UDC 517.948.33

MATHEMATICS

V. Yu. SHELEPOV

ON THE RIEMANN PROBLEM IN DOMAINS WHOSE BOUNDARY HAS BOUNDED ROTATION

(Presented by Academician I. N. Vekua on 25 X 1967)

1. Let a closed rectifiable Jordan curve \(\Gamma: t=t(s)\), \(0 \le s \le S\), have no cusps, and let the angle of its tangent with the abscissa axis be representable in the form
   \(\theta(s)=\theta_{0,1}(s)+\theta_2(s)\), \(0 \le s \le S\), where \(\theta_{0,1}(s)\) is a function of bounded variation on \([0,S]\), and \(\theta_2(s)\) satisfies the Hölder condition with exponent \(0<\alpha\le 1\). In particular, when \(\theta_2(s)\equiv 0\) we have a curve of bounded rotation (see \((^1)\), p. 98), and when \(\theta_{0,1}(s)\equiv 0\), a Lyapunov curve. Denote by \(D^+\), \(D^-\) the finite and infinite domains of the complex \(z\)-plane bounded by this curve.

In the article \((^2)\) it was proved that the singular integral operator

\[ Kf(\tau)=\frac{1}{2\pi i}\int_\Gamma \frac{f(t)}{t-\tau}\,dt,\qquad \tau\in\Gamma, \tag{1} \]

is bounded in \(L_p\), \(p>1\). In the present paper our aim is to prove its boundedness in the space \(L_p\) with a certain weight and to investigate in the domains \(D^\pm\) the Riemann boundary-value problem with discontinuous coefficients.

Suppose that on \(\Gamma\) a function \(G(t)=|G(t)|e^{i\varphi(t)}\) is defined, possessing the following properties: 1) \(0<m\le |G(t)|\le M<\infty\) almost everywhere on \(\Gamma\); 2) the function \(\varphi(t)\) is continuous on \(\Gamma\) everywhere except for at most a countable set of points \(\{t_k=t(s_k)\}_{k=1}^{\infty}\ne t_0=t(0)\), where it has jumps \(h_k\), not exceeding \(\pi\) in absolute value, and the point \(t_0\), where it has the jump \(-2\pi\chi_0\) (\(\chi_0\) is an integer); 3)

\[ \sum_{k=1}^{\infty}|h_k|<\infty \]

(whence it follows that the jumps \(h_k\) may be regarded as ordered in nonincreasing order of their absolute values); 4) \(h_k\ne 2\pi/p\), if \(h_k>0\), and \(|h_k|\ne 2\pi/p'\), if \(h_k<0\), \(1/p+1/p'=1\).

It is not difficult to show that, having prescribed an arbitrary \(\varepsilon>0\), one can find such a number \(n\) that the decomposition

\[ \varphi(t)=\psi_n(t)+\omega_n(t), \]

is valid, in which the function \(\psi_n(t)\) has jumps \(h_k\) at the points \(t_k\), \(k=1,2,\ldots,n\), and the jump \(-2\pi\chi_0\) at the point \(t_0\), and on the arcs between these points satisfies the Lipschitz condition; while the function \(\omega_n(t)\) satisfies the inequality
\[ \sup_{t\in\Gamma}|\omega_n(t)|<\varepsilon. \]

Consider the Riemann problem in the following formulation.

Find a function \(\Phi^+(z)\in E_p(D^+)\) (see \((^3)\), p. 203), and a function \(\Phi^-(z)\in E_p(D^-)\), vanishing at infinity, which almost everywhere on \(\Gamma\) satisfy the boundary condition

\[ \Phi^+(t)=G(t)\Phi^-(t)+g(t), \tag{2} \]

where \(g(t)\in L_p(\Gamma)\).

Problem (2) in the classical formulation, when \(G(t)\) and \(g(t)\) satisfy the Hölder condition and \(\Gamma\) is a Lyapunov curve, was solved by F. D. Gakhov \((^4)\). For its further generalizations see \((^4,^5)\). In the indicated formulation and under assumptions on the coefficients, problem (2), in the case where \(\Gamma\) is the unit circle, was studied by I. I. Danilyuk \((^6)\). The method of computing the index used below belongs to him. In similar assumptions, problems for Lyapunov curves were studied by I. B. Simonenko \((^7)\).

2. Let

\[ X^{\pm}(z)=\exp\left\{\frac{1}{2\pi i}\int_{\Gamma}\frac{\ln |G(t)|+i\varphi(t)}{t-z}\,dt\right\},\qquad z\in D^{\pm}. \]

From the Sokhotski formulas it follows that \(G(t)=X^{+}(t)/X^{-}(t)\), \(t\in\Gamma\). Let us study to which classes \(X^{\pm}(z)\) belong.

First of all, note that the functions

\[ \left[\exp\frac{1}{2\pi i}\int_{\Gamma}\frac{\ln |G(t)|}{t-z}\,dt\right]^{\pm1} \]

are bounded in \(D^{\pm}+\Gamma\), since, as can be proved using results of I. Radon (see \((^1)\), p. 102), the real part of the integral occurring here has this property.

Now consider

\[ W^{\pm}(z)=\exp\left\{\frac{1}{2\pi}\int_{\Gamma}\frac{\varphi(t)}{t-z}\,dt\right\},\qquad z\in D^{\pm}. \]

Lemma. For sufficiently small \(\delta>0\), the functions \(W^{\pm}(z)\) belong to the classes \(E_{\delta}(D^{\pm})\).

We shall carry out the proof for \(D^{+}\). Construct a sequence of functions \(\{\varphi_k(t)\}_{k=1}^{\infty}\), \(\sup_{t\in\Gamma,k}|\varphi_k(t)|<\infty\), satisfying the Lipschitz condition on \(\Gamma\) and converging in measure to \(\varphi(t)\). The functions

\[ W_k^{+}(z)=\exp\left\{\frac{1}{2\pi}\int_{\Gamma}\frac{\varphi_k(t)}{t-z}\,dt\right\} \]

are continuous in \(D^{+}+\Gamma\) (see \((^3)\), p. 197). Further, there exists a constant \(M\), independent of \(k\), such that

\[ \int_{\Gamma}|W_k^{+}(t)|^{\delta}\,ds\leq M. \]

This follows from Cauchy’s formula applied to the function
\([W_k^{+}(z)]^{\delta}\exp i\Phi_0(z)\), where \(\Phi_0(z)\) is a function continuous in \(D^{+}+\Gamma\) such that, for sufficiently small \(\delta>0\),

\[ \sup_{t\in\Gamma,k}\left|\delta\bigl(\varphi_k(t)/2+\operatorname{Re}K\varphi_k(t)\bigr)+(\theta(t)-\arg t)+\operatorname{Re}\Phi_0(t)\right|<\pi/2 \]

(see \((^2)\)). Denote by \(\gamma_r\) the image of the circle \(|\zeta|=r\) under the conformal mapping \(z=w(\zeta)\) of the disk \(|\zeta|<1\) onto \(D^{+}\). Then

\[ \int_{\gamma_r}|W_k^{+}(z)|^{\delta}|dz| = r\int_{0}^{2\pi}|W_k^{+}(w(re^{i\alpha}))|\sqrt[\delta]{\rho_{\delta}'}\,d\alpha\leq \]

\[ \leq \int_{0}^{2\pi}|W_k^{+}(w(e^{i\alpha}))|\sqrt[\delta]{w'(e^{i\alpha})}\,d\alpha = \int_{\Gamma}|W_k^{+}(t)|^{\delta}\,dt\leq M \]

(see \((^3)\), pp. 78, 89). Since \(W^{+}(z)=\lim_{k\to\infty}W_k^{+}(z)\) for \(z\in D^{+}\), the proof is completed by applying Fatou’s lemma. It is obvious that an analogous assertion is true for \([W^{\pm}(z)]^{-1}\). We also note that, by virtue of the property of the function \(\omega_n(t)\), for arbitrary \(1<q<\infty\) one can indi-

choose a number \(n\) such that

\[ [\Omega_n(z)]^{\pm 1} = \left[ \exp\left\{ \frac{1}{2\pi}\int_\Gamma \frac{\omega_n(t)}{t-z}\,dt \right\} \right]^{\pm 1} \in E_q(D^{\pm}). \]

1. Let \(h_1^+ \geq h_2^+ \geq \cdots\) be all the positive jumps of the function \(\varphi(t)\), and let \(h_1^- \geq h_2^- \geq \cdots\) be the moduli of the negative jumps, \(\{t_k^+\}_{k=1}^{\infty}\), \(\{t_k^-\}_{k=1}^{\infty}\) the corresponding subsets of \(\Gamma\). Define the numbers \(\varkappa^+\), \(\varkappa^-\) by the following conditions:

\[ \varkappa^+=\max\{k:\ h_k^+>2\pi/p\};\qquad \varkappa^-=\max\{k:\ h_k^->2\pi/p'\}. \]

Since the singular integral has a logarithmic singularity at a jump point of the density, it is not difficult to show that

\[ \Psi_n(\tau) \equiv |\tau-t_0|^{-\varkappa_0} \left| \frac{\displaystyle\prod_{k=1}^{\varkappa^+}|\tau-t_k^+|} {\displaystyle\prod_{k=1}^{\varkappa^-}|\tau-t_k^-|} \right| \exp\{iK\psi_n(\tau)\} = \]

\[ = \frac{ \displaystyle \prod_{k=1}^{\varkappa^+}|\tau-t_k^+|^{\,1-h_k^+/2\pi} \prod_{k=\varkappa^-+1}^{n_1}|\tau-t_k^-|^{\,h_k^-/2\pi} }{ \displaystyle \prod_{k=1}^{\varkappa^-}|\tau-t_k^-|^{\,1-h_k^-/2\pi} \prod_{k=\varkappa^++1}^{n_2}|\tau-t_k^+|^{\,h_k^+/2\pi} } \,|U_n(\tau)|, \tag{3} \]

where

\[ \sup_{\tau\in\Gamma}|U_n(\tau)|^{\pm1}<\infty, \qquad n_1>\varkappa^-, \quad n_2>\varkappa^+, \quad n_1+n_2=n. \]

For sufficiently small \(\lambda>0\) we have \(\Psi_n(\tau)\in L_{p+\lambda}(\Gamma)\), \([\Psi_n(\tau)]^{-1}\in L_{p'+\lambda}(\Gamma)\). Hence, from the lemma it follows that for some \(\delta>0\)

\[ Z^{\pm}(z) = (z-t_0)^{-\varkappa_0} \frac{\displaystyle\prod_{k=1}^{\varkappa^+}(z-t_k^+)} {\displaystyle\prod_{k=1}^{\varkappa^-}(z-t_k^-)} X^{\pm}(z) \in E_\delta(D^{\pm}) \]

and the boundary values of these functions on \(\Gamma\) are summable to the power \(p+\lambda_1\), \(\lambda_1<\lambda\). The ratio of the length of the larger of the two arcs determined by an arbitrary chord to the length of this chord is bounded above and, consequently, \(D^+\), \(D^-\) are domains of class \(C\) (see \((^8)\)). Therefore, by V. I. Smirnov’s theorem (see \((^3)\), p. 264) \(Z^{\pm}(z)\in E_{p+\lambda_1}(D^{\pm})\). Similarly we obtain that \([Z^{\pm}(z)]^{-1}\in E_{p'+\lambda_1}(D^{\pm})\).

1. Using the boundedness of the operator (1) in \(L_p(\Gamma)\), \(p>1\), and B. V. Khvedelidze’s argument (see \((^9)\), p. 24, Theorem 5), we obtain from (3):

\[ \|(Kf)\Psi_n\|_{L_p}\leq M_1\|f\Psi_n\|_{L_p}. \]

Let \(1<q<\infty\), \(0<c<1\); \(\Omega_n(\varepsilon)=|\Omega_n^+(\tau)|,\ \tau\in\Gamma\). Consider the problem

\[ \Phi^+(t)=c e^{i\omega_n(t)}\Phi^-(t)+g(t). \tag{4} \]

Here \(g(t)\in L_q(\Gamma)\), and the solutions are sought in \(E_q(D^{\pm})\).

Given arbitrary \(\eta>0\), one can choose the numbers \(c\) and \(n\) so that

\[ \sup_{t\in\Gamma}|c e^{i\omega_n(t)}-1|<\eta, \qquad [\Omega_n^{\pm}(z)]^{-1}\in E_q(D^{\pm}). \]

Then, from the results of I. B. Simonenko (see \((^{10})\); \((^7)\), p. 286), transferred to the case of our domains, it follows that problem (4) is uniquely and unconditionally solvable, and therefore

\[ \|(Kf)\Omega_n\|_{L_q}\leq M_2\|f\Omega_n\|_{L_q}. \]

We now choose numbers \(\gamma>1\) and \(\varepsilon>1\) so that
\(\gamma h_k^+ \ne 2\pi/(p-\varepsilon)\), \(\gamma h_k^- \ne 2\pi/(p-\varepsilon)\), \(k=1,2,\ldots\), and put
\(p_1=p-\varepsilon(<k)\),
\[ p_2=\frac{p(p-\varepsilon)(\gamma-1)}{p(\gamma-1)-\gamma\varepsilon}(>p),\quad 0<t=\frac{\gamma-1}{\gamma}<1,\quad \delta=\frac1t . \]
Then, for some sufficiently large \(n\),
\[ \|(Kf)\Psi_n^\gamma\|_{L_{p_1}}\le M_3\|f\Psi_n^\gamma\|_{L_{p_1}};\qquad \|(Kf)\Omega_n^\delta\|_{L_{p_2}}\le M_4\|f\Omega_n^\delta\|_{L_{p_2}}. \]

Since \(p^{-1}=(1-t)p_1^{-1}+tp_2^{-1}\), \(\Psi_n^{\gamma(1-t)}\Omega_n^{\delta t}=\Psi_n\Omega_n\), it follows immediately, by Stein’s interpolation theorem (see \((^{11})\)):

Theorem 1. The operator
\[ \frac{Z^+(\tau)}{2\pi i}\int_\Gamma \frac{g(t)}{Z^+(t)(t-\tau)}\,dt,\qquad \tau\in\Gamma, \]
is bounded in the space \(L_p(\Gamma)\), \(p>1\).

5. The results obtained make it possible to analyze problem (2):

Theorem 2. We shall call the number
\[ \varkappa=\varkappa_0-\varkappa^+ + \varkappa^- \]
the index of problem (2).

Then, for \(\varkappa\ge0\), the problem is unconditionally solvable and has \(\varkappa\) linearly independent solutions. In the case \(\varkappa<0\), the problem is solvable only when \(|\varkappa|\) conditions are satisfied:
\[ \int_\Gamma \frac{g(t)}{Z^+(t)}\,t^k\,dt=0,\qquad k=0,1,\ldots,|\varkappa|-1, \]
and in this case it has a unique solution.

In conclusion, the author takes the opportunity to express deep gratitude to Prof. I. I. Danilyuk, who supervised the work.

Donetsk Computing Center
Academy of Sciences of the Ukrainian SSR

Received
20 X 1967

CITED LITERATURE

1. I. Radon, UMN, 1, 3–4, 96 (1946).
2. I. I. Danilyuk, V. Yu. Shelepov, DAN, 174, No. 3 (1967).
3. I. I. Privalov, Boundary Properties of Analytic Functions, Moscow–Leningrad, 1950.
4. F. D. Gakhov, Boundary Value Problems, Moscow, 1963.
5. N. I. Muskhelishvili, Singular Integral Equations, Moscow, 1962.
6. I. I. Danilyuk, Sibirsk. matem. zhurn., 1, No. 2, 171 (1960).
7. I. B. Simonenko, Izv. AN SSSR, ser. matem., 28, No. 2, 277 (1964).
8. M. A. Lavrent’ev, Matem. sborn., 1 (43) (1936).
9. B. V. Khvedelidze, Tr. Tbilissk. matem. inst., 23, 3 (1956).
10. I. B. Simonenko, DAN, 124, No. 2 (1959).
11. E. M. Stein, Trans. Am. Math. Soc., 83, 224 (1956).