MATHEMATICS

A. A. BABAEV, V. V. SALAEV

ON AN ANALOGUE OF THE PLEMELJ–PRIVALOV THEOREM IN THE CASE OF NONSMOOTH CURVES AND ITS APPLICATIONS

(Presented by Academician I. N. Vekua on 31 VIII 1964)

Let a closed rectifiable Jordan curve \(\Gamma\) satisfy the metric relation

\[ s(t_1,t_2)\leq \beta(|t_1-t_2|), \tag{1} \]

where \(s(t_1,t_2)\) is the smaller of the two arcs joining the points \(t_1\) and \(t_2\), and the function \(\beta(\delta)\), \(0\leq \delta\leq l_0\) (\(l_0\) is the diameter of the curve \(\Gamma\)), satisfies the conditions: 1) \(\beta(\delta)\) is a continuous, increasing function; 2) \(\beta(0)=0\); 3) \(\beta(\delta)/\delta\) is almost decreasing, i.e., there exists a constant \(k>0\) such that for \(\delta_1>\delta_2\),
\[ \beta(\delta_1)/\delta_1 \leq k\,\beta(\delta_2)/\delta_2 . \]

By \(\alpha(\delta)\) we denote the function inverse to \(\beta(\delta)\). Introduce the modulus of continuity of the function \(f(t)\), defined on \(\Gamma\):

\[ \omega(f,\delta)=\sup_{|t_1-t_2|\leq \delta}|f(t_1)-f(t_2)|,\qquad 0<\delta\leq l_0 . \]

Consider the integral

\[ \int_{\Gamma}\frac{f(t)}{t-t_0}\,dt = \lim_{\varepsilon\to 0} \int_{\Gamma-\Gamma_\varepsilon(t_0)} \frac{f(t)}{t-t_0}\,dt, \tag{2} \]

where \(\Gamma_\varepsilon(t_0)\) denotes the part of the curve \(\Gamma\) containing the point \(t_0=t(s_0)\) and having endpoints \(t(s_0-\varepsilon)\), \(t(s_0+\varepsilon)\).

Consider the class \(\Phi\) of functions \(\varphi(\delta)\), defined on \([0,l/2]\) and satisfying the conditions: 1) \(\varphi(\delta)\) is continuous and monotonically increasing on \([0,l/2]\); 2) \(\varphi(\delta)\neq 0\) for every \(\delta\in[0,l/2]\), \(\varphi(0)=0\); 3) \(\varphi(\delta)/\delta\) is almost decreasing.

Let \(\varphi(\delta)\in\Phi\). A function \(f(t)\), defined on \(\Gamma\), is said to belong to the class \(H_\varphi\) if there exists a constant \(c>0\) such that, for \(0<\delta\leq l_0\),
\[ \omega(f,\delta)\leq c\varphi(\delta). \]

If in \(H_\varphi\) one introduces the norm

\[ \|f\|_{H_\varphi} = \max_{t\in\Gamma}|f(t_0)| + \sup_{0<\delta\leq l_0}\bigl(\omega(f,\delta)/\varphi(\delta)\bigr), \]

then, as is not difficult to see, \(H_\varphi\) becomes a Banach space.

Theorem 1. Let a closed rectifiable Jordan curve \(\Gamma\) satisfy condition (1) and

\[ \int_{\Gamma}\frac{dt}{t-t_0}=\pi i. \]

If a function \(f(t)\in H_\varphi\), defined on \(\Gamma\), is such that

\[ \int_0^{l/2}\frac{\varphi(\alpha(s))}{\alpha(s)}\,ds<+\infty, \]

then for the function

\[ g(t_0)=\int_\Gamma \frac{f(t)}{t-t_0}\,dt \]
the inequality
\[ \omega(g,\delta)\leq c\|f\|\left[\int_0^{\beta(\delta)}\frac{\varphi(\alpha(s))}{\alpha(s)}\,ds+\delta\int_{\beta(\delta)}^{l/2}\frac{\varphi(\alpha(s))}{\alpha(s)}\,ds\right] \tag{3} \]
holds for \(0<\delta\leq\delta_0\), where the constants depend on \(\Gamma\) and \(\varphi\); \(\delta_0\) depends only on the curve \(\Gamma\) (\(l\) is the length of the curve \(\Gamma\)).

Theorem 2 \((^3)\). Let \(\varphi(\delta), \psi(\delta)\in\Phi\).

1) If
\[ 0<\lim_{\delta\to0}\frac{\varphi(\delta)}{\psi(\delta)} \leq \lim_{\delta\to0}\frac{\varphi(\delta)}{\psi(\delta)}<+\infty, \]
then the classes \(H_\varphi\) and \(H_\psi\) coincide, and the norms are equivalent.

2) If
\[ \lim_{\delta\to0}\frac{\varphi(\delta)}{\psi(\delta)}=0, \]
then \(H_\varphi\) is a proper part of \(H_\psi\).

By definition, a function \(\varphi(\delta)\in\psi(\beta)\) if \(\varphi(\delta)\in\Phi\) and the relation
\[ \int_0^{\beta(\delta)}\frac{\varphi(\alpha(s))}{\alpha(s)}\,ds = O\left[\frac{\varphi(\delta)\beta(\delta)}{\delta}\right]. \tag{z} \]
is satisfied.

We also consider the class \(\psi_1(\beta)\) of functions \(\varphi(\delta)\in\Phi\) such that
\[ \int_{\beta(\delta)}^{l/2}\frac{\varphi(\alpha(s))}{\alpha^2(s)}\,ds = O\left[\frac{\varphi(\delta)\beta(\delta)}{\delta^2}\right]. \tag{z_1} \]

From the analysis in the work \((^4)\) it follows:

Lemma. 1) If \(\varphi(\delta)\in\psi(\beta)\), then condition \((z)\) is equivalent to each of the following two:

\((I)\) there exists a constant \(c>1\) such that
\[ \lim_{\delta\to0}\frac{\varphi(\alpha(c\delta))\alpha(\delta)} {\varphi(\alpha(\delta))\alpha(c\delta)} > \frac{1}{c}. \]

\((s)\) there exists a constant \(a\) \((0<a<1)\) such that the function
\[ \frac{\varphi(\delta)}{\delta}[\beta(\delta)]^{1-a} \]
is almost increasing on \([0,l/2]\).

2) Similarly, if \(\varphi(\delta)\in\psi_1(\beta)\), then condition \((z_1)\) is equivalent to each of the following:

\((I_1)\). There exists a constant \(c>1\) such that
\[ \lim_{\delta\to0}\frac{\varphi(\alpha(c\delta))\alpha^2(\delta)} {\varphi(\alpha(\delta))\alpha^2(\delta)} < \frac{1}{c}. \]

\((s_1)\) There exists a constant \(\alpha>0\) such that the function
\[ \frac{\varphi(\delta)}{\delta^2}[\beta(\delta)]^{1+\alpha} \]
is almost decreasing.

Theorem 3. Let the closed rectifiable Jordan curve \(\Gamma\) satisfy condition (1) and
\[ \int_\Gamma \frac{dt}{t-t_0}=\pi i; \]
let \(\varphi(\delta)\in \psi(\beta)\cap\psi_1(\beta)\).

Then, if \(\omega(f,\delta)=O[\varphi(\delta)]\), then
\[ \omega(g,\delta)=O\left[\varphi(\delta)\frac{\beta(\delta)}{\delta}\right], \]
where
\[ g(t_0)=\int_\Gamma \frac{f(t)}{t-t_0}\,dt\quad (t_0\in\Gamma). \]

It is not difficult to formulate Theorem 3 in terms of the operator
\[ Af=\int_\Gamma \frac{f(t)}{t-t_0}\,dt. \]

Corollary 1. If the conditions of Theorem 3 are satisfied, then the operator \(A\) acts from \(H_\varphi\) into \(H_\psi\), where

\[ \psi(\delta)=\varphi(\delta)\frac{\beta(\delta)}{\delta}, \]

and is bounded.

Corollary 2. If \(\Gamma\) is a smooth curve, \(\varphi(\delta)\in\Phi\), and there exists a constant \(c>1\) such that

\[ 1<\lim_{\delta\to 0}\frac{\varphi(c\delta)}{\varphi(\delta)} \leqslant \overline{\lim_{\delta\to 0}}\frac{\varphi(c\delta)}{\varphi(\delta)}<c, \]

then the operator \(A\) acts in \(H_\varphi\) and is bounded.

This assertion, which is a direct generalization of the Plemelj–Privalov theorem, was proved in \((^3)\).

Corollary 3. If \(\beta(\delta)=\operatorname{const}\,\delta^\gamma\), \(0<\gamma<1\), then the operator \(A\) acts boundedly from \(H_\alpha\) into \(H_{\alpha-(1-\gamma)}\) for \(1-\gamma<\alpha\), where \(H_\alpha\) \((0<\alpha<1)\) denotes \(H_{\delta^\alpha}\).

Using the theorem of N. A. Davydov \((^5)\) on boundary values of the Cauchy integral, Theorem 3, and the method of solving the Riemann problem \((^{2,6})\), one can prove the following theorem.

Theorem 4. Suppose that for a closed rectifiable Jordan curve \(\Gamma\) condition (1) is satisfied and

\[ \int_\Gamma \frac{dt}{t-t_0}=\pi i,\qquad \text{where } \beta(\delta)=\operatorname{const}\,\delta^\gamma,\quad 1/2<\gamma\leqslant 1. \]

Then for the Riemann problem

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

where \(G(t)\ne 0\), \(t\in\Gamma\), \(G(t)\in H_{2(1-\gamma)+\varepsilon}\), \(g(t)\in H_{(1-\gamma)+\varepsilon}\), \(\varepsilon>0\), all the theorems proved in \((^6)\) are valid.

Azerbaijan State University

Received
22 VIII 1964

REFERENCES

\(^1\) A. A. Alekseev, DAN, 136, No. 3 (1961).
\(^2\) N. I. Muskhelishvili, Singular Integral Equations, Moscow, 1962.
\(^3\) A. A. Babaev, Scientific Notes of Azerbaijan State University, series of physical and mathematical sciences, No. 4 (1963).
\(^4\) N. K. Bari, S. B. Stechkin, Trudy Moskov. matem. obshch., 5, 483 (1956).
\(^5\) N. A. Davydov, DAN, 64, No. 6 (1949).
\(^6\) F. D. Gakhov, Boundary Value Problems, Moscow, 1963.