MATHEMATICS

G. M. MAGOMEDOV

INVESTIGATION OF SOME SINGULAR INTEGRAL OPERATORS AND THE CORRESPONDING NONLINEAR EQUATIONS

(Presented by Academician I. N. Vekua, 20 I 1970)

In the present note we give some inequalities for the operators

\[ A_1 f \equiv \int_{\Gamma} \frac{f(t_0,\tau)}{\tau-t}\,d\tau \equiv A(t_0,t), \tag{a\(_1\)} \]

\[ A_2 f \equiv \int_{\Gamma} \frac{f(t,\tau)}{\tau-t}\,d\tau \equiv A(t,t) \tag{a\(_2\)} \]

and theorems on the existence and uniqueness of solutions of the equations

\[ u(t_0,t)=\lambda K_1u \equiv \lambda \int_{\Gamma} \frac{K[t_0,\tau,u(t_0,\tau)]}{\tau-t}\,d\tau, \tag{a\(_1^0\)} \]

\[ u(t)=\lambda K_2u \equiv \lambda \int_{\Gamma} \frac{K[t,\tau,u(\tau)]}{\tau-t}\,d\tau \tag{a\(_2^0\)} \]

in certain subsets of the space \(L_p\).

I. Let \(\Gamma\) be a closed or open rectifiable Jordan curve \(l\) such that, for any \(u(t)\in L_p(\Gamma)\), the generalized Riesz inequality holds,

\[ \|S_\varphi\|\le A_p\|\varphi\|_{L_p}, \qquad \text{where } \quad S_\varphi \equiv \int_{\Gamma}\frac{\varphi(\tau)}{\tau-t}\,d\tau . \tag{b} \]

Examples of contours for which (b) holds are given in works \((^{1-3})\). Let \(u(t_0,t)\in L_p(\Gamma\times\Gamma)\); \(\xi\) and \(\zeta\) are arc abscissas of the line \(\Gamma\). Denote

\[ \|u\|_{L_p}\equiv \left\{\int_{\Gamma}\int_{\Gamma}|u(t_0,t)|^p\,|dt_0|\,|dt|\right\}^{1/p}, \]

\[ w(u;\sigma_1,\sigma_2)= \sup_{\substack{0<h_1<\sigma_1\\0<h_2<\sigma_2}} \left\{ \int_0^{\,l-\sigma_2}\int_0^{\,l-\sigma_1} \left|u[t(\xi+h),\tau(\zeta+h_2)]-u[t(\xi),\tau(\zeta)]\right|^p \,d\xi\,d\zeta \right\}^{1/p}. \]

If \(f(t_0,t)\in L_p(\Gamma\times\Gamma)\), then from (b) it follows that

\[ \|A_1 f\|_{L_p}\le A_n\|f\|_{L_p\Gamma\times\Gamma}. \tag{1} \]

If one uses (b) and the generalized Minkowski inequality, an analogue of the Zygmund–Magaradze inequality is obtained:

\[ w(A_1f;\sigma_1,\sigma_2)\le M_1\int_0^{\sigma_2}\frac{w(f,0,\xi)}{\xi}\,d\xi + M_2\sigma_2\int_{\sigma_2}^{l}\frac{w(f,0,\xi)}{\xi^2}\,d\xi + A_p w(f,\sigma_1,0). \tag{2} \]

Definition 1. A function \(u(t_0,t)\in H_{\delta_1,\delta_2}^{p}(N,N_1,N_2)\) if

\[ \|u\|_{L_p}\le N, \tag{b\(_1\)} \]

\[ w(u;\sigma_1,\sigma_2)\le N_1\sigma_1^{\delta_1}+N_2\sigma_2^{\delta_2}. \tag{b\(_2\)} \]

It is evident that \(H_{\delta_1,\delta_2}^{p}(N,N_1,N_2)\) is a compact set in \(L_p\). Let

the function \(K(t_0,t,u)\) \((t_0,t\in\Gamma\times\Gamma,\) and \(u\) is an arbitrary point of the complex plane) satisfies the conditions:

\[ \left\{ \int_0^{\,l-\sigma_1}\int_0^{\,l-\sigma_2} \left|K\left[t(\xi+\sigma_1),t(\eta+\sigma_2),u(t(\xi),t(\eta))\right]\right.\right. \]

\[ \left.\left. {}-K\left[t(\xi),t(\eta),u(t(\xi),t(\eta))\right]\right|^p\,d\xi\,d\eta \right\}^{1/p} \le M_3\sigma_1^{\delta_1}+M_4\sigma_2^{\delta_2}, \tag{3} \]

\[ \left\{ \iint_L\iint_L \left|K[t_0,t,u(t_0,t)]-K[t_0,t,v(t_0,t)]\right|^p\,|dt|\,|dt_0| \right\}^{1/p} \le \]

\[ \le M_5 \left\{ \iint_L\iint_L \left|u(t_0,t)-v(t_0,t)\right|^p\,|dt_0|\,|dt| \right\}^{1/p} \tag{4} \]

for arbitrary \(u(t_0,t),v(t_0,t)\) from \(L_p(\Gamma\times\Gamma)\). Condition (4) is fulfilled, in particular, if for almost all \(t_0,t\) from \(\Gamma\) one has

\[ \left|K(t_0,t,u)-K(t_0,t,v)\right|\le D(t_0,t)|u-v|, \tag{4′} \]

where \(u,v\) are arbitrary complex numbers, and \(D(t_0,t)\in L_q(\Gamma\times\Gamma)\) \((1/p+1/q=1)\).

From (1) and (2) it follows

Theorem 1. If \(K(t_0,t,u)\) satisfies conditions (3), (4), then the operator \(\lambda K_1u\) acts in \(H_{\delta_1,\delta_2}^p(N,N_1,N_2)\) and is a contraction operator in the metric of the space \(L_p(\Gamma,\Gamma)\) for all \(|\lambda|<\mu_1\), where \(\mu_1\) is sufficiently small depending on \(N,N_1,N_2,(M_1-M_5),A_p\).

From this theorem, in turn, it follows

Theorem 2. If \(|\lambda|<M_2\), then equation \((a_1^0)\) has a solution in \(H_{\delta_1,\delta_2}^p(N,N_1,N_2)\), unique in all of \(L_p(\Gamma\times\Gamma)\), and this solution can be found by the method of Picard successive approximations, if the initial function \(u(t_0,t)\) is taken from \(H_{\delta_1,\delta_2}^p(N,N_1,N_2)\).

Remark 1. If the functions \(f(t_0,t)\), \(K(t_0,t,u)\), \(u(t_0,t)\) are defined with respect to \(t_0\) not on \(\Gamma\), but on an arbitrary rectifiable curve \(\Gamma_1\), situated arbitrarily relative to \(\Gamma\), then inequalities (1)—(4), and consequently both theorems, remain valid.

Remark 2. If \(\Gamma\) is an arbitrary rectifiable Jordan curve satisfying only the condition

\[ \sup_{t_1,t_2\in M}\frac{S(t_1,t_2)}{|t_1-t_2|}=M<\infty, \tag{c} \]

and \(f(t_0,t)\in H_{\delta_1,\delta_2}^p(N,N_1,N_2)\), then, using the inequalities of Hölder and Minkowski, we obtain

\[ \|A_1f\|_{L_p(\Gamma\times\Gamma)} \le M\cdot M_1\|f\|_{L_p}\ln\left(\|f\|_{L_p}\right) + \frac{2M}{\delta_2}\|f\|_{L_p}, \tag{5} \]

\[ w(A_1f;\sigma_1,\sigma_2) \le M_1\int_0^{\sigma_2}\frac{w(f,0,\xi)}{\xi}\,d\xi + M_2\sigma_2\int_{\sigma_2}^{l}\frac{w(f,0,\xi)}{\xi^2}\,d\xi + \]

\[ + M_7w(f,\sigma_1,0)\ln\frac{1+\sigma_1}{\sigma_1}. \tag{6} \]

But from these inequalities it does not follow that the operator \(\lambda A_1f\) acts in \(H_{\delta_1,\delta_2}^p(N,N_1,N_2)\) for any \(\lambda\ne0\).

II. For the operator \((a_2)\), generally speaking, on no contour is there an analogue of inequality (4). This assertion can be confirmed by examples, which we shall not give here.

Let the contour \(\Gamma\) satisfy (c). Denote

\[ \|\bar f\|_{L_p} = \left\{ \int_L |f(t,t)|^p\,|dt| \right\}^{1/p}; \qquad \|\widetilde f\|_{L_p} = \sup_{0\le \xi\le l} \left\{ \int_0^l |f(t(s),t(s+\xi))|^p\,ds \right\}^{1/p}, \]

\[ \overline{w}(f;\sigma_1,\sigma_2)= \sup_{\substack{0<h_1\leqslant\sigma_1\\ 0<h_2\leqslant\sigma_2}} \left\{\int_0^l |f(t(s+h_1),t(s+h_2))-f(t(s),t(s))|^p\,ds\right\}^{1/p}, \]

\[ \widetilde{w}(t,\sigma_1)= \sup_{\substack{0<h_1\leqslant\sigma\\ 0<\xi\leqslant l}} \left\{\int_0^l |f(t(s+h_1),t(s+\xi))-f(t(s),t(s+\xi))|^p\,ds\right\}^{1/p}. \]

Here we shall regard the function \(f[t_0(s),t(s)]\) as equal to zero outside the square \([0,l]\times[0,l]\).

If \(\|\widetilde f\|_{L_p}<\infty\) and \(\overline w(f,\sigma,\sigma)\leqslant \overline M\sigma^\alpha\), then the inequalities

\[ \|A_2\|_{L_p}\leqslant \frac{M\cdot \overline M\sigma}{\alpha}\,\|\widetilde f\|_{L_p}+\Omega_1(f), \tag{7} \]

\[ w(A_2 f,\sigma)_{L_p}\leqslant M_6\int_0^\sigma \frac{\widetilde w(f,0,\xi)}{\xi}\,d\xi+ M_9\sigma\int_0^l \frac{\widetilde w(f,0,\xi)}{\xi^2}\,d\xi+ M_{10}\widetilde w(f,\sigma,\sigma)_{L_p}+ \Omega_2(f,\sigma), \tag{8} \]

hold, where

\[ \Omega_1(f)= \begin{cases} 0, & \text{if }\|\widetilde f\|_{L_p}\geqslant l,\ p\geqslant 1,\\[6pt] \widetilde M_7\min\left\{\|\widetilde f\|_{L_p}\ln\dfrac{1+\|\widetilde f\|_{L_p}}{\|\widetilde f\|_{L_p}};\ (\|\widetilde f\|_{L_p})^{1-1/p}\right\}, & \text{for }p>1,\\[10pt] \overline M_8\|\widetilde f\|_{L_p}\ln\dfrac{1+\|\widetilde f\|_{L_p}}{\|\widetilde f\|_{L_p}}, & \text{for }p=1, \end{cases} \]

\[ \Omega_2(f)= \begin{cases} M_7\min\left\{\widetilde w(f,\sigma,\sigma)\ln\dfrac{1+\sigma}{\sigma},\ \widetilde w(f,\sigma,\sigma)\sigma^{-1/p}\right\}, & \text{for }p>1,\\[10pt] M_8\widetilde w(f,\sigma)\ln\dfrac{1+\sigma}{\sigma}, & \text{for }p=1. \end{cases} \]

Definition 2. A function \(u(t)\in H_\sigma^p(N,N_1)\) if it satisfies conditions analogous to \((b_1)\), \((b_2)\).

Suppose that, for any \(u(t)\in H_\sigma^p(N,N_1)\) and \(t,t_0\in\Gamma\times\Gamma\), the function \(K(t_0,t,u)\) satisfies the conditions:

\[ \left\{\int_0^l |K(t(s),t(s+h),u(t(s)))-K(t(s),u(t(s)))|^p\,ds\right\}^{1/p} \leqslant M_{11}h^\alpha, \tag{9} \]

\[ \sup_{0<\xi<l} \left\{\int_0^l \left|K(t(s+h),t(s+\xi),u(t(s+\xi)))- K[t(s),t(s+\xi),u(t(s+\xi))]\right|^p\,ds\right\}^{1/p} \leqslant M_{12}h^\alpha\ln^{-1}\left(1+\frac1h\right); \tag{10} \]

\[ \left\{\int_0^l |K(t(s),t(s),u(t(s)))-K(t(s),t(s),v(t(s)))|^p\,ds\right\} \leqslant M_{13}\left\{\int_L |u(t)-v(t)|^p\,ds\right\}^{1/p} \tag{11} \]

or conditions (9), (11) and

\[ \left\{\int_0^l |K[t(s+h),t(s),u(t(s))]-K[t(s),t(s),u(t(s))]|^p\,ds\right\}^{1/p} \leqslant \overline M_{10}h^{\alpha_1}, \tag{10'} \]

\[ \left\{\int_0^l\int_0^l |K[t(s+h),t(s+\xi),u(t(s+\xi)))- K(t(s),t(s+\xi),u(t(s+\xi))|^p\,d\xi\right\}^{1/p} \leqslant M_{10}h^{\alpha_2}. \tag{10''} \]

From inequalities (7), (8) it follows that

Theorem 3. If the function \(K(t_0,t,u)\) satisfies conditions (9), (10), (11), or conditions (9), (10′), (11), (10″), and \(\alpha_2-1/p>0\), then for \(|\lambda|\le M_2\), where \(M_2\) is sufficiently small, the operator \((a_2)\) acts in \(H_\delta^p(N,N_1)\) continuously in the sense of the metric of the space \(L_p\), where \(\delta=\min\{\alpha;\alpha_1;(\alpha_2-1/p)\}\).

By the Schauder topological principle, from this theorem there follows the existence of a solution of equation \((a_2^0)\) in \(H_\delta(N,N_1)\).

Remark 3. One can give examples of essentially “bad” functions \(K(t_0,t,u)\) satisfying conditions (9), (10), (11), but not satisfying conditions (10′), (10″), and there are functions satisfying conditions (9), (10′), (10″), but not satisfying condition (10).

Denote

\[ g(t,\tau,u)=K(t,\tau,u)-K(\tau,\tau,u). \]

Suppose that for all \(u(t),v(t)\in H_\delta^0\) one has

\[ \left\{\int_L\left|\int_L \frac{g(t,\tau,u(\tau))-g(t,\tau,v(\tau))}{\tau-t}\,d\tau \right|^{p_1}|dt|\right\}^{1/p_1} \le \]

\[ \le M_{13}\left\{\int_L |u(t)-v(t)|^{p_1}|dt|\right\}^{1/p_1} \tag{12} \]

at least for some \(P_1>1\). This condition is satisfied, for example, by all functions

\[ K(t,\tau,u)\equiv R(t,\tau)Q(t,u), \]

if

\[ \left\{\int_0^l |R(t(s+h),t(s+h))-R(t(s),t(s))|^{p_1}\,ds\right\}^{1/p_1} \le M_{14}h^\alpha, \]

\[ \left\{\int_L |Q(t,u(t))-Q(t,v(t))|^{p_1}|dt|\right\}^{1/p_1} \le M_{15}\left(\int_L |u(t)-v(t)|^{p_1}\right)^{1/p_1} \]

for arbitrary \(u(t),v(t)\in L_p(\Gamma)\) and \(\alpha>1/p_1,\ p_1\ge 2\).

Theorem 4. If \(K(t,t,u(t))\in L_{p_1}(\Gamma_1)\) and \(K(t,\tau,u)\) satisfies condition (12), then for all \(|\lambda|\le M_3\), where \(M_3\) is a certain constant, the operator \((a_2^c)\) will be a continuous contraction operator in \(L_{p_1}\).

Thus, under the assumptions made, equation \((a_2^c)\) has a unique solution in \(L_{p_1}\); this solution can be found by Picard’s method of successive approximations, and the successive approximations will converge in the sense of the metric of the space \(L_{p_1}\). But if, in addition, \(K(t,\tau,u)\) satisfies conditions (9), (10), (11), or conditions (9), (9″), (10″), (11), then this unique solution will lie in some \(H_{\varphi}^{p}(\bar N_1,N)\), and it can be found by the method of successive approximations if as the initial function \(u(t_0)\) one takes a function from \(H_{\varphi}^{p}(\bar N_1,N)\), with the successive approximations converging in the sense of the metric \(L_p\), even if \(p>p_1\).

Remark 4. We shall say that \(u(t)\in H_{\varphi}^{p}(N,N_1)\) if \(\|u\|_{L^p}\le N\), \(w(u',\sigma)_{L^p}\le N_1\varphi_1(\sigma)\); correspondingly we also define \(H_{\varphi_1,\varphi_2}^{p}(N,N_1,N_2)\). If one takes \(\varphi(\sigma)\in\Phi\) (for the definition of \(\Phi\), see, for example, in \((^4)\)), then under corresponding restrictions on the function \(K(t_0,t,u)\), Theorems 1, 2, and 3 also hold for the sets \(H_{\varphi_1,\varphi_2}^{p}(N,N_1,N_2)\) and \(H_{\varphi}^{p}(N,N_1)\).

Remark 5. Inequalities (2), (6), (8) in a certain special case coincide with the corresponding inequality of paper \((^5)\).

Dagestan State University
named after V. I. Lenin
Makhachkala

Received
26 XI 1969

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