On the Riemann-Hilbert problem in multiply connected domains

Abstract We proved the existence of multivalent solutions with the infinite number of branches for the Riemann-Hilbert problem in the general settings of finitely connected domains bounded by mutually disjoint Jordan curves, measurable coefficients and measurable boundary data. The theorem is formulated in terms of harmonic measure and principal asymptotic values. It is also given the corresponding reinforced criterion for domains with rectifiable boundaries stated in terms of the natural parameter and nontangential limits. Furthermore, it is shown that the dimension of the spaces of these solutions is infinite.


Introduction
This note is a continuation of the paper [1] where the Riemann-Hilbert problem was solved in these general settings for simply connected domains. In the present paper, on the basis of [1] and a theorem due to Poincare, see e.g. Section VI.1 in [2], it is given a solution of the problem for finitely connected domains.
Recall that boundary value problems for analytic functions are due to the Riemann dissertation (1851), also to works of Hilbert (1904Hilbert ( , 1912Hilbert ( , 1924 and Poincaré (1910). The Riemann dissertation contained a general setting of a problem on finding analytic functions with a connection between their real and imaginary parts on the boundary.
The first concrete problem of such a type has been proposed by Hilbert (1904) and called by the Hilbert problem or the Riemann-Hilbert problem. That consists in finding an analytic function f inside of a domain bounded by a rectifiable Jordan curve C with the boundary condition where it was assumed by him that the functions and ' are continuously differentiable with respect to the natural parameter s on C and, moreover, j j ¤ 0 everywhere on C . Hence without loss of generality one can assume that j j Á 1 on C . The first way for solving this problem based on the theory of singular integral equations was proposed by Hilbert (1904), see [3]. This attempt was not quite successful because the theory of singular integral equations had not been developed enough at that time yet. However, just that way became the main approach in this research direction, see e.g. the monographs [4][5][6]. In particular, the existence of solutions to this problem was in that way proved for Hölder continuous and ', see e.g. [4]. But subsequent weakening conditions on and ' in this way led to strengthening conditions on the contour C , say to the Lyapunov curves or to the Radon condition of bounded rotation or even to smooth curves.
However, Hilbert (1905) has solved his problem with the above settings to (1) in the second way based on the reduction it to solving the corresponding two Dirichlet problems, see e.g. [7]. It has been recently shown in [1] that the latter approach makes possible to obtain perfectly general results in the problem for the arbitrary Jordan domains with coefficients and boundary data ' that are only measurable with respect to the harmonic measure.
The key was the following Gehring result on the Dirichlet problem for harmonic functions: if ' W R ! R is 2 -periodic, measurable and finite a.e. with respect to the Lebesgue measure, then there is a harmonic function in the unit disk D D fz 2 C W jzj < 1g such that u.z/ ! '.#/ for a.e. # as z ! e i# along any nontangential path, see [8], see also [9]. But the way of the reduction of the Riemann-Hilbert problem to the corresponding 2 Dirichlet problems was original in [1].

The case of circular domains
Let us start from the simplest kind of multiply connected domains. Recall that a domain D in C D C [ f1g is called circular if its boundary consists of finite number of mutually disjoint circles and points. We call such a domain nondegenerate if its boundary consists only of circles. along any nontangential path to a.e. 2 @D .
Proof. Indeed, by the Poincare theorem, see e.g. Theorem VI.1 in [2], there is a locally conformal mapping g of the unit disc D D fz 2 C W jzj < 1g onto D . Let h W D ! D be the corresponding multivalent analytic function that is inverse to g. D without a finite number of cuts is simply connected and hence h has there only single-valued branches that are extended to the boundary by the Caratheodory theorem. By Section VI.2 in [2], @D without a countable set of its points consists of a countable collection of arcs every of which is a one-to-one image of a circle in @D without its one point under every extended branch of h. Note that by the reflection principle g is conformally extended into a neighborhood of every such arc and, thus, nontangential paths to its points go into nontangential paths to the corresponding points of circles in @D and inversely.
Setting ƒ D ı g andˆD ' ı g with the extended g on the given arcs of @D we obtain measurable functions on @D. Thus, by Theorem 2.1 in [1] there exist analytic functions along any nontangential path to a.e. Á 2 @D. By the above arguments, we see that f D F ı h are desired multivalent analytic solutions of (2).
In particular, choosing Á 1 in (2), we obtain the following statement.

The case of rectifiable Jordan curves
To resolve the Riemann-Hilbert problem in the case of domains bounded by a finite number of rectifiable Jordan curves we should extend to this case the known results of Caratheodory (1912), Lindelöf (1917), F. and M. Riesz (1916) and Lavrentiev (1936) for Jordan's domains.
Lemma 3.1. Let D be a bounded domain in C whose boundary components are Jordan curves, D be a bounded nondegenerate circular domain in C and let ! W D ! D be a conformal mapping. Then (i) ! can be extended to a homeomorphism of D onto D ; (ii) arg OE!. / !.z/ arg OE z ! const as z ! whenever @D has a tangent at 2 @D; Proof. (i) Indeed, we are able to transform D into a simply connected domain D through a finite sequence of cuts. Thus, we come to the desired conclusion applying the Caratheodory theorems to simply connected domains D and D WD ! 1 .D /, see e.g. Theorem 9.4 in [10] and Theorem II.C.1 in [11].
(ii) In the construction from the previous item, we may assume that the point is not the end of the cuts in D generated by the cuts in D under the extended mapping ! 1 . Thus, we come to the desired conclusion twice applying the Caratheodory theorems, the reflection principle for conformal mappings and the Lindelöf theorem for the Jordan domains, see e.g. Theorem II.C.2 in [11].
Points (iii) and (iv) are proved similarly to the last item on the basis of the corresponding results of F. and M. Riesz and Lavrentiev for Jordan domains with rectifiable boundaries, see e.g. Theorem II.D.2 in [11], and [12], see also the point III.1.5 in [13].
Theorem 3.2. Let D be a bounded multiply connected domain in C whose boundary components are rectifiable Jordan curves and W @D ! C, j . /j Á 1, and ' W @D ! R be measurable functions with respect to the natural parameter on @D. Then there exist multivalent analytic functions f W D ! C with the infinite number of branches such that along any nontangential path with respect to the natural parameters of the boundary components of D.
Proof. This case is reduced to the case of a bounded nondegenerate circular domain D in the following way. First, there is a conformal mapping ! of D onto a circular domain D , see e.g. Theorem V.6.2 in [2]. Note that D is not degenerate because isolated singularities of conformal mappings are removable due to the well-known Weierstrass theorem, see e.g. Theorem 1.2 in [10]. Without loss of generality, we may assume that D is bounded. By point (i) in Lemma 3.1 ! can be extended to a homeomorphisms of D onto D . If @D is rectifiable, then by point (iii) in Lemma 3.1 length ! 1 .E/ D 0 whenever E @D with jEj D 0, and by (iv) in Lemma 3.1, conversely, j!.E/j D 0 whenever E @D with length E D 0.
In the last case ! and ! 1 transform measurable sets into measurable sets. Indeed, every measurable set is the union of a sigma-compact set and a set of measure zero, see e.g. Theorem III(6.6) in [14], and continuous mappings transform compact sets into compact sets. Thus, a function ' W @D ! R is measurable with respect to the natural parameter on @D if and only if the functionˆD ' ı ! 1 W @D ! R is measurable with respect to the natural parameter on @D .
By point (ii) in Lemma 3.1, if @D has a tangent at a point 2 @D, then arg OE!. / !.z/ arg OE z ! const as z ! . In other words, the conformal images of sectors in D with a vertex at is asymptotically the same as sectors in D with a vertex at w D !. /. Thus, nontangential paths in D are transformed under ! into nontangential paths in D and inversely. Finally, a rectifiable Jordan curve has a tangent a.e. with respect to the natural parameter and, thus, Theorem 3.2 follows from Theorem 2.1.
In particular, choosing Á 1 in (5), we obtain the following statement. along any nontangential path with respect to the natural parameters of the boundary components of @D.

The case of arbitrary Jordan curves
The conceptions of a harmonic measure introduced by R. Nevanlinna in [15] and a principal asymptotic value based on one nice result of F. Bagemihl [16] make possible with a great simplicity and generality to formulate the existence theorems for the Dirichlet and Riemann-Hilbert problems. First of all, given a measurable set E Â @D and a point z 2 D, a harmonic measure of E at z relative to D is the value at z of the bounded harmonic function u in D with the boundary values 1 a.e. on E and 0 a.e on @D n E. In particular, by the mean value theorem for harmonic functions, the harmonic measure of E at 0 relative to D is equal to jEj=2 . In general, the geometric sense of the harmonic measure of E at z 0 relative to D is the angular measure of view of E from the point z 0 in radians divided by 2 . Hence the harmonic measure on @D has also the corresponding probabilistic interpretation. The harmonic measure in domains D bounded by finite collections of Jordan curves is defined in a similar way.
Next, a Jordan curve generally speaking has no tangents. Hence we need a replacement for the notion of a nontangential limit. In this connection, recall Theorem 2 in [16], see also Theorem III.1.8 in [17], stating that, for any function W D ! C, for all pairs of arcs 1 and 2 in D terminating at 2 @D, except a countable set of 2 @D, where C. ; / denotes the cluster set of at along , i.e., C. ; / D fw 2 C W .z n / ! w; z n ! ; z n 2 g : Applying the Poincare mapping, branches of its inverse mapping and their boundary behavior, see e.g. Theorem VI.1 and Section VI.2 in [2], we extend this result to arbitrary domains D bounded by a finite number of Jordan curves, cf. the proof of Theorem 2.1. Now, given a function W D ! C and 2 @D, denote by P . ; / the intersection of all cluster sets C. ; / for arcs in D terminating at . Later on, we call the points of the set P . ; / principal asymptotic values of at . Note that, if has a limit along at least one arc in D terminating at a point 2 @D with the property (7), then the principal asymptotic value is unique. Re fƒ.Á/ F .w/g Dˆ.Á/ along any nontangential path to a.e. Á 2 @D .
By the construction the functions f WD F ı ! are desired multivalent analytic solutions of (8) in view of the Bagemihl result.
In particular, choosing Á 1 in (8), we obtain the following consequence. with respect to harmonic measures in D in the sense of the unique principal asymptotic value.

On dimension of spaces of solutions
Recall that by the Lindelöf maximum principle, see e.g. Lemma 1.1 in [18], it follows the uniqueness theorem for the Dirichlet problem in the class of bounded harmonic functions on the unit disk. Our multivalent analytic solutions are generally speaking not bounded and the latter explains the following fact. Remark 5.2. Of course, results concerning the countable dimension of the space of solutions are not new and treated in terms of the infinite index of the Riemann-Hilbert problem, see e.g. [19] and [20]. By the general theory of boundary value problems, each additional singularities, including singularities on the boundary contour, increases the index. Hence the author results can be also interpreted as the case of the infinite index. Note that the considered situation admits the boundary functions with uncountable singularities. Such examples are given by Poisson-Stiltjes integral with the Cantor type functions under its differential, see e.g. [9], and the corresponding examples of analytic functions in the simplest case of the Riemann-Hilbert problem under D 1 and ' D 0 a.e.
Indeed, a Cantor type set C is perfect, i.e. it is closed and without isolated points. Hence C is of the continuum cardinality by the well-known W.H. Young theorem, see [21]. The corresponding Cantor type function has the symmetric Lebesgue derivative C1 at every point in C except ends, see e.g. the survey [22]. Then by the Fatou theorem, see e.g. Theorem I.D.3.2 in [11], the corresponding harmonic function has the radial limit C1 on the set of the continuum (maximal possible) cardinality.
In this connection, it would be also interesting to study the problem on a maximal possible cardinality of the dimension of the spaces of multivalent solutions with the infinite number of branches as here for the Riemann-Hilbert problem in multiply connected Jordan domains.