UDC 517.9

MATHEMATICS

I. Ts. Gohberg, N. Ya. Krupnik

SYSTEMS OF SINGULAR INTEGRAL EQUATIONS IN WEIGHTED \(L_p\) SPACES

(Presented by Academician N. I. Muskhelishvili on 27 XI 1968)

1. Let \(\Gamma\) be a simple smooth closed oriented curve in the plane, surrounding the point \(\lambda=0\); let \(t_1,\ldots,t_n\) be certain points on \(\Gamma\); and let \(p,\beta_1,\ldots,\beta_n\) be real numbers satisfying the relations
   \[ 1<p<\infty;\qquad -1<\beta_k<p-1 \quad (k=1,\ldots,n). \tag{1} \]

By \(L_p(\Gamma,\rho)\) we denote the space \(L_p\) on \(\Gamma\) with weight \(\rho(t)=\prod |t-t_k|^{\beta_k}\), and by \(L_p^m(\Gamma,\rho)\) the space of vector-functions \(\varphi=(\varphi_1,\ldots,\varphi_m)\) with components \(\varphi_j\in L_p(\Gamma,\rho)\).

Consider in \(L_p^m(\Gamma,\rho)\) the matrix singular integral operators
\[ (K\varphi)(t)=C(t)\varphi(t)+D(t)(S\varphi)(t), \qquad (L\varphi)(t)=C(t)\varphi(t)+(SD\varphi)(t), \]
where \(C(t)\) and \(D(t)\) are matrix-functions \((t\in\Gamma)\), and \(S\) is the matrix operator of singular integration
\[ (S\varphi)(t)=\frac{1}{\pi i}\int_\Gamma \frac{\varphi(\tau)}{\tau-t}\,d\tau . \]

The operators \(K\) and \(L\), in the case of continuous coefficients \(C\) and \(D\), are \(\Phi\)-operators if and only if the determinant \(\det(C+D)(C-D)\) does not vanish anywhere on \(\Gamma\). In the case of piecewise-continuous coefficients this condition, while remaining necessary, ceases to be sufficient for \(K\) and \(L\) to be \(\Phi\)-operators.

In the present note, for the case of piecewise-continuous coefficients \(C(t)\) and \(D(t)\), necessary and sufficient conditions are established under which \(K\) and \(L\) are \(\Phi\)-operators. The results obtained are a generalization to the case of systems of equations of the results of the authors \(\left({}^{1}\right)\) (see also \(\left({}^{2-4}\right)\)).

The operators \(K\) and \(L\), under the condition that the matrices \(C\) and \(D\) are composed of piecewise Hölder functions satisfying a Lipschitz condition in one-sided neighborhoods of the points of discontinuity, were studied by N. P. Vekua \(\left({}^{5}\right)\). In particular, he obtained sufficient conditions under which the operators \(K\) and \(L\) are \(\Phi\)-operators in certain classes of Hölder functions. Under the same restrictions on the coefficients \(C\) and \(D\), the operators \(K\) and \(L\) in the spaces \(L_p^m(\Gamma,\rho)\) were considered by B. V. Khvedelidze \(\left({}^{6}\right)\), who also obtained certain sufficient conditions under which \(K\) and \(L\) are \(\Phi\)-operators. In the paper of E. Shamir \(\left({}^{7}\right)\), operators of the type \(K\) and \(L\) are considered for the case when the contour \(\Gamma\) coincides with the real axis, while the matrices \(C\) and \(D\) are piecewise differentiable and satisfy certain conditions at infinity. Under these assumptions and the additional assumption of the nonsingularity of the matrices \(C+D\) and \(C-D\), he found necessary and sufficient conditions for the operators \(K\) and \(L\) to be \(\Phi\)-operators in the spaces \(L_p\).

Sufficient conditions for \(K\) and \(L\) to be \(\Phi\)-operators in \(L_2^m(\Gamma)\), in the case of bounded measurable coefficients \(C\) and \(D\), were obtained by I. B. Simonenko \(\left({}^{8}\right)\).

We note that the method of investigation presented below differs from the methods of the authors listed above.

1. Denote by \(\Lambda_m(\Gamma)\) the set of all matrix-functions of order \(m\) that are piecewise continuous and continuous from the left on \(\Gamma\). Let \(G(t)\in\Lambda_m(\Gamma)\); \(t_1,\ldots,t_n\in\Gamma\) are all points of discontinuity of the matrix \(G\), and \(\omega=(p,\beta_1,\ldots,\beta_n)\) is a vector whose coordinates satisfy relations (1).

To the matrix-function \(G(t)\) and the vector \(\omega\) we assign the continuous matrix curve \(V(G,\omega)\), obtained by adding to the set of values \(G(t)\) \(n\) matrix arcs \(W_k(\mu)\) \((0\leq\mu\leq1)\), where*

\[ W_k(\mu)= \frac{e^{i\mu\theta_k}\sin(1-\mu)\theta_k}{\sin\theta_k}\,G(t_k) + \frac{e^{i(\mu-1)\theta_k}\sin\mu\theta_k}{\sin\theta_k}\,G(t_k+0) \]

\[ \left(\theta_k=\pi-\frac{2\pi(1+\beta_k)}{p}\right). \]

We shall call the matrix-function \(G(t)\) \(\omega\)-regular if the curve \(\det V(G,\omega)\) does not pass through zero. The continuous closed curve \(\det V\) in the plane is oriented so that, at points of continuity of the matrix \(G(t)\), motion along the curve \(\det V(G,\omega)\) is determined by the motion of \(t\) along \(\Gamma\) in the positive direction, while along the additional arcs it corresponds to the change of \(\mu\) from 0 to 1.

Introduce the following notation: \(C(t)+D(t)=A(t)\), \(C(t)-D(t)=B(t)\), \(I+S=2P\), \(I-S=2Q\), where \(I\) is the identity operator; then the operators \(K\) and \(L\) may be written in the form \(K=AP+BQ\), \(L=PA+QB\).

Theorem 1. Let \(A\) and \(B\in\Lambda_m\); \(t_1,\ldots,t_n\in\Gamma\) are all points of discontinuity of the matrices \(A\) and \(B\); \(\rho=\prod |t-t_k|^{\beta_k}\), where \(-1<\beta_k<p-1\) \((1<p<\infty)\). In order that the operator \(K=AP+BQ\) \((L=PA+QB)\) be a \(\Phi\)-operator in \(L_p^m(\Gamma,\rho)\), it is necessary and sufficient that the following two conditions be fulfilled:

\[ \alpha)\quad \det B(t\pm0)\ne0\quad \text{for all } t\in\Gamma; \]

\[ \beta)\quad \text{the matrix } B^{-1}A\ (AB^{-1}) \text{ is } \omega\text{-regular}. \]

If conditions \(\alpha)\) and \(\beta)\) are fulfilled, then the index of the operator \(K(L)\) is determined by the equality

\[ \varkappa=-\operatorname{ind}\det V(B^{-1}A,\omega) \quad \left(\varkappa=-\operatorname{ind}\det V(AB^{-1},\omega)\right). \tag{*} \]

The proof of this theorem is based on two lemmas.

Lemma 1. Let \(M(t)\) and \(N(t)\) be regular matrix-functions of order \(m\), continuous on \(\Gamma\), and let \(X(t)\) be a bounded measurable matrix-function of the same order. In order that the operator \(MXNP+Q\) \((PMXN+Q)\) be a \(\Phi\)-operator in \(L_p^m(\Gamma,\rho)\), it is necessary and sufficient that the operator \(XP+Q\) \((PX+Q)\) be a \(\Phi\)-operator in \(L_p^m(\Gamma,\rho)\).

Proof. Since, for any continuous matrix-function \(C(t)\), the operators \(CP-PC\) and \(CQ-QC\) are completely continuous in \(L_p^m(\Gamma,\rho)\), and the operators \(PN+QM^{-1}\) and \(MP+N^{-1}Q\) are \(\Phi\)-operators, the assertion of the lemma follows from the following easily verified equalities:

\[ MXNP+Q = M\bigl[(XP+Q)(PN+QM^{-1})+X(NP-PN)+ \]

\[ +\,M^{-1}Q-QM^{-1}\bigr], \]

\[ PMXN+Q = \bigl[(MP+N^{-1}Q)(PX+Q)+(PM-MP)X+ \tag{2} \]

\[ +\,QN^{-1}-N^{-1}Q\bigr]N. \]

Lemma 2. Every piecewise-continuous matrix-function \(G(t)\) \((t\in\Gamma)\), satisfying at each point of discontinuity \(t_k\) \((k=1,\ldots,n)\) the condition \(\det G(t_k\pm0)\ne0\), can be represented in the form

\[ G(t)=M(t)X(t)N(t), \tag{3} \]

\[ \text{* If } 2\beta_k=p-2,\text{ then as } W_k(\mu) \text{ one takes the matrix segment } W_k=G(t_k)(1-\mu)+G(t_k+0)\mu. \]

where \(M\) and \(N\) are continuous (nonsingular) matrix-functions, and \(X\) is a piecewise-continuous triangular matrix-function.

Proof. Choose nonsingular constant matrices \(N_k\) so that \(N_kG^{-1}(t_k+0)G(t_k-0)N_k^{-1}\) are upper triangular matrices. Let \(N(t)\) be a continuous nonsingular matrix-function on \(\Gamma\), subject to the sole condition \(N(t_k)=N_k\) \((k=1,\ldots,n)\). As \(X(t)\) take any nonsingular upper triangular matrix-function, continuous at all points except the points \(t_k\), at which \(X(t_k-0)\) is the identity matrix and
\[ X(t_k+0)=N_kG^{-1}(t_k-0)G(t_k+0)N_k^{-1}. \]
Then
\[ G(t_k+0)N_k^{-1}X^{-1}(t_k+0)=G(t_k-0)N_k^{-1}X^{-1}(t_k-0), \]
and, consequently, the matrix-function \(M(t)=G(t)N^{-1}(t)X^{-1}(t)\) is continuous on \(\Gamma\). The lemma is proved.

Proof of Theorem 1. Let \(\det B(t\pm0)\ne0\). The matrix \(G=B^{-1}A\) can be represented in the form (3). Clearly, together with the matrix \(G\), the matrix \(X\) is also \(\omega\)-nonsingular; consequently, all diagonal elements of the matrix-function \(X\) are \(\omega\)-nonsingular functions. From (1) it follows that all diagonal elements of the matrix operator \(XP+Q\) are \(\Phi\)-operators in \(L_p(\Gamma,\rho)\). Hence it follows immediately that \(XP+Q\) are \(\Phi\)-operators in \(L_p^m(\Gamma,\rho)\), and together with it the operator \(AP+BQ\) is also a \(\Phi\)-operator in \(L_p^m(\Gamma,\rho)\).

Formula (*) can be derived without difficulty from the equalities (2), (3) and the results of the work \((^1)\).

We proceed to the proof of the necessity of the conditions of the theorem. We first show that \(\det A(t\pm0)\ne0\) and \(\det B(t\pm0)\ne0\). Suppose that the operator \(AP+BQ\) is a \(\Phi\)-operator and that, for at least one of the matrices (for example, for \(A\)), this condition is not satisfied. Then it is not difficult to choose a continuous matrix-function \(F(t)\) satisfying the following conditions:

1) The norm \(\|FP\|\) is so small that the operator \((A+F)P+BQ\) is a \(\Phi\)-operator.

2) On \(\Gamma\) one can find an arc \(t't''(=\gamma)\) such that \(A(t)\) is continuous on \(\gamma\),
\[ \det(A+F)(t'+0)\ne0,\quad \det(A+F)(t''-0)\ne0, \]
and
\[ \det(A+F)(t_0)=0, \]
where \(t_0\) is some point on \(\gamma\).

Let \(Y(t)\) be some continuous matrix-function coinciding with \(A+F\) on \(\gamma\) and nonsingular from \(\Gamma\setminus\gamma\); then \(A+F\) can be represented in the form of a product
\[ A+F=ZY,\qquad Z\in\Lambda_m. \]
Since \((A+F)P+BQ\) is a \(\Phi\)-operator, it follows from the equality
\[ (A+F)P+BQ=(ZP+BQ)(PY+Q)+Z(YP-PY) \]
that \(PY+Q\) is a \(\Phi_+\)-operator, and this contradicts the fact that \(\det Y(t_0)=0\).

We show that the matrix \(B^{-1}A\) is \(\omega\)-nonsingular. The matrix \(B^{-1}A\) can be represented in the form (3). Since \(B^{-1}AP+Q\) is a \(\Phi\)-operator, \(XP+Q\) is also a \(\Phi\)-operator. Hence we already obtain (using the results of the work \((^1)\)) that \(X\) (and therefore also \(B^{-1}A\)) is an \(\omega\)-nonsingular matrix-function. For the operator \(K\) the theorem is proved. It is proved analogously for the operator \(L\) as well.

Theorem 2. Let \(A(t)\) and \(B(t)\in\Lambda_m(\Gamma)\); \(t_1,\ldots,t_n\ (\in\Gamma)\) are all discontinuity points of the matrices \(A\) and \(B\);
\[ \rho=\prod |t-t_k|^{\beta_k}\quad (-1<\beta_k<p-1;\ 1<p<\infty). \]
In order that the operator \(K=AP+BQ\) \((L=PA+QB)\) be a \(\Phi\)-operator in \(L_p(\Gamma,\rho)\), it is necessary and sufficient that the following two conditions be fulfilled:

\[ \gamma)\quad \det A(t\pm0)B(t\pm0)\ne0\quad \text{for all } t\in\Gamma; \]

\[ \delta)\quad \text{for each eigenvalue } \lambda_j^{(k)} \text{ of the matrix} \]

\[ A^{-1}(t_k+0)B(t_k+0)B^{-1}(t_k)A(t_k) \quad \bigl(B(t_k+0)A^{-1}(t_k+0)A(t_k)B^{-1}(t_k)\bigr) \]

the relation
\[ \beta_k\ne p\alpha_j^{(k)}-1\qquad (j=1,\ldots,m;\ k=1,\ldots,n) \]
holds, where
\[ \alpha_j^{(k)}=(\arg \lambda_j^{(k)})/2\pi\qquad (0\le \alpha_j^{(k)}<1). \]

The formulated theorem obviously follows from Theorem 1 and the following easily proved lemma.

Lemma 3. In order that the matrix \(G(t)\) \((\in \Lambda_m(\Gamma))\) be \(\omega\)-nonsingular, it is necessary and sufficient that the following two conditions be satisfied:

\(1^\circ.\) \(\det G(t \pm 0) \ne 0\) for all \(t \in \Gamma\).

\(2^\circ.\) For each eigenvalue \(\lambda_j^{(k)}\) of the matrix \(G^{-1}(t_k+0)G(t_k)\) the relation
\[ \beta_k \ne p\alpha_j^{(k)} - 1 \quad (j=1,\ldots,m;\ k=1,\ldots,n), \]
holds, where
\[ \alpha_j^{(k)}=(\arg \lambda_j^{(k)})/2\pi \quad (0 \leq \alpha_j^{(k)} < 1). \]

1. As in \((^{1-3})\), the results obtained above can be generalized to the general case when \(\Gamma\) consists of a finite number of closed and open contours.

The results of the present note carry over to the operators \(K\) and \(L\) acting in certain symmetric spaces (see \((^3)\)). From the theorems of this note one naturally derives necessary and sufficient conditions for the operator corresponding to a system of Wiener—Hopf equations (a discrete analogue) with piecewise-continuous symbol to be a \(\Phi\)-operator in \(l_2^m\) \((h_p^m)\).

Institute of Mathematics with Computing Center
Academy of Sciences of the MSSR
Kishinev State University

Received
18 XI 1968

REFERENCES

\(^1\) I. Ts. Gokhberg, N. Ya. Krupnik, DAN, 185, No. 4 (1969).
\(^2\) I. Ts. Gokhberg, N. Ya. Krupnik, Matem. issl., Kishinev, 3, 1 (1968).
\(^3\) I. Ts. Gokhberg, N. Ya. Krupnik, Studia Math., 31, 347 (1968).
\(^4\) I. Ts. Gokhberg, Functional Analysis and Its Applications, 1, 2 (1967).
\(^5\) N. P. Vekua, Systems of Singular Integral Equations, Moscow—Leningrad, 1950.
\(^6\) B. V. Khvedelidze, Tr. Tbilissk. matem. inst., 23, 3 (1956).
\(^7\) E. Shamir, DAN, 167, No. 5 (1966).
\(^8\) I. Yu. Simonenko, Izv. AN SSSR, ser. matem., 28, No. 2, 277 (1964).