UDC 517.948.32

MATHEMATICS

A. A. BABAEV

SOME PROPERTIES OF A SINGULAR INTEGRAL WITH DISCONTINUOUS DENSITY AND ITS APPLICATIONS

(Presented by Academician I. N. Vekua on 26 X 1965)

1. Let \(\Gamma\) be a closed Jordan rectifiable curve, at each point of which a tangent exists, and let \(S(t_1,t_2)\) denote the smaller of the lengths of the arcs joining the points \(t_1,t_2 \in \Gamma\). Suppose that

\[ S(t_1,t_2)\leqslant \beta(|t_1-t_2|), \tag{1} \]

where \(\beta(\delta)\) is a continuous, increasing function on \((0,l_0]\) (\(l_0\) is the diameter of \(\Gamma\)); \(\lim_{\delta\to 0}\beta(\delta)=0\), and \(\beta(\delta)/\delta\) is almost decreasing. Let \(\alpha(\delta)\) be the inverse function of \(\beta(\delta)\); \(l\) the length of the curve \(\Gamma\). Denote by \(\Phi\) the class of functions \(\varphi(\delta)\), defined on \((0,l_0]\) and having the following properties: 1) \(\varphi(\delta)\) is continuous and monotonically increasing on \((0,l_0]\); 2) \(\varphi(\delta)\ne 0\) and \(\lim_{\delta\to 0}\varphi(\delta)=0\); 3) \(\varphi(\delta)/\delta\) is almost decreasing, i.e. \(\varphi(\delta_2)/\delta_2 \leqslant C_\varphi \varphi(\delta_1)/\delta_1\) for \(\delta_2>\delta_1\).

Introduce the modulus of continuity of a function \(f(t)\), defined on \(\Gamma\):

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

Theorem 1. If \(\Gamma\) satisfies the conditions stated above and

\[ \omega(f,\delta)\leqslant C_f\varphi(\delta),\qquad \varphi(\delta)\in\Phi,\qquad \int_0^{l/2}\frac{\varphi(\alpha(s))}{\alpha(s)}\,ds<+\infty, \]

then

\[ \omega(g,\delta)\leqslant CC_f\left[ \int_0^{\beta(\delta)}\frac{\varphi(\alpha(s))}{\alpha(s)}\,ds + \delta\int_{\beta(\delta)}^{l/2}\frac{\varphi(\alpha(s))}{\alpha^2(s)}\,ds \right], \qquad 0<\delta\leqslant \widetilde l_0\leqslant l_0, \]

where

\[ g(t_0)=\frac{1}{\pi i}\int_\Gamma \frac{f(t)}{t-t_0}\,dt, \]

and the constant \(C\) depends only on the curve \(\Gamma\) and on the constant \(C_\varphi\) (\(\widetilde l_0\) depends only on \(\Gamma\)).

Remark. This theorem* was given in paper (1). Here it is included in order to note the dependence of \(C\) only on \(\Gamma\) and \(C_\varphi\), which will be used essentially in what follows.

With the aid of theorem (1) and the remark, one proves

Theorem 2. Let \(\Gamma\) be a closed Jordan rectifiable curve, having a tangent at every point and satisfying the condition

\[ S(t_1,t_2)\leqslant K|t_1-t_2|,\qquad K=\mathrm{const}. \tag{2} \]

Then, if

\[ \int_0^{l_0}\frac{\omega(f,\tau)}{\tau}\,d\tau<+\infty, \]

then for the function

\[ g(t_0)=\frac{1}{\pi i}\int_\Gamma \frac{f(t)}{t-t_0}\,dt \]

* In paper (1), in inequality (3), in the second integral, \(\alpha(s)\) was printed in the denominator; it should read \(\alpha^2(s)\).

there is the inequality

\[ \omega(g,\delta)\leq C\left[\int_0^\delta \frac{\omega(f,\tau)}{\tau}\,d\tau+ \delta\int_\delta^{l_0}\frac{\omega(f,\tau)}{\tau^2}\,d\tau\right], \qquad 0<\delta\leq l_0, \tag{3} \]

where \(C\) depends only on \(\Gamma\).

An inequality close to inequality (3), in the case of smooth curves, was first obtained by L. G. Magnaradze \((^2)\).

II. Denote by \(\Psi\) the class of positive, continuous functions \(\psi(\delta)\), defined on \((0,l_0]\) and having the properties: 1)

\[ \int_0^{l_0}\psi(u)\,du=+\infty; \]

2)

\[ \int_0^{l_0}u\psi(u)\,du<+\infty. \]

In this section it is assumed that \(\Gamma\) satisfies the conditions of theorem (2).

Let \(\psi(\delta)\in\Psi\). Denote by \(J_\psi\) the class of functions \(f(t)\), defined on \(\Gamma\), for which

\[ \int_0^{l_0}\omega(f,\tau)\psi(\tau)\,d\tau<+\infty. \]

The following theorem partially solves the question of the classification of \(J_\psi\).

Theorem 3. Let \(\psi_1(\delta),\psi_2(\delta)\in\Psi\). If

\[ 0<\underline{\lim}_{\delta\to0}\bigl(\psi_1(\delta)/\psi_2(\delta)\bigr) \leq \overline{\lim}_{\delta\to0}\bigl(\psi_1(\delta)/\psi_2(\delta)\bigr)<+\infty, \]

then \(J_{\psi_1}\) and \(J_{\psi_2}\) coincide, while if

\[ \lim_{\delta\to0}\bigl(\psi_1(\delta)/\psi_2(\delta)\bigr)=+\infty, \]

then \(J_{\psi_1}\) is a proper part of \(J_{\psi_2}\).

With the aid of theorems 2 and 3 one proves

Theorem 4. Let \(\psi(\delta)\in\Psi\) and

\[ \lim_{\delta\to0}\delta^2\psi(\delta)=0, \qquad \lim_{\delta\to0}\left(\delta\psi(\delta)\bigg/\int_\delta^{l_0}\psi(\tau)\,d\tau\right)=K \quad (0<K<1). \]

Then \(J_\psi\) is invariant with respect to the operator

\[ Af=\frac{1}{\pi i}\int_\Gamma \frac{f(t)}{t-t_0}\,dt. \]

It is not difficult to verify that the functions
\(\psi(\delta)=1/\delta^{1+\varepsilon}\),
\(\psi(\delta)=\ln|1/\delta|/\delta^{1+\varepsilon}\)
\((0<\varepsilon<1)\) satisfy the conditions of theorem 4, and the classes \(J_\psi\) generated by them, by virtue of theorem 3, are different for different \(\varepsilon\).

For the further arguments the following is useful

Lemma. Let \(\psi(\delta)\in\Psi\) and

\[ \lim_{\delta\to0}\delta^2\psi(\delta)=0, \qquad \lim_{\delta\to0}\left(\delta\psi(\delta)\bigg/\int_\delta^{l_0}\psi(\tau)\,d\tau\right)=0. \tag{4} \]

Then the function

\[ \frac{1}{\delta}\int_\delta^{l_0}\psi(\tau)\,d\tau=\psi_1(\delta)\in\Psi \]

also satisfies conditions (4).

With the aid of theorems 2 and 3 and the lemma one proves

Theorem 5. Let \(\psi(\delta)\in\Psi\) and satisfy conditions (4). Then the operator \(A\) maps \(J_{\psi_{i+1}}\) into \(J_{\psi_i}\), where

\[ \psi_{i+1}(\delta)=\frac{1}{\delta}\int_\delta^{l_0}\psi_i(\tau)\,d\tau \quad (i=0,1,2,\ldots),\qquad \psi_0(\delta)=\psi(\delta), \]

and \(J_{\psi_{i+1}}\) is a proper part of \(J_{\psi_i}\).

This result in the case \(\psi(\delta)=1/\delta\) was obtained by L. G. Magnaradze \((^2)\). Theorem 5 makes it possible to construct a sequence \(\{J_{\psi_i}\}\) different from the sequence \(\{J_i\}\) constructed by L. G. Magnaradze in the same paper.*

We note that, by virtue of Zygmund’s estimate \((^3)\), Theorems 4 and 5 are also valid for trigonometrically conjugate functions.

Remark. Let us point out that X. Sjoe-Mou \((^4)\) succeeded in proving analogues of the theorems of L. G. Magnaradze expressing the relation between the modulus of continuity of \(\varphi(t)\) in \(L_p\) \((p>1)\) and the modulus of continuity in \(L_p\) of the angular boundary values of the Cauchy-type integral

\[ F(z)=\frac{1}{2\pi i}\int_{\Gamma}\frac{\varphi(t)}{t-z}\,dt \]

(\(\Gamma\) is a closed Jordan rectifiable curve satisfying condition (2)).

By virtue of the estimate, obtained in the same paper, expressing the relation between the modulus of continuity of \(\varphi(t)\) in \(L_p\) and the modulus of continuity in \(L_p\) of the angular boundary values \(F(z)\), Theorems 4 and 5 are valid in this case as well.

III. Consider the singular integral

\[ \Phi(t_0,\tau)=\int_{\Gamma}\frac{f(t,\tau)}{t-t_0}\,dt, \]

where \(t_0\in\Gamma,\ \tau\in D\) (\(D\) is some bounded set in the complex plane). Denote

\[ \omega_t(f,\delta)=\sup_{\tau}\ \sup_{|t_1-t_2|\le \delta}\left|f(t_1,\tau)-f(t_2,\tau)\right|, \]

\[ \omega_\tau(f,\delta)=\sup_t\ \sup_{|\tau_1-\tau_2|\le \delta}\left|f(t,\tau_1)-f(t,\tau_2)\right|. \]

With the aid of Theorem 1 one proves

Theorem 6. Let \(\Gamma\) be a closed Jordan rectifiable curve, having a tangent at every point and satisfying condition (1). If

\[ \omega_t(f,\delta)=O[\varphi(\delta)],\qquad \omega_\tau(f,\delta)=O[\widetilde{\varphi}(\delta)], \]

\[ \varphi(\delta)\in\Psi[\beta(\delta)]\cap\Psi_1[\beta(\delta)]^{**},\qquad \widetilde{\varphi}(\delta)\int_{\beta(\delta)}^{1/2}\frac{ds}{\alpha(s)} =O\left[\varphi(\delta)\frac{\beta(\delta)}{\delta}\right] \]

(\(\widetilde{\varphi}(\delta)\) is a positive function), then

\[ \omega_{t_0}(\Phi,\delta)=O\left[\varphi(\delta)\frac{\beta(\delta)}{\delta}\right],\qquad \omega_\tau(\Phi,\delta)=O\left[\varphi(\delta)\frac{\beta(\delta)}{\delta}\right]. \]

Let us note one important particular case of this theorem, which is a generalization of a theorem of N. I. Muskhelishvili \((^5)\) on a singular integral containing a parameter.

Theorem 7. Let \(\Gamma\) be a closed Jordan rectifiable curve, having a tangent at every point and

\[ S(t_1,t_2)\le \mathrm{const}\,|t_1-t_2|^\gamma \qquad (0<\gamma\le 1). \]

If

\[ \omega_t(f,\delta)=O[\delta^\alpha],\qquad \omega_\tau(f,\delta)=O[\delta^{\alpha_1}],\qquad 1-\gamma<\alpha<\alpha_1\le 1, \]

then

\[ \omega_{t_0}(\Phi,\delta)=O[\delta^{\alpha-(1-\gamma)}],\qquad \omega_\tau(\Phi,\delta)=O[\delta^{\alpha-(1-\gamma)}]. \]

IV. Denote \(\displaystyle \bigcup_{\alpha>\beta} H_\alpha\) by \(M_\beta\) \((0\le \beta<1)\), where \(H_\alpha\) is the class of functions,

* For example, it follows from Theorem 3 that, if \(\psi(\delta)=|\ln|\ln(1/\delta)||/\delta\), then for every \(i=1,2,\ldots\) the class \(J_{\psi_i}\) will be strictly contained between the classes \(J_{i+1}\) and \(J_i\) \((J_{i+1}\subset J_{\psi_i}\subset J_i)\).

** The definition of the classes \(\Psi[\beta(\delta)]\) and \(\Psi_1[\beta(\delta)]\) is given in \((^1)\).

satisfying on \(\Gamma\) the Hölder condition with exponent \(\alpha\). Consider the singular integral equation

\[ R\varphi=A(t_0)\varphi(t_0)+\frac{B(t_0)}{\pi i}\int_{\Gamma}\frac{\varphi(t)}{t-t_0}\,dt+\frac{1}{\pi i}\int_{\Gamma}N(t_0,t)\varphi(t)\,dt=f(t_0) \]

and its adjoint equation

\[ R'\psi=A(t_0)\psi(t_0)-\frac{1}{\pi i}\int_{\Gamma}\frac{B(t)\psi(t)}{t-t_0}\,dt+\frac{1}{\pi i}\int_{\Gamma}N(t,t_0)\psi(t)\,dt=g(t_0). \]

With the aid of Theorem 7 and the Carleman–Vekua method \({}^{(6)}\), the following is proved.

Theorem 8. Let \(\Gamma\) satisfy the conditions of Theorem 7, \(2/3<\gamma\leq 1\). If \(A^2(t_0)-B^2(t_0)\neq 0,\ t_0\in\Gamma,\ A(t_0), B(t_0)\in M_{3(1-\gamma)};\ N(t_0,t)\), in both arguments, uniformly respectively in \(t\) and \(t_0\), belongs to \(M_{2(1-\gamma)}\), then the following assertions are true:

1. The number of linearly independent solutions of the equations \(R\varphi=0\) and \(R'\psi=0\) in \(M_{1-\gamma}\) is finite.

2. For \(f(t_0)\in M_{2(1-\gamma)}\), in order for the equation \(R\varphi=f\) to be solvable in \(M_{1-\gamma}\), it is necessary and sufficient that

\[ \int_{\Gamma} f(t)\psi_k(t)\,dt=0\qquad (k=1,\ldots,m'), \]

where \(\psi_1(t),\ldots,\psi_{m'}(t)\) is a complete system of linearly independent solutions of the adjoint homogeneous equation \(R'\psi=0\) in \(M_{1-\gamma}\).

1. If by \(m\) and \(m'\) we denote respectively the number of linearly independent solutions of \(R\varphi=0\) and \(R'\psi=0\) in \(M_{1-\gamma}\), then

\[ m-m'=\frac{1}{2\pi i}\left[\ln\frac{A-B}{A+B}\right]_{\Gamma}, \]

where \([\ ]_{\Gamma}\) denotes the increment of the expression in brackets when traversing \(\Gamma\) in the positive direction.

Azerbaijan State University
named after S. M. Kirov

Received
12 X 1965

REFERENCES

\({}^{1}\) A. A. Babaev, V. V. Salaev, DAN, 161, No. 2 (1965).
\({}^{2}\) L. G. Magnaradze, Communications of the Academy of Sciences of the Georgian SSR, 8, No. 8 (1947).
\({}^{3}\) N. K. Bari, S. B. Stechkin, Trudy Moskov. Mat. Obshch., 5 (1956).
\({}^{4}\) U. Soe-Mou, RZhMat, No. 10, 11508 (1960).
\({}^{5}\) I. N. Muskhelishvili, Singular Integral Equations, Moscow, 1962.
\({}^{6}\) I. N. Vekua, Trudy Tbilissk. Mat. Inst., 10 (1941).