Mathematics

V. V. IVANOV

SOME PROPERTIES OF SINGULAR INTEGRALS OF CAUCHY TYPE AND THEIR APPLICATIONS

(Presented by Academician I. N. Vekua on 14 IV 1958)

Let us take a function \(f(t)\), defined on a simple closed Lyapunov curve \(L\), situated in the complex plane with the origin inside \(L\), and consider the integral \(Sf=\dfrac{1}{\pi i}\int_L \dfrac{f(t)\,dt}{t-t_0}\), \(t_0\in L\), understood as an improper integral in the sense of the Cauchy principal value. Introduce the classes of continuous functions \(H(p+\alpha,L)\), where \(p\) is an integer, \(p\ge 0\), \(0\le \alpha\le 1\). A function \(f(t)\in H(p+\alpha,L)\) if its \(p\)-th derivative \(f^{(p)}(t)\) satisfies on \(L\) the condition

\[ f^{(p)}(t_1)-f^{(p)}(t_2)=O\left(|t_1-t_2|^\alpha\right),\qquad t_1,\ t_2\in L. \]

The well-known theorem of Plemelj–Privalov \((^1)\) states that from \(f\in H(p+\alpha,L)\) it follows that \(Sf\in H(p+\alpha,L)\), and conversely, if \(0<\alpha<1\). For \(\alpha=0,1\) this is no longer so.

The present note is devoted to the study of the limiting cases of the Plemelj–Privalov theorem \((\alpha=0,1)\). The results obtained are then applied to the question of the structural properties of the solution of the Riemann boundary-value problem and of the singular integral equation of normal type.

Let \(Z(p,L)\) be the class of functions defined on \(L\) whose \(p\)-th derivative satisfies the relation

\[ f^{(p)}[t,(s+h)]+f^{(p)}[t(s-h)]-2f^{(p)}[t(s)]=O(h), \]

where \(s\) is the arc abscissa of the point \(t(s)\in L\). Denote by \(\rho_n(f,L)\) the magnitude of the best approximation of the function \(f\), given on \(L\), by means of polynomials of degree \(n\) in the positive and negative powers of \(t\):

\[ \rho_n(f,L)=\sup_{P_n\in H_n}\inf_{t\in L}|f(t)-P_n(t)|, \]

where \(H_n\) is the totality of all polynomials of the form \(\sum_{-n}^{n} a_k t^k\).

Theorem 1. If \(f\in Z(p,L)\), then \(Sf\in Z(p,L)\), and conversely.

Theorem 2. If \(f\in Z(p,L)\), then \(\rho_n(f,L)=O(n^{-p-1})\), and conversely.

Theorem 3. If \(\rho_n(f,L)=O(n^{-p-\alpha})\), then \(\rho_n(Sf,L)=O(n^{-p-\alpha})\), \(0<\alpha\le 1\), and conversely.

Theorem 4. If \(f\in C\), i.e., is continuous on \(L\), then \(\exp(Sf)\in L_p\), \(p>0\), i.e., is summable on \(L\) with any power \(p>0\).

The proof of all these theorems is based on the representation

\[ Sf=\frac{1}{\pi i}\int_{|w|=1}\frac{f[\psi(w)]}{w-w_0}\,dw +\frac{1}{\pi i}\int_{|w|=1} f[\psi(w)] \left[ \frac{\psi'(w)}{\psi(w)-\psi(w_0)}-\frac{1}{w-w_0} \right]dw, \]

in which \(t=\psi(w)\) are the boundary values of a function conformally mapping the exterior of the circle \(|w|=1\) onto the exterior of \(L\), \(t_0=\psi(w_0)\), and also to the corresponding (when \(L\) is the circle: \(|w|=1\)) or similar results of the authors \((^{2-4})\).

Let us now consider the Riemann boundary-value problem \((^{1,5,6})\): to find a piecewise-holomorphic function of the form \(\dfrac{1}{2\pi i}\int_L \dfrac{\varphi(\tau)\,d\tau}{\tau-z}\), whose angular boundary values \(\varphi^+\) and \(\varphi^-\), respectively from inside and from outside \(L\), satisfy the condition

\[ \varphi^+ = G\varphi^- + f \tag{1} \]

almost everywhere on \(L\), where \(G\) is continuous, and \(f\in L_r,\ r>1\), on \(L\), the index \(\varkappa=\dfrac{1}{2\pi}[\arg G]_L=0\).

We shall regard two functions as identical if their values coincide almost everywhere on \(L\). Following the reasoning of F. D. Gakhov \((^5)\), represent \(G\) in the form \(G=\psi^+/\psi^-\), where

\[ \psi^\pm=\exp\left[\pm \frac{1}{2}\ln G+\frac{1}{2}S(\ln G)\right] \tag{2} \]

and \((\psi^\pm)^{-1}\), by virtue of Theorem 4, belongs to \(L_p\) for every \(p>0\). After this, relation (1) is given the form

\[ \frac{\varphi^+}{\psi^+}-\frac{\varphi^-}{\psi^-}=\frac{f}{\psi^+}. \tag{3} \]

By Hölder’s inequality \(\dfrac{f}{\psi^+}\in L_{r-\varepsilon}\), where \(\varepsilon\) is any arbitrarily small positive number. Solving problem (3) (see \((^6)\)), we obtain

\[ \varphi^\pm=\psi^\pm\left[\pm \frac{1}{2}\frac{f}{\psi^+}+\frac{1}{2}S\left(\frac{f}{\psi^+}\right)\right]. \tag{4} \]

Thus, under the assumptions made, there exists a unique solution of the Riemann problem belonging to \(L_{r-\varepsilon}\).*

By modifying and generalizing the above arguments, one can similarly investigate the Riemann problem when the index \(\varkappa\ne0\) and when \(f(t)\) is a piecewise-continuous function with a finite number of discontinuities of the first kind. The results obtained can also be applied to the study of a general singular integral equation of normal type.

From Theorem 3 and formulas (2), (4) it follows: if \(\rho_n(G,L)\) and \(\rho_n(f,L)\) have order \(n^{-p-\alpha}\) (this will be the case if, for example, \(G\) and \(f\) belong to \(H(p+\alpha,L)\)), then \(\rho_n(\varphi^\pm,L)\) also have order \(n^{-p-\alpha}\). This last assertion may find application in investigating the rate of convergence of an approximate solution of the singular integral equation \((^7)\).

Rostov-on-Don Institute
of Agricultural-Machinery Construction

Received
14 IV 1958

CITED LITERATURE

\({}^1\) N. I. Muskhelishvili, Singular Integral Equations, Moscow–Leningrad, 1946.
\({}^2\) A. Zygmund, Duke Math. J., 12, 47 (1945).
\({}^3\) V. I. Smirnov, Math. Ann., 107, 313 (1932).
\({}^4\) S. Ya. Alper, Izv. Akad. Nauk SSSR, Ser. Math., 9, No. 6 (1955).
\({}^5\) F. D. Gakhov, Matem. sbornik, 2, No. 4 (1937).
\({}^6\) B. V. Khvedelidze, Tr. Tbilisi Math. Inst., 23 (1956).
\({}^7\) V. V. Ivanov, DAN, 114, No. 5 (1957).

* I. B. Simonenko showed that in the case \(r=2\) in the conclusion formulated, \(\varepsilon\) can be omitted.