)(\varphi ^{1}(w)) \in L^r(\varphi (\Gamma ))\). Using Lemma 3.9 again, we conclude that \(fg/\varphi , fg \in \dot{L}^r(\Gamma )\). Part (b) follows.$$\begin{aligned} \Vert fg\Vert _{L^r(\Gamma )} \le \Vert f\Vert _{L^p(\Gamma )} \Vert g\Vert _{L^q(\Gamma )} < \infty . \end{aligned}$$
3.6 The Cauchy singular operator
Theorem 2.1 can be used to establish boundedness of the Cauchy singular operator \(\mathcal {S}_\Gamma \) on \(\dot{L}^p(\Gamma )\) if \(1< p < \infty \) and \(\Gamma \) is Carleson.
Proposition 3.11
Let \(1< p < \infty \) and let \(\Gamma \) be a Carleson curve. Then \(\mathcal {S}_\Gamma \) generates a bounded operator \(\tilde{\mathcal {S}}_\Gamma \) on \(\dot{L}^p(\Gamma )\). Moreover, if \(h \in \dot{L}^p(\Gamma )\), then the limit in (2.3) exists and \((\mathcal {S}_\Gamma h)(z) = (\tilde{\mathcal {S}}_\Gamma h)(z)\) for a.e. \(z \in \Gamma \).
Proof
Let \(z_0 \notin \Gamma \) and let \(w(z) = z  z_0^{1  \frac{2}{p}}\). The result follows from Theorem 2.1 if we can show that \(w \in A_p(\Gamma )\). If \(p =2\), this is an immediate consequence of \(\Gamma \) being Carleson. Thus suppose \(p \ne 2\).
Our next objective is to determine how \(\mathcal {S}_\Gamma \) transforms under the change of variables \(w = 1/(z  z_0)\). We need the following lemma.
Lemma 3.12
Proof
Proposition 3.13
Proof
We will show that \(\mathcal {S}_\Gamma h = \Phi ^{1} \mathcal {S}_{\varphi (\Gamma )} \Phi h\) a.e. on \(\Gamma \) whenever \(h \in C_0^\infty (\Gamma )\). Since \(C_0^\infty (\Gamma )\) is dense in \(\dot{L}^p(\Gamma )\) and the operators \(\mathcal {S}_\Gamma \) and \(\Phi ^{1} \mathcal {S}_{\varphi (\Gamma )} \Phi \) are bounded on \(\dot{L}^p(\Gamma )\) by Lemma 3.9 and Proposition 3.11, this will prove (3.24).
4 Cauchy integrals over Carleson jump contours
The following two theorems generalize Theorems 2.2 and 2.3 to the case where \(\Gamma \) is a general Carleson jump contour.
Theorem 4.1
 (a)Let \(1 \le p < \infty \). If \(f \in \dot{E}^p(D_+)\), then the nontangential limits of f(z) as z approaches the boundary exist a.e. on \(\Gamma \); if \(f_+(z)\) denotes the boundary function, then \(f_+ \in \dot{L}^p(\Gamma )\) andIf \(f \in \dot{E}^p(D_)\), then the nontangential limits of f(z) as z approaches the boundary exist a.e. on \(\Gamma \). If \(f_(z)\) denotes the boundary function, then \(f_ \in \dot{L}^p(\Gamma )\) and$$\begin{aligned} (\mathcal {C} f_+)(z) = {\left\{ \begin{array}{ll} f(z), &{} z \in D_+, \\ 0, &{} z \in D_. \end{array}\right. } \end{aligned}$$(4.1)In particular, \(f = \mathcal {C}(f_+  f_)\) for all \(f \in \dot{E}^p(D_+ \cup D_)\).$$\begin{aligned} (\mathcal {C} f_)(z) = {\left\{ \begin{array}{ll} 0, &{} z \in D_+, \\  f(z), &{} z \in D_. \end{array}\right. } \end{aligned}$$(4.2)
 (b)Let \(1< p < \infty \). Then the Cauchy singular operator \(\mathcal {S}_\Gamma :\dot{L}^p(\Gamma ) \rightarrow \dot{L}^p(\Gamma )\) defined in (2.3) satisfies \(\mathcal {S}_\Gamma ^2 = I\). Moreover, if \(h \in \dot{L}^p(\Gamma )\), then$$\begin{aligned} \mathcal {C}h_{D_+} \in \dot{E}^p(D_+), \qquad \mathcal {C}h_{D_} \in \dot{E}^p(D_). \end{aligned}$$(4.3)
Theorem 4.1 implies that if \(\Gamma \) is a Carleson jump contour and \(h \in \dot{L}^p(\Gamma )\) for some \(1< p < \infty \), then the left and right nontangential boundary values of \(\mathcal {C}h\), which we denote by \(\mathcal {C}_+ h\) and \(\mathcal {C}_ h\), lie in \(\dot{L}^p(\Gamma )\). This allows us to define two linear operators \(\mathcal {C}_\pm :h \mapsto \mathcal {C}_\pm h\) on \(\dot{L}^p(\Gamma )\).
Theorem 4.2
 The SokhotskiPlemelj formulasare valid.$$\begin{aligned} \mathcal {C}_+ = \frac{1}{2}(I + \mathcal {S}_\Gamma ), \qquad \mathcal {C}_ = \frac{1}{2}(I + \mathcal {S}_\Gamma ), \end{aligned}$$(4.4)
 \(\mathcal {C}_\pm \) are orthogonal projections on \(\dot{L}^p(\Gamma )\) in the sense thatand$$\begin{aligned} \dot{L}^p(\Gamma ) = \mathcal {C}_+\dot{L}^p(\Gamma ) \oplus \mathcal {C}_\dot{L}^p(\Gamma ) \end{aligned}$$$$\begin{aligned} \mathcal {C}_+  \mathcal {C}_ = \mathbb {I}, \qquad \mathcal {C}_+^2 = \mathcal {C}_+, \qquad \mathcal {C}_^2 = \mathcal {C}_, \qquad \mathcal {C}_+\mathcal {C}_ = \mathcal {C}_\mathcal {C}_+ = 0. \end{aligned}$$
 If \(h = \mathcal {C}_+h  \mathcal {C}_h \in \dot{L}^p(\Gamma )\), then$$\begin{aligned} (\mathcal {C}h)_{D_+} = (\mathcal {C}\mathcal {C}_+h)_{D_+} \in \dot{E}^p(D_+), \qquad (\mathcal {C}h)_{D_} = (\mathcal {C}\mathcal {C}_h)_{D_} \in \dot{E}^p(D_). \end{aligned}$$(4.5)

The map \(h \mapsto (\mathcal {C}h)_{D_+}\) is a bijection \(\mathcal {C}_+\dot{L}^p(\Gamma ) \rightarrow \dot{E}^p(D_+)\) with inverse \(f \mapsto f_+\).

The map \(h \mapsto (\mathcal {C}h)_{D_}\) is a bijection \(\mathcal {C}_\dot{L}^p(\Gamma ) \rightarrow \dot{E}^p(D_)\) with inverse \(f \mapsto f_\).
In the special case of a jump contour \(\Gamma \) consisting of a single rectifiable Jordan curve, Theorems 4.1 and 4.2 reduce to Theorems 2.2 and 2.3, respectively.
4.1 Proof of Theorem 4.1
4.1.1 Proof of (a)
Suppose first that \(\infty \notin \Gamma \), so that \(\Gamma \subset \mathbb {C}\) is bounded. Let \(f \in \dot{E}^p(D_+)\). Represent \(\Gamma \) as the union of finitely many arcs each pair of which have at most endpoints in common. If \(z \in \Gamma \) is not one of these finitely many endpoints, then z belongs to \(\partial D_j^+\) for exactly one component \(D_j^+\) of \(D_+\). Since Theorem 2.2 implies that \(f_{D_j^+}\) has nontangential limits a.e. on \(\partial D_j^+\), it follows that f has nontangential limits a.e. on \(\Gamma \). Another application of Theorem 2.2 shows that \(f_+_{\partial D_j^+} \in L^p(\partial D_j^+)\) for each j. Hence \(f_+ \in L^p(\Gamma ) = \dot{L}^p(\Gamma )\).
4.1.2 A convergence lemma
For the proof of (b), we need the following lemma.
Lemma 4.3
 (i)
If \(\{f_n\}_1^\infty \) is a sequence of functions in \(E^p(D_+)\) such that \(f_{n+} \rightarrow h\) in \(L^p(\Gamma )\), then there exists a function \(f \in E^p(D_+)\) such that \(f_n \rightarrow f\) uniformly on compact subsets of \(D_+\) and \(f_+ = h\).
 (ii)
If \(\{f_n\}_1^\infty \) is a sequence of functions in \(E^p(D_)\) such that \(f_{n} \rightarrow h\) in \(L^p(\Gamma )\), then there exists a function \(f \in E^p(D_)\) such that \(f_n \rightarrow f\) uniformly on compact subsets of \(D_\) and \(f_ = h\). If \(\{f_n\}_1^\infty \subset \dot{E}^p(D_)\), then \(f \in \dot{E}^p(D_)\).
Proof
4.1.3 Proof of (b)
Equation (4.7) implies that \(\mathcal {S}_\Gamma ^2 r = r\) for every \(r \in R(\Gamma )\). Since \(R(\Gamma )\) is dense in \(L^p(\Gamma )\) (see Lemma 9.14 in [4]), it follows that \(\mathcal {S}_\Gamma ^2h = h\) for every \(h \in L^p(\Gamma )\).
4.2 Proof of Theorem 4.2
The Sokhotski–Plemelj formulas (4.4) together with the fact that \(\mathcal {S}_\Gamma ^2 = I\) immediately imply that \(\mathcal {C}_\pm \) are bounded orthogonal projections on \(\dot{L}^p(\Gamma )\).
5 Riemann–Hilbert problems
With Theorems 4.1 and 4.2 at our disposal, we can introduce a notion of \(L^p\)RH problem for Carleson jump contours. Throughout this section, \(\Gamma \subset \hat{\mathbb {C}}\) will denote a Carleson jump contour, \(D_\pm \subset \hat{\mathbb {C}}\) will denote the associated open sets such that \(\partial D_+ = \partial D_ = \Gamma \), and we will assume that \(1< p <\infty \). We let \(D = D_+ \cup D_\).
5.1 Definition
Let \(n \ge 1\) be an integer. Given an \(n \times n\)matrix valued function \(v: \Gamma \rightarrow GL(n, \mathbb {C})\), we define a solution of the \(L^p\)RH problem determined by \((\Gamma , v)\) to be an \(n \times n\)matrix valued function \(m \in I + \dot{E}^p(D)\) such that the nontangential boundary values \(m_\pm \) satisfy \(m_+ = m_ v\) a.e. on \(\Gamma \).
5.2 Properties of \(m_\pm \)
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 \(L^p\)RH problem if and only if the boundary functions \(m_+\) and \(m_\) satisfy the properties (RH1)(RH2) below.
Proposition 5.1
 (RH1)There exists a function \(h \in \dot{L}^p(\Gamma )\) such that$$\begin{aligned} m_\pm  I = \mathcal {C}_\pm h \quad \text {in}\quad \dot{L}^p(\Gamma ). \end{aligned}$$(5.1)
 (RH2)
\(m_+ = m_ v\) a.e. on \(\Gamma \).
Proof
Theorem 4.1 implies that if \(m \in I + \dot{E}^p(D)\) satisfies the \(L^p\)RH problem determined by \((\Gamma , v)\), then \(m_\pm \in I + \dot{L}^p(\Gamma )\) and \(m = I + \mathcal {C}(m_+  m_)\). Thus (RH1) is satisfied with \(h = m_+  m_\). The property (RH2) holds by definition.
Conversely, suppose \(m_\pm \in I + \dot{L}^p(\Gamma )\) satisfy (RH1) and (RH2). By (RH1), \(m_\pm \in I + \mathcal {C}_\pm \dot{L}^p(\Gamma )\). Thus, Theorems 4.1 and 4.2 imply that \(m_\pm \) are the nontangential boundary values of the function m defined by \(m = I + \mathcal {C}(m_+  m_) \in I + \dot{E}^p(D)\). It follows that m satisfies the \(L^p\)RH problem determined by \((\Gamma , v)\). \(\square \)
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 \(m_\pm \in I + L^p(\Gamma )\) satisfying (RH1)(RH2) (or properties very similar to (RH1)(RH2)); the associated function m(z) is then referred to as the ‘extension of \(m_\pm \)’. 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 setup provided by the spaces \(\dot{L}^p(\Gamma )\) and \(\dot{E}^p(D)\), the definitions in terms of m and \(m_\pm \) are equivalent.
Remark 5.3
Condition (RH1) is equivalent to the condition that \(m_\pm \in I + \mathcal {C}_\pm \dot{L}^p(\Gamma )\).
5.3 Uniqueness results
We will show that the solution of the \(L^p\)RH problem determined by \((\Gamma , v)\) is unique provided that \(\det v = 1\) and \(n \le p\).
Lemma 5.4
Suppose \(v: \Gamma \rightarrow GL(n, \mathbb {C})\). Let \(1< p < \infty \) and define q by \(1/p + 1/q =1\). Let \(m, \tilde{m} \in I + \dot{E}^p(D)\) be two solutions of the \(L^p\)RH problem determined by \((\Gamma ,v)\). If \(m^{1} \in I + \dot{E}^q(D)\), then \(m(z) = \tilde{m}(z)\) for all \(z \in D\).
Proof
Remark 5.5
The assumption in Lemma 5.4 that \(m^{1} \in I + \dot{E}^q(D)\) implies that \(m_\pm \) deliver a socalled \(L^p\)canonical factorization of v; the uniqueness of the latter is known, see [18, 24].
Theorem 5.6
 (a)
If m is a solution of the \(L^p\)RH problem determined by \((\Gamma ,v)\), then \(\det m(z) = 1\) for all \(z \in D\).
 (b)
The solution of the \(L^p\)RH problem determined by \((\Gamma ,v)\) is unique if it exists.
Proof
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 \(\Gamma \) and any \(1 \le n \le p\). As an application, we note that the case \(n=3\) is relevant for the \(3 \times 3\)matrix RH problem associated with the Degasperis–Procesi equation, see Fig. 1.
5.4 A singular integral equation
Proposition 5.8
Given \(v^\pm : \Gamma \rightarrow GL(n, \mathbb {C})\), let \(v = (v^)^{1}v^+\), \(w^+ = v^+  I\), and \(w^ = I  v^\). Suppose \(v^\pm , (v^\pm )^{1} \in I + \dot{L}^p(\Gamma ) \cap L^\infty (\Gamma )\). If \(m \in I + \dot{E}^p(D)\) satisfies the \(L^p\)RH problem determined by \((\Gamma , v)\), then \(\mu = m_+ (v^+)^{1} = m_ (v^)^{1} \in I + \dot{L}^p(\Gamma )\) satisfies (5.3). Conversely, if \(\mu \in I + \dot{L}^p(\Gamma )\) satisfies (5.3) , then \(m = I + \mathcal {C}(\mu (w^+ + w^)) \in I + \dot{E}^p(D)\) satisfies the \(L^p\)RH problem determined by \((\Gamma , v)\).
Proof
5.5 Fredholm properties
Given \(v: \Gamma \rightarrow GL(n, \mathbb {C})\), we define a solution of the homogeneous \(L^p\)RH problem determined by \((\Gamma , v)\) to be an \(n \times n\)matrix valued function \(m \in \dot{E}^p(D)\) such that \(m_+ = m_ v\) a.e. on \(\Gamma \).
Lemma 5.9
 (a)
The map \(I  \mathcal {C}_w: \dot{L}^p(\Gamma ) \rightarrow \dot{L}^p(\Gamma )\) is bijective.
 (b)
The \(L^p\)RH problem determined by \((\Gamma , v)\) has a unique solution.
 (c)
The homogeneous \(L^p\)RH problem determined by \((\Gamma , v)\) has only the zero solution.
 (d)
The map \(I  \mathcal {C}_w: \dot{L}^p(\Gamma ) \rightarrow \dot{L}^p(\Gamma )\) is injective.
Proof
Let \(C(\Gamma )\) denote the set of restrictions to \(\Gamma \) of continuous functions \(\hat{\mathbb {C}} \rightarrow \mathbb {C}\). If \(\Gamma \subset \hat{\mathbb {C}}\) is given the subspace topology, Tietze’s extension theorem implies that \(C(\Gamma )\) coincides with the set of continuous functions \(\Gamma \rightarrow \mathbb {C}\). We will show that if \(w^\pm \in C(\Gamma )\) then the operator \(I  \mathcal {C}_w\) is Fredholm. If, in addition, \(w^\pm \) are nilpotent, the Fredholm index of this operator is zero, so that all four statements (a)(d) of Lemma 5.9 are equivalent.
For a Banach space X, let \(\mathcal {K}(X) \subset \mathcal {B}(X)\) denote the set of compact operators on X. The set of Fredholm operators \(\mathcal {F}(X)\) is open in \(\mathcal {B}(X)\) and the index map \({{\mathrm{Ind}}}:\mathcal {F}(X) \rightarrow \mathbb {Z}\) is constant on the connected components of \(\mathcal {F}(X)\). If \(X = \dot{L}^p(\Gamma )\), we define \(\mathcal {B}(X)\), \(\mathcal {K}(X)\), and \(\mathcal {F}(X)\) as the set of bounded, compact, and Fredholm operators on \(L^p(\Gamma , w)\) where \(w(z) = z  z_0^{1  \frac{2}{p}}\) and \(z_0\) is any point of \(\mathbb {C}{\setminus } \Gamma \).
Theorem 5.10
 (a)
The operator \(I  \mathcal {C}_w:\dot{L}^p(\Gamma ) \rightarrow \dot{L}^p(\Gamma )\) is Fredholm.
 (b)
If \(w^\pm \) are nilpotent matrices, then \(I  \mathcal {C}_w\) has Fredholm index zero; in this case, each of the four statements (a)(d) of Lemma 5.9 implies the other three.
Proof
Since \(\Gamma \subset \hat{\mathbb {C}}\) is compact, there exists a c such that \(\det v^\pm  \ge c > 0\) on \(\Gamma \). Thus \((v^\pm )^{1} \in C(\Gamma )\). Let \(\tilde{w}^+ = (v^+)^{1}  I\) and \(\tilde{w}^ = I  (v^)^{1}\). Then \(\mathcal {C}_{w}\) and \(\mathcal {C}_{\tilde{w}}\) are bounded \(\dot{L}^p(\Gamma ) \rightarrow \dot{L}^p(\Gamma )\).
Assume first that \(\infty \notin \Gamma \).
5.6 Reversal of subcontours
5.7 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.
Lemma 5.11
Let D be the union of any number of components of \(\hat{\mathbb {C}} {\setminus } \Gamma \), where \(\Gamma \) is a Carleson jump contour. Let \(E^\infty (D)\) denote the space of bounded analytic functions in D. If \(f \in \dot{E}^p(D)\) and \(g \in E^\infty (D)\), then \(fg \in \dot{E}^p(D)\).
Proof
The result is immediate when \(\infty \notin \Gamma \). The case of \(\infty \in \Gamma \) can be reduced to the case of \(\infty \notin \Gamma \) by means of Proposition 3.6. \(\square \)
Let \(\hat{\Gamma } = \Gamma \cup \gamma \) denote the union of the Carleson jump contour \(\Gamma \) and a curve \(\gamma \in \mathcal {J}\), see Figs. 5 and 6. Suppose that, reversing the orientation on a subcontour if necessary, \(\hat{\Gamma }\) is a Carleson jump contour. To be definite, we henceforth fix an orientation on the contour \(\hat{\Gamma }\) which turns it into a Carleson jump contour, and we endow the contours \(\Gamma \) and \(\gamma \) with the orientations they inherit as subsets of \(\hat{\Gamma }\). Then \(\Gamma \) is a Carleson jump contour up to reorientation; we define a solution of the \(L^p\)RH problem determined by \((\Gamma , v)\) as in Sect. 5.6.
Let \(B_+\) and \(B_\) denote the two components of \(\hat{\mathbb {C}} {\setminus } \gamma \). Without loss of generality, we may assume that \(\infty \in \bar{B}_\). Let \(\hat{D}_\pm \) be the open sets such that \(\hat{\mathbb {C}} {\setminus } \hat{\Gamma } = \hat{D}_+ \cup \hat{D}_\) and \(\partial \hat{D}_+ =  \partial \hat{D}_ = \hat{\Gamma }\). Let \(U_\pm = \hat{D}_\pm \cap B_+\). Let \(\hat{D} = \hat{D}_+ \cup \hat{D}_\) and \(U = U_+ \cup U_\). Let \(\gamma _+\) and \(\gamma _\) be the parts of \(\gamma \) that belong to the boundary of \(U_+\) and \(U_\), respectively. The orientations of \(\gamma _\pm \) are such that \(B_+\) lies to the left of \(\gamma _+\), whereas \(B_+\) lies to the right of \(\gamma _\).
Theorem 5.12
Proof
6 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 \(\Gamma \) which are the union of a finite number of possibly unbounded simple Carleson curves. Several results wellknown 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 \(m_\pm \) as in [11, 12, 16, 29]). Moreover, we have established uniqueness of the \(L^p\)RH problem for \(n\times n\) matrices for any \(1 \le n \le p\) (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 \(\mathcal {S}_\Gamma \), which is essential in the RH formalism, is known to be bounded on \(L^p(\Gamma )\), \(1< p < \infty \), if and only if \(\Gamma \) is a Carleson curve [4].
The presented results can be used to determine rigorously the longtime 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 initialboundary value problems for integrable PDEs. For example, the analysis of the Degasperis–Procesi equation on the halfline leads to a RH problem with an unbounded jump contour involving nontransversal intersections, see Fig. 1.
Notes
Acknowledgements
The author is grateful to Prof. I. M. Spitkovsky for valuable remarks on a first version of the manuscript. Support is acknowledged from the EPSRC, UK, the European Research Council, Consolidator Grant No. 682537, the Swedish Research Council, Grant No. 201505430, and the Göran Gustafsson Foundation, Sweden.
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