MATHEMATICS

Yu. L. RODIN

ALGEBRAIC THEORY OF GENERALIZED ANALYTIC FUNCTIONS ON CLOSED RIEMANN SURFACES

(Presented by Academician I. N. Vekua, September 21, 1961)

Let \(B(z)\) be a covariant, with respect to \(\bar z\), continuous except for a finite number of points and lines of discontinuity, defined on a closed Riemann surface \(R\) of genus \(p>2\). The differential equation

\[ U_{\bar z}=B(z)\,\overline{U} \tag{1} \]

is reduced to the integral equation

\[ U(z)+\frac{1}{\pi}\iint_R B(t)\,\overline{U(t)}\,A(t,z)\,dT=\Phi(z), \tag{2} \]

where \(\Phi(z)\) is an analytic function. Here \(A(t,z)\) is the Cauchy kernel constructed in \((^7)\). Its properties have been studied in more detail in the author’s paper \((^3)\). The kernel \(A(t,z)\) has a pole with residue \(+1\) when \(P[t]=P[z]\) and poles of first order in \(z\), determined by the divisor \(\delta\), \(\operatorname{ord}(\delta)=p\), \(\dim(W-\delta)=0\), called characteristic. The principal parts of the expansions at the points \(\delta\) are covariants of the first kind with respect to \(t\), forming a basis. With respect to \(t\), at the points \(\delta\) the kernel has zeros and a pole of first order at the point \(P_0\).

It is not difficult to introduce a metric in which the operator

\[ KU\equiv-\frac{1}{\pi}\iint_R B(t)\,\overline{U(t)}\,A(t,z)\,dT \tag{3} \]

will be completely continuous.

The adjoint to (2) turns out to be the covariant equation

\[ V(z)+\frac{1}{\pi}\iint_R B(t)\,\overline{V(t)}\,A(z,t)\,dT=\Psi(z), \tag{4} \]

where \(\Psi(z)\) is an analytic covariant. Its solutions are covariants satisfying the differential equation

\[ V_{\bar z}+\overline{B(z)}\,\overline{V}=0, \tag{5} \]

which we shall call the adjoint of (1).

For the solvability of equation (2), it is necessary and sufficient that

\[ \operatorname{Re}\iint_R \overline{B(t)}\,\overline{V_j(t)}\,\Phi(t)\,dt=0 \quad (j=1,2,\ldots,k), \tag{6} \]

where \(V_j(z)\) \((j=1,2,\ldots,k)\) is a complete system of solutions of the homogeneous equation (4).

For the solvability of equation (4), it is necessary and sufficient that

\[ \operatorname{Re}\iint_R B(t)\,\overline{U_j(t)}\,\Psi(t)\,dT=0 \quad (j=1,2,\ldots,k), \tag{7} \]

where \(U_j(z)\) \((j=1,2,\ldots,k)\) is a complete system of solutions of the homogeneous equation (2).

As shown in \((^3)\), such solutions, generally speaking, exist.

From the integral representation

\[ U(z)=\varphi(z)\exp\left\{-\frac{1}{\pi}\iint_T B(t)\frac{\overline{U(t)}}{U(t)}A(t,z)\,dT\right\}\quad \text{in }T, \tag{8} \]

established by I. I. Danilyuk \((^2)\), the Liouville theorem easily follows in the following formulation: a solution of equation (2), regular everywhere on \(R\) and having at least one zero, is identically equal to zero. Hence it follows that the number \(k_0\) of solutions of equation (1) regular on \(R\) (in what follows we call them generalized constants) does not exceed two. The number \(k_1\) of covariants regular on \(R\) satisfying equation (5) (we call them generalized covariants of the first kind) does not exceed \(2p\).

Examples show that the numbers \(k_0\) and \(k_1\) may vary depending on the choice of \(B(z)\). In particular, for \(B(z)\equiv 0\), \(k_0=2\).

Let us construct an example with \(k_0<2\). In the domain \(R-T\) consider an analytic covariant having a pole of first order at \(P_0\) and zeros in \(\delta\). Extend it continuously into \(T\). Put \(B(z)=-\overline{V_z(z)}/V(z)\), where \(V(z)\) is the covariant constructed by us. Then it satisfies the integral equation

\[ V(z)+\frac{1}{\pi}\iint_T \overline{B(t)}\,\overline{V(t)}\,A(z,t)\,dT=Z'(z), \tag{9} \]

where \(Z'(z)\) is an Abelian covariant of the first kind having a zero in \(\delta\). As a consequence of the condition \(\dim(W-\delta)=0\), \(Z'(z)\equiv 0\). In view of the fact that the residue of \(V(z)\) at the point \(P_0\)

\[ -\frac{1}{\pi}\iint_T \overline{B(t)}\,\overline{V(t)}\,dT\ne 0, \tag{10} \]

at least one of the equations is insoluble,

\[ U(z)+\frac{1}{\pi}\iint_T B(t)\overline{U(t)}\,A(t,z)\,dT=1\ (i). \tag{11} \]

It follows that \(k_0<2\), since generalized constants, as is easy to see, satisfy one of these equations.

Theorem 1. The difference between the number \(k_0\) of generalized constants and the number \(k_1\) of generalized covariants of the first kind is equal to

\[ k_1-k_0=2p-2. \tag{12} \]

Let \(k_0=2\). Then both equations are soluble,

\[ U(z)+\iint_R B(t)\overline{U(t)}\,A(t,z)\,dT=1\ (i), \tag{13} \]

and, consequently, for all solutions of the equation

\[ V(z)+\frac{1}{\pi}\iint_R \overline{B(t)}\,\overline{V(t)}\,A(z,t)\,dT=0 \tag{14} \]

the condition

\[ \iint_R \overline{B(t)}\,\overline{V(t)}\,dT=0 \tag{15} \]

is fulfilled. Thus all \(k\) solutions of equation (14) turn out to be generalized covariants of the first kind. Let us consider the solvability conditions for the equation

\[ V(z)+\frac{1}{\pi}\iint_R \overline{B(t)}\,\overline{V(t)}\,A(z,t)\,dT=\sum_{i=1}^{2p} x_i Z_i'(z)^{*}, \tag{16} \]

\[ ^{*}\ Z_i'(z)\ (i=1,2,\ldots,2p) \]
is a real basis of the Abelian differentials of the surface \(R\).

among whose solutions all the remaining generalized covariants of the first kind are contained. We arrive at the system

\[ \sum_{i=1}^{2p} x_i a_{ij}=0 \qquad (j=1,2,\ldots,k), \tag{17} \]

where

\[ a_{ij}=\operatorname{Re}\iint_R B(t)\,\overline{U_j(t)}\,Z_i'(t)\,dT \qquad (i=1,2,\ldots,k), \tag{18} \]

\(U_j(t)\) \((j=1,2,\ldots,k)\) is a complete system of solutions of the homogeneous equation (2). It is proved that \(\operatorname{rang}\|a_{ij}\|=k\), and, consequently, equation (16) has \(2p-k\) linearly independent solutions, and all of them prove to be generalized covariants of the first kind. The total number of generalized covariants of the first kind is equal to \(2p\), and formula (12) is valid.

When \(k_0=1\) or \(k_0=0\), among the solutions of equation (14) there are one or two having poles at \(P_0\), and the number of generalized covariants of the first kind proves to be equal to \(2p-1\) or \(2p-2\). Formula (12) also remains valid in this case.

Theorem 2 (Riemann–Roch). The difference between the number \(M\) of functions satisfying equation (1) and divisible by the divisor \(-\Delta\), and the number \(N\) of covariants satisfying equation (5) and divisible by \(\Delta\), is equal to

\[ M-N=2\,\operatorname{ord}(\Delta)-2p+2. \tag{19} \]

Divide \(R\) into domains \(T^+\) and \(T^-\) so that all points of \(\Delta\) lie in \(T^-\). We have the representations

\[ \begin{aligned} U(z)&=U^+(z) &&\text{in } T^+,\\ U(z)&=\Delta(z)U^-(z) &&\text{in } T^-, \end{aligned} \]

where \(\Delta(z)\) is a function analytic in \(T^-\) whose poles are determined by the divisor \(\Delta\); \(U^\pm(z)\) are functions regular in \(T^\pm\) and satisfying the equation \(U_z=B_1(z)\overline{U}\), where

\[ B_1(z)= \begin{cases} B(z), & \text{in } T^+,\\[4pt] B(z)\dfrac{\overline{\Delta(z)}}{\Delta(z)}, & \text{in } T^-. \end{cases} \tag{20} \]

On the contour \(\Gamma\) we arrive at the Riemann problem

\[ U^+(t)=\Delta(t)U^-(t) \tag{21} \]

of index \(\chi=\operatorname{ord}(\Delta)\).

For covariants we obtain the problem

\[ V^+(t)=\frac{1}{\Delta(t)}V^-(t) \tag{22} \]

for the conjugate equation.

The assertion of the theorem follows from the results of work \((^5)\).

In conclusion, consider the equation

\[ \partial_{\bar z}\omega_\nu=A\omega_\nu+B\overline{\omega_\nu} \tag{23} \]

for differentials of dimension \(\nu\).

Conjugate to the Riemann problem

\[ \omega_\nu^+(t)=G(t)\omega_\nu^-(t)+g_\nu(t), \qquad (\omega_\nu)+\Delta\geqslant 0, \tag{24} \]

for differentials belonging to equation (23) and divisible by \(-\Delta\) (see \((^4)\)), is called the problem

\[ \eta_{1-\nu}^+(t)=\frac{1}{G(t)}\eta_{1-\nu}^-(t), \qquad (\eta_{1-\nu})-\Delta\geqslant 0, \tag{25} \]

for differentials of dimension \(1-\nu\), multiples of \(\Delta\), and satisfying the equation

\[ \partial_z \eta_{1-\nu}+A\eta_{1-\nu}+\overline{B}\,\overline{\eta}_{1-\nu}=0, \tag{26} \]

conjugate to (23).

Using the results of work \({}^{6}\), one can prove the following assertions:

Theorem 3. The difference between the number \(l\) of solutions of the homogeneous problem (24) and the number \(l'\) of solutions of the conjugate problem (25) is equal to

\[ l-l' = 2\varkappa+(2\nu-1)(2p-2)+2\,\operatorname{ord}(\Delta), \qquad \varkappa=\operatorname{ind}_{\Gamma} G. \tag{27} \]

For solvability of the nonhomogeneous problem it is necessary and sufficient that

\[ \operatorname{Im}\int_{\Gamma} g_\nu(\tau)\eta_{1-\nu}^{+(j)}(\tau)=0 \qquad (j=1,2,\ldots,l'), \tag{28} \]

where \(\eta_{1-\nu}^{+(j)}\) \((j=1,2,\ldots,l')\) is a complete system of solutions of problem (25).

Theorem 4. The difference between the number \(M\) of differentials of dimension \(\nu\), multiples of \(-\Delta\) and satisfying equation (23), and the number \(N\) of differentials of dimension \(1-\nu\), satisfying the conjugate equation (26) and multiples of \(\Delta\), is equal to

\[ M-N = 2\,\operatorname{ord}(\Delta)+(2\nu-1)(2p-2). \tag{29} \]

Perm Polytechnic
Institute

Received
24 II 1961

REFERENCES CITED

\({}^{1}\) I. N. Vekua, Generalized Analytic Functions, 1959.
\({}^{2}\) I. I. Danilyuk, DAN, 109, No. 1 (1956).
\({}^{3}\) Yu. L. Rodin, DAN, 130, No. 1 (1960).
\({}^{4}\) Yu. L. Rodin, DAN, 132, No. 5 (1960).
\({}^{5}\) Yu. L. Rodin, DAN, 142, No. 4 (1961).
\({}^{6}\) Yu. L. Rodin, Theses of the All-Union Conference on the Theory of Functions of a Complex Variable, Yerevan, 1960.
\({}^{7}\) H. Behnke, K. Stein, Math. Ann., 120, 430 (1948).