MATHEMATICS

S. G. Mikhlin

SINGULAR INTEGRAL EQUATIONS IN CLASSES OF LIPSCHITZ FUNCTIONS

(Presented by Academician V. I. Smirnov on 10 I 1961)

In the present note we retain the terminology and notation of the papers \((^1,^2)\).

\(1^\circ\). The purpose of the present note is to establish sufficient conditions under which the solution (if it exists) of the multidimensional singular integral equation

\[ a(x)u(x)+\int_{E_m}\frac{f(x,\theta)}{r^m}u(y)\,dy=g(x), \]

\[ r=|y-x|,\qquad \theta=\frac{y-x}{r}, \tag{1} \]

satisfies a Lipschitz condition with some positive exponent. We assume that the free term satisfies the following condition: the product
\[ \left(\frac{1+|x|^2}{2}\right)^{m/2} g(x)\in \operatorname{Lip}_{\alpha}(\Sigma). \]
Here \(\Sigma\) is the sphere into which the Euclidean space \(E_m\) is transformed under stereographic projection. From the external condition it follows, in particular, that \(g(x)\in L_2(E_m)\). The matter is reduced to imposing sufficient conditions on \(a(x)\) and \(f(x,\theta)\). As usual, we assume that the symbol of equation (1) nowhere vanishes.

The problem posed here was considered by Giraud \((^3)\), who, however, investigated a very special class of equations whose characteristics are spherical functions of the first order.

Below we shall use the following, essentially well-known, lemma.

Lemma. Let \(\Omega\) be a finite domain of the space \(E_m\), and let in this domain the function \(A(x,y)\) satisfy the inequalities

\[ |A(x,y)|\leq C,\qquad |A(x+h,y)-A(x,y)|\leq N|h|^\alpha,\qquad 0<\alpha<1, \tag{2} \]

where \(C,N,\alpha\) are constants. Then the integral operator with weak singularity

\[ v(x)=\int_{\Omega}\frac{A(x,y)}{r^\lambda}u(y)\,dy,\qquad 0\leq \lambda<m, \tag{3} \]

maps every bounded function \(u(x)\) into a function \(v(x)\in \operatorname{Lip}_{\beta}(\Omega)\), where \(\beta=\min(\alpha,m-\lambda)\).

\(2^\circ\). We assume that \(a(x)\) and \(f(x,\theta)\) in equation (1) satisfy the following requirements:

a) \(a(x)\in C^{(1)}(\Sigma)\).

b) \(f(x,\theta)\in \hat W_2^{(l)}(S)\), \(l\geq m+2\).

c) Let \(\omega(x,r,\theta)\) be an arbitrary function of its arguments. Denote by \(\partial'\omega/\partial x_j\) the derivative computed under the assumption that

\(r\) and \(\theta\) do not depend on \(x\), and by \(\partial''\omega/\partial x_j\) we mean the derivative computed under the assumption that only \(r\) and \(\theta\) depend on \(x\), so that
\[ \frac{\partial\omega}{\partial x_j}=\frac{\partial'\omega}{\partial x_j}+\frac{\partial''\omega}{\partial x_j}. \]

It is easy to see that a formula of the form
\[ \frac{\partial''}{\partial x_j}\left[\frac{f(x,\theta)}{r^m}\right] = \frac{f_j(x,\theta)}{r^{m+1}} . \]

We require that
\[ \frac{\partial' f(x,\theta)}{\partial x_j}\,\hat{\in}\,W_2^{(l-1)}(S), \qquad f_j(x,\theta)\,\hat{\in}\,W_2^{(l-1)}(S) \]
and that these functions, as well as the function \(f(x,\theta)\), be continuous on \(\Sigma\times S\).

From a)—c) the following properties follow:

c) \(f(x',\theta)-f(x,\theta)=\rho^\varkappa F(\xi',\xi,\theta)\). Here \(\xi'\) and \(\xi\) are the images of the points \(x'\) and \(x\) under stereographic transformation; \(\rho\) is the distance between \(\xi'\) and \(\xi\); \(\varkappa\) is a constant, \(0<\varkappa<1\), and
\[ \left|F(\xi_1,\eta_1,\theta)-F(\xi_2,\eta_2,\theta)\right| \leq N_1\left(|\xi_1-\xi_2|^\sigma+|\eta_1-\eta_2|^\sigma\right), \]
\[ N_1=\mathrm{const},\qquad 0<\sigma<1. \]

d)
\[ a(x')-a(x)=\rho^\varkappa a(\xi',\xi),\qquad a(\xi',\xi)\in \mathrm{Lip}_\sigma(\Sigma). \]

From the conditions listed here and in item \(1^\circ\) it follows that if the function \(g(x)\) is orthogonal to all solutions of the homogeneous equation adjoint to equation (1), then the latter has a solution (possibly nonunique) in \(L_2(E_m)\). We shall prove that, under the above conditions, any such solution satisfies the condition
\[ \left(\frac{1+|x|^2}{1}\right)^{m/2}u(x)\in \mathrm{Lip}_\delta(\Sigma), \]
where \(\delta\) is determined by the data of the problem.

On the basis of the results of paper \((^4)\), from conditions b) and c) the following follows: if the characteristic is expanded in a series in spherical functions
\[ f(x,\theta)=\sum_{n=1}^{\infty}\sum_{k=1}^{k_n} a_n^{(k)}(x)Y_{n,m}^{(k)}(\theta), \]
then
\[ \sum_{n=1}^{\infty}\sum_{k=1}^{k_n} n^{2l}\left|a_n^{(k)}(x)\right|^2\leq C_1, \qquad \sum_{n=1}^{\infty}\sum_{k=1}^{k_n} n^{2l-2} \left|\frac{\partial a_n^{(k)}}{\partial x_j}\right|^2\leq C_1, \qquad C_1=\mathrm{const}. \]

Let us form the symbol of equation (1):
\[ \Phi(x,\theta)=a(x)+\sum_{n=1}^{\infty}\sum_{k=1}^{k_n} \gamma_{n,m}a_n^{(k)}(x)Y_{n,m}^{(k)}(\theta) = \]
\[ = a(x)+\int_S f(x,\theta')P(\cos\gamma)\,dS', \]
where \((^5)\)
\[ P(\cos\gamma)=\ln|\cos\gamma|^{-1}-\frac{i\pi}{2}\operatorname{sign}\cos\gamma_x \]
and \(\gamma\) is the angle between the radii of the sphere \(S\) drawn to the points \(\theta\) and \(\theta'\). Hence
\[ \frac{\partial'\Phi}{\partial x_j} = \frac{\partial a}{\partial x_j} + \int_S \frac{\partial'f(x,\theta)}{\partial x_j}P(\cos\gamma)\,dS'. \]

Thus, if \(\partial'f(x,\theta)/\partial x_j\) is regarded as the characteristic and \(\partial a(x)/\partial x_j\) as the nonintegral coefficient of a certain singular operator, then its symbol will be the function \(\partial'\Phi(x,\theta)/\partial x_j\); hence, from the results—

of the results of paper \((^4)\) it follows that \(\partial'\Phi/\partial x_j \in \hat W_2^{(l-1+[m/2])}(S)\). The same is true also for the function \(\partial'\Phi^{-1}/\partial x_j=-\Phi^{-2}\partial'\Phi/\partial x_j\), since \(\inf|\Phi|>0\).

Let now

\[ \Phi^{-1}(x,\theta)=\sum_{n=0}^{\infty}\sum_{k=1}^{k_n} b_n^{(k)}(x)Y_{n,m}^{(k)}(\theta). \]

Then, by virtue of the same results of paper \((^4)\),

\[ \sum_{n=0}^{\infty}\sum_{k=1}^{k_n} n^{2(l-1+[m/2])} \left|\frac{\partial b_n^{(k)}}{\partial x_j}\right|^2 \leq C_2=\mathrm{const}, \]

and, consequently,

\[ \sum_{n=0}^{\infty}\sum_{k=1}^{k_n} n^{2l-3}|\gamma_{n,m}|^{-2} \left|\frac{\partial b_n^{(k)}}{\partial x_j}\right|^2 \leq C_3=\mathrm{const}. \]

Hence, in any case, it follows that \(\operatorname{grad}'\psi(x,\theta)\in \hat W_2^{(l-2)}(S)\), where \(\varphi(x,\theta)\) is the characteristic corresponding to the symbol \(\Phi^{-1}(x,\theta)\). At the same time, as again follows from the assertion of paper \((^4)\), \(\operatorname{grad}''\varphi(x,\theta)\in \hat W_2^{(l-2)}(S)\). Since \(l-2\geq m\), then, as can be derived from the theorems of paper \((^4)\), the series obtained by differentiating the series for the function \(\varphi(x,\theta)\) with respect to the Cartesian coordinates of the point \(\theta\) converge absolutely and uniformly. From the formula for the symbol it follows, by conditions b) and c), that the functions \(\Phi(x,\theta)\) and \(\partial'\Phi(x,\theta)/\partial x_i\) are continuous on \(\Sigma\times S\). But then on \(\Sigma\times S\) the functions \(\Phi^{-1}(x,\theta)\) and \(\partial'\Phi^{-1}(x,\theta)/\partial x_i\) are also continuous, and consequently so are the coefficients \(b_n^{(k)}(x)\) and their derivatives with respect to \(x_j\). But then

\[ \left|\operatorname{grad}_{\xi}\varphi(x,\theta)\rho^{-m}\right| \leq C_4\rho^{-m-1},\qquad C_4=\mathrm{const}. \tag{4} \]

Here \(\rho=|\xi-\eta|\), where \(\xi\) and \(\eta\) are the images of the points \(x\) and \(y\) under stereographic transformation. Using the corresponding theorem of Giraud \((^3)\), one can prove that the singular operator

\[ b_0^{(1)}(x)u(x)+\int_{E_m} L(x,x-y)u(y)\,dy,\qquad L(x,x-y)=\frac{\varphi(x,\theta)}{r^m}, \tag{5} \]

maps every function \(u(x)\) satisfying the condition

\[ \left(\frac{1+|x|^2}{2}\right)^{m/2}u(x)\in \operatorname{Lip}_{\alpha}(\Sigma), \tag{6} \]

into a function satisfying the same condition.

\(3^\circ\). Apply to both sides of equation (1) the operator (5), whose symbol is equal to \(\Phi^{-1}(x,\theta)\). This will lead us to an equation of Riesz–Schauder type

\[ u(x)+Tu=F(x), \tag{7} \]

which is satisfied by all solutions of equation (1). From what was said above it follows that \(F(x)\) satisfies condition (6).

In equation (7) put

\[ \left(\frac{1+|x|^2}{2}\right)^{m/2}u(x)=\tilde u(\xi),\qquad \left(\frac{1+|x|^2}{2}\right)^{m/2}=\tilde F(\xi),\qquad \left(\frac{1+|x|^2}{2}\right)^{m/2}Tu=\tilde T\tilde u; \]

then this equation becomes the following:
\[ \widetilde{u}(\xi)+\widetilde{T}\widetilde{u}=\widetilde{F}(\xi), \tag{8} \]
in which the operator \(\widetilde{T}\) is completely continuous in \(L_2(\Sigma)\).

It can be proved that
\[ \widetilde{T}\widetilde{u}= \int_{\Sigma} \left\{ [a(y)-a(x)]\frac{\varphi(x,\theta)}{\rho^m} + \int_{\Sigma} \frac{[f(z,\theta_{yz})-f(x,\theta_{yz})]\varphi(x,\theta_{xz})} {|\xi-\eta|^m|\eta-\zeta|^m}\,d\Sigma_\zeta \right\} \widetilde{u}(\eta)\,d\Sigma_\eta . \tag{9} \]

Here \(\zeta\) is the image of the point \(z\) under stereographic transformation,
\[ \theta_{yz}=\frac{z-y}{|z-y|},\qquad \theta_{xz}=\frac{z-x}{|z-x|}. \]
From conditions г) and д) and from the results of Giraud \({}^{(3)}\) it follows that the kernel of the operator (9) satisfies the conditions of the lemma of the present note. In particular, equation (8) is an integral equation with a weak singularity. The function \(\widetilde{F}(\xi)\in \operatorname{Lip}_{\alpha}(\Sigma)\), and hence is bounded; then every solution of equation (8) belonging to \(L_2(\Sigma)\) is bounded. By the lemma, \(\widetilde{T}\widetilde{u}\in \operatorname{Lip}_{\beta}(\Sigma)\), where \(\beta\) is determined by the conditions of the problem, and therefore
\[ \widetilde{u}=(\widetilde{F}-\widetilde{T}\widetilde{u})\in \operatorname{Lip}_{\delta}(\Sigma), \]
\(\delta=\min(\alpha,\beta)\).

\(4^\circ\). The results also extend to the case when the equation has the form
\[ a(\xi)u(\xi)+\int_{\Gamma}K(\xi,\eta)u(\eta)\,d\Gamma_\eta = g(\xi), \qquad g(\xi)\in \operatorname{Lip}_{\alpha}(\Gamma), \]
where \(\Gamma\) is a closed sufficiently smooth manifold of dimension \(m\), and the kernel is subject, for example, to the following condition: if some neighborhood of the point \(\xi\in\Gamma\) is mapped sufficiently smoothly onto some domain of the space \(E_m\), and if \(x\) and \(y\) are the images of the points \(\xi\) and \(\eta\) under this transformation, then
\[ K(\xi,\eta)=\frac{f(x,\theta)}{r^m}+\frac{f_0(x,y)}{r^{m-\lambda}}, \]
where the singular kernel \(r^{-m}f(x,\theta)\) satisfies all the conditions listed above, the exponent \(\lambda>0\), and the function \(f_0(x,y)\) is continuously differentiable with respect to the coordinates of both points \(x\) and \(y\).

The extension of the results to systems of singular equations is obvious.

Leningrad State University
named after A. A. Zhdanov

Received
5 I 1961

REFERENCES

\({}^{1}\) S. G. Mikhlin, Vestn. LGU, No. 1 (1956).
\({}^{2}\) S. G. Mikhlin, DAN, 131, No. 5 (1960).
\({}^{3}\) G. Giraud, Ann. Sci. de l’École Norm. Sup., 51, fasc. 3 et 4, 251 (1934).
\({}^{4}\) S. G. Mikhlin, DAN, 126, No. 2 (1959).
\({}^{5}\) A. P. Calderon, A. Zygmund, Am. J. Math., 78, No. 2 (1956).