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UDC 517.544
MATHEMATICS
R. N. ABDULAEV
A DISCONTINUOUS PROBLEM OF LINEAR CONJUGATION FOR ANALYTIC FUNCTIONS ON RIEMANN SURFACES
(Presented by Academician I. N. Vekua on 8 IV 1966)
Let on a closed Riemann surface \(R\) of genus \(\rho\) there be given a contour \(\Gamma\), consisting of a finite number of mutually nonintersecting closed oriented curves satisfying the Lyapunov condition. Suppose, further, that measurable functions \(G\) and \(g\) are given on \(\Gamma\). It is required to determine a function \(\Phi\), holomorphic in \(R-\Gamma\), belonging to the class \(H_p,\ p \geq 1\)* on each component of \(R-\Gamma\), whose boundary values almost everywhere on \(\Gamma\) satisfy the equality
\[ \Phi^{+}=G\Phi^{-}+g. \tag{1} \]
This problem has been well studied in the planar case under various restrictions on \(G\) and \(g\) (see, for example, \((^{2-5})\)). On a Riemann surface it was studied for \(G\) and \(g\) satisfying the Hölder condition \((^{6,7})\) and when \(G\) has discontinuities of the first kind \((^{8,9})\).
In the present note a method is given, analogous to the well-known alternating method of Schwarz, which makes it possible to investigate problem (1) on a Riemann surface under restrictions on \(G\) and \(g\) adopted in the papers \((^{2-5})\).
Let on \(R\) there be given two domains \(K_0\) and \(K_1\), \(\overline{K}_1 \subset K_0\), bounded by analytic curves and conformally equivalent to planar circular rings. Suppose, further, that in \(K_0-K_1\) a harmonic function \(u_0\) is given.
Theorem 1. The condition
\[ \int_{\partial K_0} d u_0^{*}=0 \tag{2} \]
is necessary and sufficient for the existence of a function \(u\) with the following properties:
A. \(u\) is single-valued and harmonic on \(R-K_1\).
B. \(u-u_0\) is the restriction to \(K_0-K_1\) of a function single-valued and harmonic in \(K_0\).
The necessity of condition (2) is obvious.
Sufficiency. Consider sequences of harmonic functions \(\{u_n\}\) and \(\{v_n\}\), defined in the domains \(R-K_1\) and \(K_0\), respectively, by the boundary conditions
\[ u_n=v_{n-1}+u_0 \quad \text{on } \partial(R-K_1), \tag{3} \]
\[ v_n=u_n-u_0 \quad \text{on } \partial K_0 \tag{4} \]
\[ (n=1,2,\ldots;\ v_0\equiv 0). \]
Map the domain \(K_0\) by means of an analytic function \(\tau\) one-to-one and conformally onto the planar annulus \(1<|z|<r\) and choose two numbers \(r_1,r_2,\ 1<r_1<r_2<r\), with \(\ln r_1=\ln r/r_2\), so that \(\tau^{-1}(r_2<|z|<r_1)\) and \(\tau^{-1}(r_2<|z|<r)\) lie in different components of \(K_0-K_1\). On the basis—
* For the definition of the Hardy classes \(H_p\) on finite Riemann surfaces, see \((^1)\).
of equalities (2), (3), and (4) it is easy to show that
\[ \int_{0}^{2\pi}\left[v_n\left(e^{i\varphi}\right)-v_{n-1}\left(e^{i\varphi}\right)+v_n\left(re^{i\varphi}\right)-v_{n-1}\left(re^{i\varphi}\right)\right]\,d\varphi=0. \tag{5} \]
Consequently,
\[
\min_{K_0}(v_n-v_{n-1})\leq 0,\qquad
\max_{K_0}(v-v_{n-1})\geq 0.
\]
Denote by \(s(w,M)\) the oscillation of the function \(w\) on the set \(M\). From the last inequalities it follows that
\[ |v_n-v_{n-1}|\leq s(v_n-v_{n-1},K_0). \tag{6} \]
According to Lemma 3 of [10],
\[ s(v_n-v_{n-1},K_1)\leq q\,s(v_n-v_{n-1},K_0), \tag{7} \]
where \(q<1\) is a constant depending only on \(K_0\) and \(K_1\). From (3), (4), and (7) we obtain
\[ s(v_{n+1}-v_n,K_0)\leq q^n s(v_1,K_0). \tag{8} \]
The last inequality, together with inequality (6), entails the uniform convergence of the sequence \(\{v_n\}\) in \(K_0\). From the convergence of \(\{v_n\}\) there follows the convergence of \(\{u_n\}\) in \(R-K_1\). Denote \(v=\lim_{n\to\infty} v_n\), \(u=\lim_{n\to\infty} u_n\). Passing to the limit in equalities (3) and (4), we obtain that the functions \(v\) and \(u-u_0\) coincide in \(K_0-K_1\). The theorem is proved.
Let now a function \(h\in L_p\), \(p\geq 1\), be given on \(\Gamma\). Consider the so-called Sokhotski problem: it is required to determine on \(R-\Gamma\) a function \(F\in H_p\) whose boundary values satisfy on \(\Gamma\) the condition
\[ F^{+}-F^{-}=h. \tag{9} \]
If a solution of the problem exists, then, as is easy to see, \(\int_{\Gamma} h\,dw=0\) for any differential of the first kind \(dw\).
Choose for each \(\Gamma_j\) a doubly connected neighborhood \(K_j\), map it by means of an analytic function \(\tau_j\) onto a plane annulus, and construct in the plane a piecewise-holomorphic function \(f_j\in H_p\) with respect to the boundary condition prescribed on \(\gamma_j=\tau_j(\Gamma_j)\),
\[ f_j^{-}-f_j^{+}=h\circ\tau_j^{-1}. \tag{10} \]
The function \(F_j=f_j\circ\tau_j\), defined in \(K_j-\Gamma_j\), is holomorphic, belongs to \(H_p\), and satisfies condition (9) on \(\Gamma_j\). According to Theorem 1, there exists a function \(u_j\), harmonic in \(R-\Gamma_j\), such that \(u_j-\operatorname{Re}F_j\) is the restriction to \(K_j-\Gamma_j\) of a single-valued function harmonic in \(K_j\). On each component of \(R-\Gamma_j\) construct the function
\[ F'_j=u_j+iu_j^{*}. \tag{11} \]
The restriction of any branch \(F'_j\) to \(K_j-\Gamma_j\) belongs to \(H_p\) and satisfies on \(\Gamma_j\) the condition
\[ F_j^{\prime +}-F_j^{\prime -}=h+iC_j, \tag{12} \]
where \(C_j\) is a real constant depending on the chosen branch of \(F'_j\).
Form the differential \(dF=\sum_{j=1}^{m}dF_j\) and compute its periods on \(R-\Gamma\). The calculation of the periods is carried out with the aid of the bilinear relation, taking (12) into account. The following assertion holds: for any cycle \(l\subset R-\Gamma\) there exists a differential of the first kind \(dZ_l\), depending only on \(l\), such that
\[
\int_l dF=\operatorname{Im}\int_{\Gamma}h\,dZ_l.
\]
Let \(\int_l dF=0\) for any \(l\subset R-\Gamma\). Then the analytic function
\[
F=\int dF
\]
is single-valued on \(R-\Gamma\), belongs to \(H_p\), and satisfies the boundary-
condition
\[ F^+ - F^- = h + iB, \tag{13} \]
where \(B\) is a real function, equal to a constant on each \(\Gamma_j\). The value \(B\) on a nondividing component of the contour \(\Gamma\) can be made zero by adding a constant to \(F\). The value \(B_j\) of the function \(B\) on a nondividing \(\Gamma_j\) can be computed. To this end one must take an Abelian differential of the first kind \(dZ_{\Gamma_j}\), whose real part has a single nonzero period, equal to one along \(\Gamma_j\).
From Cauchy’s theorem it follows that
\[ B_j = \operatorname{Im} \int_{\Gamma} h\,dZ_{\Gamma_j}. \]
All that has been said confirms the validity of the following theorem.
Theorem 2. For the solvability of the Sokhotski problem it is necessary and sufficient that
\[ \int_{\Gamma} h\,dw_\nu = 0 \qquad (\nu = 1,2,\ldots,\rho), \]
where \(dw_1, dw_2,\ldots,dw_\rho\) is a basis of the space of Abelian differentials of the first kind.
As mentioned above, the investigation of problem (1) is carried out under the assumptions adopted in papers \((^2\!-\!^5)\). In the planar case it follows from these assumptions that:
I. \(\ln G \in L_{p_1},\ p_1 \geq 1\).
II. \(G = X^+(t)/X^-(t),\ t \in \Gamma\), where \(X^\pm(z)\in H_p,\ 1/X^\pm(z)\in H_q,\ q=p/(p-1)\), and \(X(z)\) has neither zeros nor poles outside \(\Gamma\), except at the point at infinity, where the order of \(X(z)\) is equal to the index of the problem.
III. The general solution of problem (1) is given by the formula \(\Phi = XF + XP\), where \(F\) is a solution of the Sokhotski problem determined by the jump \(g/X^+\), and \(P\) is a polynomial.
Thus, without formulating the restrictions imposed on \(G\) and \(g\), we shall assume that the functions \(G \circ \tau_j^{-1}\) and \(g \circ \tau_j^{-1}\) satisfy conditions I–III on \(\gamma_j\).
Let us first consider the homogeneous problem
\[ \Phi^+ = G\Phi^-. \tag{14} \]
For the given \(G\) one can construct a function \(G_0\), satisfying the Hölder condition on \(\Gamma\), such that
\[ \frac{1}{2\pi i}\int_{\Gamma_j} d\ln G_0 = \operatorname{ind}_{\gamma_j}\bigl(G\circ\tau_j^{-1}\bigr) \qquad (j=1,2,\ldots,m), \]
\[ \frac{1}{2\pi i}\int_{\Gamma} \ln G_0\,dw_\nu = \frac{1}{2\pi i}\int_{\Gamma} \ln G\,dw_\nu \qquad (\nu=1,2,\ldots,\rho). \]
The functions \(\dfrac{G}{G_0}\circ\tau_j^{-1}\) satisfy conditions I, II and
\[
\operatorname{ind}_{\gamma_j}\left(\frac{G}{G_0}\circ\tau_j^{-1}\right)=0.
\]
Taking this into account, on the basis of Theorems 1 and 2 it is not difficult to establish that the homogeneous problem with coefficient \(G/G_0\) is solvable. Its solution \(X_0\) is single-valued and nowhere vanishes outside \(\Gamma\). Dividing both sides of (14) by \(X_0\), we arrive at the boundary-value problem defined by the boundary condition
\[ \Phi_0^+ = G_0\Phi_0^-. \tag{15} \]
If (15) is solvable in the class \(H_p\), then its solution is automatically continuously extendable to \(\Gamma\). Problems (14) and (15) are equivalent. The general solution of problem (14) is given by the formula \(\Phi = X_0\Phi_0\). The number of linearly independent solutions of problem (15), and hence of problem (14), depends on the numbers
\[ \frac{1}{2\pi i}\int_{\Gamma} \ln G\,dw_\nu \qquad (\nu=1,2,\ldots,\rho) \quad \text{(see \((^{11,12})\)).} \]
To study the solvability of the nonhomogeneous problem, let us consider the conjugate problem associated with it, consisting in finding on \(R-\Gamma\) a holomorphic differential \(d\Psi \in H_q\), whose boundary values satisfy on \(\Gamma\) the condition
\[ d\Psi^+ = \frac{1}{G}\,d\Psi^- . \tag{16} \]
Dividing (1) and multiplying (16) by \(X_0^+\), we obtain two problems defined by the boundary conditions
\[ \Phi_1^+ = G_0\Phi_1^- + \frac{g}{X_0^+}, \tag{17} \]
\[ d\Psi_1^+ = \frac{1}{G_0}\,d\Psi_1^- . \tag{18} \]
The solution of problem (17) is sought in the class \(H_p\), and that of problem (18) in the class of differentials continuously extendable to \(\Gamma\). Using assumption III on the functions \(G\) and \(g\), one can show that problems (1) and (17) are equivalent and their solutions are related by the relation \(\Phi=X_0\Phi_1\). The equivalence of problems (16) and (18) and the validity of the relation \(d\Psi_1=X_0d\Psi\) are obvious.
Theorem 3. For problem (1) to be solvable, it is necessary and sufficient that
\[ \int_{\Gamma} g\,d\Psi^+ = 0 \tag{19} \]
for every solution \(d\Psi\) of the conjugate problem.
Necessity follows from the Cauchy theorem.
Sufficiency. We rewrite condition (19) in the form
\[ \int_{\Gamma} \frac{g}{X_0^+}\,d\Psi_1^+ = 0. \tag{20} \]
It is not difficult to show that on \(\Gamma\) one can construct a function \(g_1\), satisfying the Hölder condition, such that the problem determined by the boundary condition
\[ \Phi_2^+ = G_0\Phi_2^- + g/X_0^+ + g_1, \]
will have a solution in the class \(H_p\). From the necessity of condition (19) it follows that
\[ \int_{\Gamma}\left(\frac{g}{X_0^+}+g_1\right)d\Psi_1^+ = \int_{\Gamma} g_1 d\Psi_1^+ = 0. \tag{21} \]
But, as is known, from (21) there follows the existence of a function \(\Phi_3\), holomorphic on \(R-\Gamma\) and continuously extendable to \(\Gamma\), whose boundary values satisfy the condition
\[ \Phi_3^+ = G_0\Phi_3^- + g_1. \]
A direct verification shows that \(\Phi=\Phi_2-\Phi_3\) is a solution of problem (17). Thus problem (17), and consequently problem (1), is solvable.
Remark. All those facts of the theory of boundary-value problems on Riemann surfaces with Hölder coefficients to which we have referred in the present note can be established starting from Theorem 1.
Tbilisi Mathematical Institute
named after A. M. Razmadze
Academy of Sciences of the Georgian SSR
Received
2 IV 1966
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