UDC 517.947.42

THE HEIGHTENING PROPERTY OF DIRECT VALUES OF GENERALIZED POTENTIALS FOR THE COMPARISON FUNCTION OF A GENERAL ELLIPTIC OPERATOR OF SECOND ORDER

WANG TUN

§ 1. NOTATION AND STATEMENT OF RESULTS

1°. Notation and definitions.

1. Let, in the \(m\)-dimensional Euclidean space \(R_m\), there be given a certain open domain \(g\), bounded by a closed surface \(\Gamma\).

We shall say that a function \(f(x)\), defined in \(g+\Gamma\), belongs to the class \(C^{(k)}[C^{(k,\lambda)}]\), \(k\ge 0\), \(0<\lambda\le 1\), if its derivatives of order \(k\) in \(g+\Gamma\) are continuous (satisfy the Hölder condition with exponent \(\lambda\)). The function \(f\) in \(g\) belongs to the class \(C^{(k)}[C^{(k,\lambda)}]\) if it belongs to this class in every closed subdomain of the domain \(g\).

1. We shall say that the surface \(\Gamma\) belongs to the class \(A^{(k)}[A^{(k,\lambda)}]\), if for each of its points \(x\) there exists a ball \(\Omega_x^d\) (a Lyapunov ball) of radius \(d\) (\(d=\mathrm{const}\) for all \(x\)) with center at the point \(x\); that the intersection \(\Gamma\cap\Omega_x^d\), in some local coordinate system with origin at \(x\), admits a representation of the form

\[ \xi_m=f(\xi_1,\ldots,\xi_{m-1}),\quad (\xi_1,\ldots,\xi_{m-1})\in \Lambda \]

and \(f\in C^{(k)}[C^{(k,\lambda)}]\) in the domain \(\Lambda\), where \(f(0)=\partial f(0)/\partial \xi_i=0,\ i=1,\ldots,m-1\), \(\Lambda\) is the projection of \(\Gamma\cap\Omega_x^d\) onto the plane \(\pi\) tangent to \(\Gamma\) at \(x\).

1. By \(F_i\) we denote the sum of the maxima of the moduli of all derivatives of order \(i\) of the function \(f\); by \(F_{i\lambda}\), the sum of the Hölder coefficients of all derivatives of order \(i\) of the function \(f\).

The norm of a function \(f\) from the class \(C^{(k)}(g+\Gamma)\) is defined as follows:

\[ \|f\|_{C^{(k)}(g+\Gamma)} =\max_{g+\Gamma}|f|+\max_{g+\Gamma}|Df|+\cdots+\max_{g+\Gamma}|D^{(k)}f| =\sum_{i=0}^{k}F_i; \]

the norm of a function \(f\) from the class \(C^{(k,\lambda)}(g+\Gamma)\) is defined in the following way:

\[ \|f\|_{C^{(k,\lambda)}(g+\Gamma)} =\|f\|_{C^{(k)}(g+\Gamma)}+F_{k\lambda}. \]

When \(\Gamma\) is a Lyapunov surface, i.e. \(\Gamma\in A^{(1,\lambda)}\), \(0<\lambda\le 1\), the above norms are equivalent to the following [1]:

\[ \|f\|_{C^{(k)}(g+\Gamma)}=F_0+F_k, \]

\[ \|f\|_{C^{(k,\lambda)}(g+\Gamma)}=F_0+F_{k\lambda}. \]

1. We shall say that a function \(R(xy)\), continuous jointly in \((xy)\) everywhere in \(g+\Gamma\) for \(x\ne y\), belongs to the class \(N^{(\alpha)}\), \(\alpha<m\), if for it the estimate

\[ R(xy)=O\bigl(2_{xy}^{\alpha-m}\bigr), \]

holds.

uniformly in \(g+\Gamma\); \(R(xy)\) belong to the class \(N^{(m)}\), if the estimate

\[ R(xy)=O\left(\ln \frac{2D}{2xy}\right), \]

holds uniformly in \(g+\Gamma\), where \(D\) is the diameter of the domain \(g\); finally, \(R(xy)\) belongs to the class \(N^{(\alpha)}\), \(\alpha>m\), if it is continuous also for \(x=y\).

1. In the domain \(g\) we shall consider a uniformly elliptic operator of the second order

\[ Mu=\sum_{i,j=1}^{m} a_{ij}(x)\frac{\partial^2 u}{\partial x_i\partial x_j} +\sum_{i=1}^{m} b_i(x)\frac{\partial u}{\partial x} +c(x)u, \tag{1.1} \]

i.e., one such that for all \(x=(x_1,\ldots,x_m)\in g\)

\[ a_{ij}=a_{ji},\qquad \sum_{i,j=1}^{m} a_{ij}\eta_i\eta_j \geq a_0\sum_{i=1}^{m}\eta_i^2 \tag{1.2} \]

for arbitrary real \(\eta_1,\ldots,\eta_m\), where \(a_0>0\) is the ellipticity constant. It is known that to every operator (1.1) one can associate its comparison function

\[ H(xy)= \begin{cases} \dfrac{1}{(m-2)\omega_m\sqrt{A(y)}}\,\rho^{\,2-m}, & m\geq 3,\\[1.2ex] \dfrac{1}{2\pi\sqrt{A(y)}}\,\ln|\rho|, & m=2, \end{cases} \tag{1.3} \]

where

\[ \rho^2=\sum_{i,j=1}^{m} A_{ij}(y)(x_i-y_i)(x_j-y_j), \]

and \(A_{ij}(x)\) is the ratio of the algebraic complement of the element \(a_{ij}(x)\) in the determinant \(A(x)=|a_{ij}(x)|\) to the determinant \(A(x)\) itself, \(\omega_m\) is the area of the unit spherical surface of \(m-1\) dimensions.

1. For the comparison function \(H(xy)\) we shall consider the volume potential

\[ U(x)=\int_g \mu(y)H(xy)\,dy, \tag{1.4} \]

the single-layer potential

\[ V(x)=\int_{\Gamma} \nu(y)H(xy)\,ds_y \tag{1.5} \]

and the double-layer potential

\[ W(x)=\int_{\Gamma} z(y)P_yH(xy)\,ds_y, \tag{1.6} \]

where \(\mu(x)\), \(\nu(x)\), and \(z(x)\) are the densities of the corresponding potential,

\[ P_yH(xy)=\frac{\partial}{\partial \nu_y}H(xy) =\frac{1}{a(y)}\sum_{i,j=1}^{m} a_{ij}(y)\frac{\partial H(xy)}{\partial y_i}\cos(ny_j) \tag{1.7} \]

is the derivative of \(H(xy)\) in the direction of the conormal \(\nu\) to \(\Gamma\), while \(n\) is the exterior normal to \(\Gamma\), and

\[ a(y)=\left[\sum_{i=1}^{m}\left(\sum_{j=1}^{m} a_{ij}(y)\cos(ny_j)\right)^2\right]^{1/2}. \]

The direct value \(W_{\mathrm{pr}}\) of the double-layer potential and the direct value \(V_{\mathrm{pr}}\) of the conormal derivative of the single-layer potential are defined as follows:

\[ W_{\mathrm{pr}}=W(x)\big|_{x\in\Gamma} =\int_{\Gamma} z(y)\,[P_yH(xy)]_{x\in\Gamma}\,ds_y, \tag{1.8} \]

\[ V_{\mathrm{pr}}=[P_xV(x)]_{x\in\Gamma} =\int_\Gamma \nu(y)[P_xH(xy)]_{x\in\Gamma}\,ds_y . \tag{1.9} \]

\(2^\circ\). Formulation of the results.
Let the following conditions A be satisfied:

1. \(\Gamma\) belongs to the class \(A^{(n+2,\lambda)}\), \(n\geqslant 0,\ 0<\lambda\leqslant 1\);
2. \(a_{ij}(x)\) belongs to the class \(C^{(n+1,\lambda)}\) in \(g+\Gamma\) for all \(i\) and \(j:\ 1\leqslant i,j\leqslant m\);
3. \(z(x)\) belongs to the class \(C^{(n,\lambda)}\) on the surface \(\Gamma\).

Then the following holds.

Theorem 1. If conditions A are satisfied, the direct value \(W_{\mathrm{pr}}\) on \(\Gamma\) of the double-layer potential for the comparison function \(H(xy)\) belongs to the class \(C^{(n+1,\lambda')}\) on \(\Gamma\), where \(0<\lambda'<\lambda\) is arbitrary, and the norm of \(W_{\mathrm{pr}}\) in \(C^{(n+1,\lambda')}(\Gamma)\) is estimated in terms of the norm of the density \(z(x)\) in \(C^{(n,\lambda)}(\Gamma)\) as follows:

\[ \|W_{\mathrm{pr}}\|_{C^{(n+1,\lambda')}(\Gamma)} =O\bigl(\|z\|_{C^{(n,\lambda)}(\Gamma)}\bigr); \tag{1.10} \]

here the constant entering the term \(O\) depends only on the norms in \(C^{(n+1,\lambda)}(g+\Gamma)\) of the coefficients \(\|Q_{ij}\|_{C^{(n+1,\lambda)}}\), on the ellipticity constant \(\alpha_0\), on the choice of \(\lambda'\), and on the properties of the surface \(\Gamma^*\), but does not depend on the density \(z(x)\).

Remark 1. Theorem 1 remains valid if in (1.8) \(P\) is taken in the more general form:

\[ P_xu(x)=\alpha(x)\frac{\partial u(x)}{\partial \nu_x}+\beta(x)u(x), \]

where \(\alpha(x)\) and \(\beta(x)\) are two arbitrary functions of the class \(C^{(n,\lambda)}(\Gamma)\). Then the constant entering the term \(O\) on the right-hand side of (1.10) also depends on the norms \(\|\alpha\|_{C^{(n,\lambda)}}\) and \(\|\beta\|_{C^{(n,\lambda)}}\).

Let now the following conditions B be satisfied:

1. \(\Gamma\in A^{(n+2,\lambda)}\), \(n\geqslant 0,\ 0<\lambda\leqslant 1\);
2. \(a_{ij}\in C^{(n+1,\lambda)}(g+\Gamma)\), \(1\leqslant i,j\leqslant m\);
3. \(\nu(x)\in C^{(n,\lambda)}\) on \(\Gamma\).

Then the following holds.

Theorem 2. If conditions B are satisfied, the direct value \(V_{\mathrm{pr}}\) on \(\Gamma\) of the conormal derivative of the simple-layer potential for the comparison function \(H(xy)\) belongs to the class \(C^{(n+1,\lambda')}\) on \(\Gamma\), where \(0<\lambda'<\lambda\) is arbitrary, and the norm of \(V_{\mathrm{pr}}\) in \(C^{(n+1,\lambda')}(\Gamma)\) is estimated in terms of the norm of the density \(\nu(x)\) in \(C^{(n,\lambda)}(\Gamma)\) as follows:

\[ \|V_{\mathrm{pr}}\|_{C^{(n+1,\lambda')}(\Gamma)} =O\bigl(\|\nu\|_{C^{(n,\lambda)}(\Gamma)}\bigr); \tag{1.11} \]

here the constant entering the term \(O\) depends only on the norms of the coefficients \(\|a_{ij}\|_{C^{(n+1,\lambda)}(g+\Gamma)}\), on the ellipticity constant \(\alpha_0\), on the choice of \(\lambda'\), and on the properties of the surface \(\Gamma\), but does not depend on the density \(\nu(x)\).

Remark 2. Theorem 2 remains valid if in (1.9) \(P\) is taken in the more general form:

\[ P_xu(x)=\gamma(x)\frac{\partial u(x)}{\partial \nu_x}+\xi(x)u(x), \]

where \(\gamma(x)\) and \(\xi(x)\) are two arbitrary functions of the class \(C^{(n+1,\lambda)}(\Gamma)\). Then the constant entering the term \(O\) on the right-hand side of (1.11) also depends on the norms \(\|\gamma\|_{C^{(n+1,\lambda)}(\Gamma)}\) and \(\|\xi\|_{C^{(n+1,\lambda)}(\Gamma)}\).

Remark 3. For the Laplace operator, Theorems 1 and 2 were obtained by Kh. L. Smolitskii [3].

Let now the following conditions C be satisfied:

1. \(g\) is an arbitrary bounded open domain;
2. \(a_{ij}\in C^{(n,\lambda)}\) in \(g+\Gamma\), \(1\leqslant i,j\leqslant m\), \(n\geqslant 0,\ 0<\lambda\leqslant 1\);
3. \(\mu(x)\in C^{(n,\lambda)}\) in \(g+\Gamma\).

Then the following holds

*) More precisely, it depends on the norm in \(C^{(n+2,\lambda)}\) of the function which, in local coordinates, defines the surface \(\Gamma\).

Theorem 3. Under conditions B, the volume potential \(U(x)\) for the comparison function \(H(xy)\) belongs to the class \(C^{(n+2,\lambda')}\) in \(g\), where \(0<\lambda'<\lambda\) is arbitrary, and the norm of \(U\) in \(C^{(n+2,\lambda')}(g')\) is estimated in terms of the norm of the density \(\mu(x)\) in \(C^{(n,\lambda)}(g+\Gamma)\):

\[ \|U\|_{C^{(n+2,\lambda')}(g')}=O\left(\|\mu\|_{C^{(n,\lambda)}(g+\Gamma)}\right), \tag{1.12} \]

where \(g'\) is an arbitrary interior subdomain of \(g\), whose distance from the boundary \(\Gamma\) of the domain \(g\) is equal to \(d\), and the constant occurring in the \(O\)-term depends only on the norms of the coefficients \(\|a_{ij}\|_{C^{(n,\lambda)}(g+\Gamma)}\), the ellipticity constant \(a_0\), the choice of \(\lambda'\), and the number \(d\), but does not depend on the density \(\mu(x)\).

§ 2. GENERAL REMARK ON THE METHOD

In this section we shall give, in general outline, the justification of the method which will be used repeatedly in the proofs.

Let the integral

\[ \Psi(x)=\int_g \rho(y)R(xy)\,dy \tag{2.1} \]

depend on \(x\), which varies in the closed domain \(g+\Gamma\). Suppose further that the kernel of the integral (2.1) has the following properties:

\[ \begin{aligned} &1)\quad R(xy)\in N^{(1)}(g+\Gamma),\\ &2)\quad \partial R(xy)/\partial x_i \in N^{(0)}(g+\Gamma),\quad i=1,\ldots,m. \end{aligned} \tag{2.2} \]

Under these conditions we do not know whether the derivatives of \(\Psi(x)\) exist in the closed domain \(g+\Gamma\), for after formal differentiation under the integral sign a divergent integral may appear. However, when the kernel \(R(xy)\) has certain additional properties and, besides this, the function \(\rho(x)\) is differentiable, it is possible to compute the derivatives of \(\Psi(x)\) in the open domain by means of the method of smoothing.

Indeed, suppose that, in addition to (2.2), it is also known that

\[ r^{m-1}R(xy)\big|_{y=x+r\varphi}=F(r;x,\varphi) \tag{2.3} \]

has continuous derivatives with respect to \(x\) everywhere in \(g+\Gamma\). The symbol \(\big|_{y=x+r\varphi}\) denotes the following change of variables: first put \(x'=x,\ y'=y-x\), and then write \(y'\) in spherical coordinates according to the formula

\[ \begin{aligned} y_1&=r g_1(\varphi)=r g_1(\varphi_1,\ldots,\varphi_{m-1})=r\cos\varphi_1,\\ y_2&=r g_2(\varphi)=r g_2(\varphi_1,\ldots,\varphi_{m-1})=r\sin\varphi_1\cos\varphi_2,\\ &\cdots\\ y_{m-1}&=r g_{m-1}(\varphi)=r\sin\varphi_1\sin\varphi_2,\ldots,\sin\varphi_{m-2}\cos\varphi_{m-1},\\ y_m&=r g_m(\varphi)=r\sin\varphi_1\sin\varphi_2,\ldots,\sin\varphi_{m-2}\sin\varphi_{m-1}, \end{aligned} \tag{2.4} \]

(the prime will be omitted below), where \(\varphi=(\varphi_1,\ldots,\varphi_{m-1})\) is an abbreviated notation for the spherical angles.

If conditions (2.2) and (2.3) are satisfied, we can compute the derivatives of \(\Psi(x)\) in \(g\) by means of the method of smoothing, which consists in the following.

Let \(g'\) be any domain lying together with the boundary \(\Gamma'\) inside \(g\). Compute the derivatives of \(\Psi(x)\) in \(g'+\Gamma'\). Take yet another arbitrary subdomain \(g''\) of the domain \(g\) such that \(g'+\Gamma'\subset g''\), \(g''+\Gamma''\subset g\). Then we may smooth the kernel \(R(xy)\), i.e. extend it with respect to \(y\) in a smooth manner to the whole space \(R_m\), so that the extended kernel \(R_{\mathrm{cr}}(xy)\) satisfies the following requirements:

\[ \begin{aligned} R_{\mathrm{cr}}(xy)&\equiv R(xy) &&\text{for } x,y\in g''+\Gamma'',\\ R_{\mathrm{cr}}(xy)&\equiv 0 &&\text{for } x\in g,\ y\in g+\Gamma, \end{aligned} \tag{2.5} \]

\[ \frac{\partial}{\partial x_i}R_{\mathrm{cr}}(xy)\quad \text{are continuous in }x\text{ in }g''+\Gamma''\text{ for }y\in g-(g''+\Gamma''). \]

We shall call the kernel \(R_{\mathrm{cr}}(xy)\) smoothed over the domain \(g''\).

Such smoothing is usually carried out by multiplying by a cutoff function \(\theta(x)\), which is infinitely differentiable in \(R_m\) and is defined as follows:

\[ \theta(x) \begin{cases} \equiv 1, & \text{for } x\in g''+\Gamma'',\\ \equiv 0, & \text{for } x\in \overline g+\Gamma,\\ \text{decreases smoothly from } 1 \text{ to } 0, & \text{for } x\in g-(g''+\Gamma''). \end{cases} \tag{2.6} \]

After smoothing we have

\[ \Psi(x)= \int_{g-g''} R(xy)\rho(y)\,dy+ \int_{R_m} R_{\mathrm{сг}}(xy)\rho(y)\theta(y)\,dy- \int_{g-g''} R_{\mathrm{сг}}(xy)\rho(y)\theta(y)\,dy . \tag{2.7} \]

It is evident that, by virtue of (2.5) and (2.7), the first and third integrals on the right-hand side of (2.7) have continuous derivatives with respect to \(x_i\), which can be obtained by differentiating under the integral sign.

In the second integral on the right-hand side of (2.7) we pass to a system of spherical coordinates with pole at \(x\) [4]:

\[ \int_{R_m} R_{\mathrm{сг}}(xy)\rho(y)\theta(y)\,dy = \int_{R_m} R_{\mathrm{сг}}(xy)\rho_{\mathrm{сг}}(y)\,dy = \]

\[ = \int_{0}^{\pi}\sin\varphi_1^{\,m-2}\,d\varphi_1 \int_{0}^{\pi}\sin\varphi_2^{\,m-3}\,d\varphi_2 \cdots \int_{0}^{\pi}\sin\varphi_{m-2}\,d\varphi_{m-2} \int_{0}^{2\pi}d\varphi_{m-1} \times \tag{2.8} \]

\[ \times \int_{0}^{\infty} F(x;r,\varphi)\rho_{\mathrm{сг}}(x+r\varphi)\,dr, \]

where

\[ F(x;r,\varphi)=r^{m-1}R_{\mathrm{сг}}(xy)\big|_{y=x+r\varphi} \]

has continuous derivatives with respect to \(x_i\) everywhere in \(g''+\Gamma''\) for arbitrary \(r\) and \(\varphi\). Then (2.8) can be differentiated with respect to \(x_i\) under the integral sign, since the integral in fact extends over a finite region.

Thus, we have computed the derivatives of \(\Psi(x)\) in \(g'\) by means of the smoothing method. This method was used by Kh. L. Smolitskii [3] in the case when (1.1) was the three-dimensional Laplace operator. In this case the comparison function \(H(xy)\) has the simplest form:

\[ H(xy)=\frac{1}{4\pi r}. \]

This circumstance facilitates the derivation of a number of estimates for \(P_yH(xy)\), using which Kh. L. Smolitskii proved Theorems 1 and 2 for the Laplace operator.

In the general case it is no longer possible to rely on the special form of the comparison function \(H(xy)\). However, owing to the uniform ellipticity of the operator (1.1), the distance \(\rho(xy)\) taken in the corresponding Riemannian metric is equivalent to the Cartesian distance \(r_{xy}\) in the sense that there is a constant \(A>0\) such that

\[ r_{xy}A^{-1}\leqslant \rho(xy)\leqslant Ar_{xy}. \]

Consequently, after overcoming certain technical difficulties, under a certain smoothness of the coefficients of the operator \(\mathbf M\), we can still establish a number of estimates for \(P_yH(xy)\). With the aid of these estimates and by means of the method set forth above, we shall prove Theorems 1 and 2 in the general case.

As an example, consider the special case when the density in (2.1) is \(\rho(y)\equiv 1\) and

\[ R(xy)=[\rho(xy)]^{1-m}. \]

Suppose that \(A_{ij}(y)\) is continued through \(\Gamma\) to the whole space \(R_m\) in such a way that the operator (1.1) is elliptic in \(R_m\). Then one may take as \(R_{\mathrm{сг}}(xy)\)

\[ R_{\mathrm{сг}}(xy)=R(xy)\theta(y)= \left[ \sum_{i,j=1}^{m} A_{ij}(y)(x_i-y_i)(x_j-y_j) \right]^{\frac{1-m}{2}}\theta(y). \]

and

\[ F(x; r,\varphi)=\left. r^{m-1} R_{\mathrm{cir}}(xy)\right|_{y=x+r\varphi} = \left[ \sum_{ij=1}^{m} A_{ij}(x+r\varphi) g_i(\varphi)g_j(\varphi) \right]^{\frac{1-m}{2}} \theta(x+r\varphi). \]

Suppose \(A_{ij}(x)\) is differentiable \(n\) times; then \(F(x;r,\varphi)\) can be differentiated the same number of times for arbitrary \(r\) and \(\varphi\). But the limits of integration in (2.8) do not depend on \(x\); therefore (7.8) can be differentiated \(n\) times under the integral sign.

Thus, in order to compute the derivatives of (2.1), we first smoothed the subintegral function in (2.1), so that in (2.8) the limits of integration would not depend on \(x\); then, with the aid of the replacement (2.4), we “hid” the singularity of the function \(R(xy)\), and thereby placed “the entire burden of differentiation” on the coefficients \(A_{ij}\). After this, (2.8) can be differentiated under the integral sign as many times as the coefficients \(A_{ij}\) are differentiable.

§ 3. THE DIRECT VALUE OF THE DOUBLE-LAYER POTENTIAL FOR THE COMPARISON FUNCTION

In this section we shall prove Theorem 1 for the case \(m\geqslant 3\). The case \(m=2\) will be considered separately in § 5.

\(1^\circ\). Let \(x_0\) be an arbitrary point belonging to the surface \(\Gamma\). We take it as the origin of the local coordinates \((\xi_1,\ldots,\xi_m)\). Suppose further that the radius of the Lyapunov sphere \(\Omega_{x_0}^{d}\) at the point \(x_0\) is \(d\), and \(\Sigma=\Gamma\cap\Omega_{x_0}^{d}\). Then \(\Sigma\) is given by the equation

\[ \xi_m=f(\xi_1,\ldots,\xi_{m-1}),\qquad (\xi_1,\ldots,\xi_{m-1})\in\sigma, \]

where \(\sigma\) is the projection of \(\Sigma\) onto the plane \(\pi\) tangent to \(\Gamma\) at \(x_0\); the axis \(\xi_m\) is directed along the normal \(N_0\) to \(\Gamma\) at \(x_0\).

Let \(d_0\) be so small that the \((m-1)\)-dimensional ball \(\sigma_1\) of radius \(2d_0\) with center at \(x_0\) on \(\pi\) is contained in \(\sigma\), and let \(\sigma_0\) be the \((m-1)\)-dimensional ball of radius \(d_0\) with center at \(x_0\). Denote by \(\Sigma_0\) that part of \(\Sigma\) whose projection onto \(\pi\) is \(\sigma_0\).

Then, by virtue of condition A,

\[ f(\xi_1,\ldots,\xi_{m-1})\in C^{(n+2,\lambda)} \quad \text{in } \sigma \]

and

\[ \bar z(x_1,\ldots,x_{m-1}) = z(x_1,\ldots,x_{m-1}, f(x_1,\ldots,x_{m-1})) \in C^{(n,\lambda)} \quad \text{in } \sigma. \]

We shall consider \(W_{\mathrm{pr}}(x)\) in local coordinates and, for simplicity, assume that \(x=(x_1,\ldots,x_m)\) are the local coordinates of the point \(x\). Our aim is to prove that \(W_{\mathrm{pr}}(x)\), as a function on the surface \(\Gamma\), belongs to the class \(C^{(n+1,\lambda')}\), and to estimate its norm. For this, it is obviously sufficient to prove that

\[ \overline{W}_{\mathrm{pr}}(x_1,\ldots,x_{m-1}) = W_{\mathrm{pr}}(x_1,\ldots,x_{m-1}, f(x_1,\ldots,x_{m-1})) \]

belongs to the class \(C^{(n+1,\lambda')}\) in \(\sigma_0\), and to estimate its norm in \(C^{(n+1,\lambda')}(\sigma_0)\) in terms of the norm \(\|z\|_{C^{(n,\lambda)}(\Gamma)}\).

Write \(W_{\mathrm{pr}}(x)\) in the form

\[ W_{\mathrm{pr}}(x) = \int_{\Sigma} z(y)\,[P_y H(xy)]_{x\in\Sigma_0}\,ds_y + \int_{\Gamma-\Sigma} z(y)\,[P_y H(xy)]_{x\in\Sigma_0}\,ds_y . \tag{3.1} \]

It is clear that the second term can be differentiated with respect to \(x_i\) \((i=1,\ldots,m)\) under the integral sign arbitrarily many times when \(x\in\Sigma_0\), i.e. \(x=(x_1,\ldots,x_{m-1})\in\sigma_0\). Therefore, by virtue of \(f\in C^{(n+2,\lambda)}\), it belongs to the class \(C^{(n+1,\lambda')}\) for any \(0<\lambda'<1\). It is also clear that the norm in \(C^{(n+1,\lambda')}(\sigma_0)\)

of the second term in (3.1) has order \(O C\|z\|_{C^{(n,\lambda)}(\Gamma)}\), where \(O\) depends only on \(\|a_{ij}\|_{C^{(n+1,\lambda)}}\), \(\|f\|_{C^{(n+2)}}\), and the numbers \(d\) and \(d_0\). Thus, in order to prove Theorem 1, it remains to consider only the first term in (3.1), which we denote by \(\omega(x)\).

\(2^\circ\). It is easy to verify that

\[ \frac{\partial H(xy)}{\partial y_i}=\sum_{k=1}^{3} H_{ki}(xy), \]

where

\[ H_{1i}(xy) = -\frac{1}{2}\frac{\partial A(y)}{\partial y_i} \frac{1}{(m-2)\omega_m[A(y)]^{3/2}}\rho^{2-m}, \]

\[ H_{2i}(xy) = -\frac{\rho^{-m}}{2\omega_m\sqrt{A(y)}} \sum_{s,l=1}^{m} \frac{\partial A_{sl}(y)}{\partial y_i} (x_s-y_s)(x_l-y_l), \]

\[ H_{3i}(xy) = \frac{\rho^{-m}}{\omega_m\sqrt{A(y)}} \sum_{l=1}^{m} A_{il}(y)(x_l-y_l). \]

\(\omega(x)\) is accordingly decomposed into three integrals: \(\omega(x)=\omega_1(x)+\omega_2(x)+\omega_3(x)\), where

\[ \omega_k(x)= \int_{\Sigma} z(y)a(y)\sum_{ij=1}^{m} a_{ij}(y)H_{ki}(xy)\cos(ny_j)\,ds_y, \quad k=1,2,3. \tag{3.2} \]

Let us consider separately each integral \(\omega_k(x)\). Items \(3^\circ\)—\(10^\circ\) are devoted to the integral \(\omega_1(x)\), item \(11^\circ\) to \(\omega_2(x)\), and items \(12^\circ\)—\(16^\circ\) to \(\omega_3(x)\).

\(3^\circ\). Consider

\[ \omega_1(x)= \int_{\Sigma} z^*(y)\,[\rho^{2-m}]_{x\in\Sigma_0}\,ds_y, \tag{3.3} \]

where

\[ z^*(y)= -\frac{z(y)}{2(m-2)a(y)\omega_m[A(y)]^{3/2}} \sum_{ij=1}^{m} a_{ij}(y)\frac{\partial A(y)}{\partial y_i}\cos(ny_j). \]

From conditions A it is not difficult to obtain that \(z^*(y)\in C^{(n,\lambda)}(\Gamma)\). Thus, it is necessary to prove that if \(z^*(y)\in C^{(n,\lambda)}(\Gamma)\), then \(\omega_1(x)\) belongs to the class \(C^{(n+1,\lambda')}(\sigma_0)\) and the norm \(\|\omega_1\|_{C^{(n+1,\lambda')}(\sigma_0)}\) can be estimated in terms of the norm \(\|z^*\|_{C^{(n,\lambda)}(\Gamma)}\).

For this purpose we pass in (3.3) to integration in the plane \(\pi\):

\[ \omega_1(x)= \int_{\sigma} z^*(y)\,[\rho^{2-m}]_{x\in\Sigma_0} \frac{1}{\cos(ny_m)}\,dy_1,\ldots,dy_{m-1}. \tag{3.4} \]

From the property of a Lyapunov surface it follows that, for sufficiently small \(d\), \(\cos(ny_m)>\frac{1}{2}\). Therefore (3.4) can be rewritten in the form

\[ \omega_1(\bar x)= \int_{\sigma} [\bar z(y)\rho^{2-m}]_{\substack{x_m=f(\bar x)\\ y_m=f(\bar y)}}\,d\bar y, \tag{3.5} \]

where \(\bar x=(x_1,\ldots,x_{m-1})\), \(\bar y=(y_1,\ldots,y_{m-1})\), \(d\bar y=dy_1,\ldots,dy_{m-1}\), \(z(\bar y)=z^*(y)/\cos(ny_m)\).

When \(x,y\in\Sigma\), then

\[ \rho^2=\sum_{ij=1}^{m-1} A_{ij}(\bar y,+t(\bar y))(x_i-y_i)(x_j-y_j)+ \]
\[ +2\sum_{i=1}^{m-1} A_{im}(\bar y,f(\bar y))\bigl(f(\bar y)-f(\bar x)\bigr)(y_i-x_i)+ \]
\[ +A_{mm}(\bar y,f(\bar y))[f(\bar y)-f(\bar x)]^2,\qquad \bar x,\bar y\in\sigma . \tag{3.6} \]

By virtue of the uniform ellipticity of the operator (1.1), for any \(\bar x,\bar y\in\sigma\) one has

\[ \sum_{ij=1}^{m-1} A_{ij} f(\bar y,f(\bar y))(x_i-y_i)(x_j-y_j)>a_0\sum_{i=1}^{m-1}(x_i-y_i)^2 . \tag{3.7} \]

We are now dealing with the computation of the derivatives of the integral (3.5). From (3.6) and (3.7) it is easy to see that the kernel of this integral has property (2.2) (in \((m-1)\)-dimensional space). In view of this we apply the smoothing method to (3.5). In this case, as \(g\), \(g'\), and \(g''\) one may take, respectively, the balls \(\sigma_1\), \(\sigma_0\), and \(\sigma_2\): \(r_{x_0\bar y}\le \dfrac{3}{2}d_0\), where \(r_{x_0\bar y}\) is the distance between the points \(x_0\) and \(\bar y\) lying on the plane \(\pi\).

Let us dwell on the details of this smoothing.

The smoothing of the density \(z(\bar y)\) and of the function \(f(\bar y)\) can be carried out by means of the cutoff function indicated in the preceding section. For what follows it is necessary to smooth in different ways the functions \(A_{ij}(\bar y)=A_{ij}(\bar y,f(\bar y))\), \(1\le i,j\le m-1\), and the functions \(A_{im}(\bar y)=A_{im}(\bar y,f(\bar y))\), \(1\le i\le m\).

By virtue of (3.7), following G. Giraud [5], we can extend \(A_{ij}(\bar y)\) \((1\le i,j\le m-1)\) through the boundary of the ball \(\sigma_1\) to the whole plane \(\pi\) so that the extended functions \(A_{ij}(\bar y)\) satisfy the following requirements:

1) \(A_{ij}(\bar y)\in C^{(n+1,\lambda)}\) on \(\pi\), \(\|A_{ij}(\bar y)\|_{C^{(n+1,\lambda)}(\pi)}\) is estimated in terms of \(\|A_{ij}(\bar y)\|_{C^{(n+1,\lambda)}(\sigma_1)}\);

2) for all \(\bar x,\bar y\in\pi\), (3.7) holds.

The functions \(A_{im}(\bar\varphi)\) \((1\le i\le m)\) are smoothed in the usual way, i.e., by multiplication by the cutoff function \(\theta(\bar y)\).

According to § 2, after smoothing it remains to consider only the integral (see (2.7))

\[ \Phi(\bar x)=\int_0^\pi \sin\varphi_1^{m-3}\,d\varphi_1 \int_0^\pi \sin\varphi_2^{m-4}\,d\varphi_2\cdots \int_0^\pi \sin\varphi_{m-3}\,d\varphi_{m-3}\times \]
\[ \times\int_0^{2\pi} d\varphi_{m-2}\int_0^\infty \bar z(\bar x+r\varphi)\,r^{m-2}\rho^{2-m}\,dr= \]
\[ =\int_\theta d\theta\int_0^\infty \bar z(\bar x+r\varphi)\,F(\bar x;r,\varphi)\,dr, \tag{3.8} \]

where

\[ F(\bar x;r,\varphi)=(r/\rho)^{m-2}. \]

in which one must everywhere replace \(y_m\) by \(f(\bar y)\), and \(\bar y\) by \(\bar x+r\varphi\) according to formula (2.4), where \(r\) is the distance between \(\bar x\) and \(\bar y\); \(r\varphi\) is an abbreviated notation for the right-hand side of (2.4) in the \(m-1\)-dimensional case.

\(4^\circ\). Let us consider the differentiability of \(z(\bar x+r\varphi)\) and \(F(\bar x;r,\varphi)\).

Obviously, \(z(\bar x+r\varphi)\) has continuous derivatives with respect to \(x_1,\ldots,x_{m-1}\) up to order \(n\) inclusive.

We shall prove that \(F(\bar x;r,\varphi)\) has continuous derivatives with respect to \(x_1,\ldots,x_{m-1}\) up to order \(n+1\) inclusive, and that the norm \(\|F\|_{C^{(n+2)}(\sigma_1)}\) can be estimated in terms of the norms \(\|a_{ij}\|_{C^{(n+1)}}\), \(\|f\|_{C^{(n+1)}}\), uniformly with respect to all \(r\) and \(\varphi\).

Indeed, when \(x,y\in\Sigma\), then

\[ y_m-x_m=f(\bar y)-f(\bar x) = \sum_{i=1}^{m-1}(y_i-x_i)\int_0^1 \frac{\partial f(\bar x+tr\varphi)}{\partial x_i}\,dt, \tag{3.9} \]

therefore

\[ \rho^2=\sum_{i,j=1}^{m-1}B_{ij}(\bar x,\bar y)(y_i-x_i)(y_j-x_j), \]

where

\[ \begin{aligned} B_{ij}(\bar x,\bar y) &= A_{ij}(\bar y,f(\bar y)) + 2A_{im}(\bar y,f(\bar y)) \int_0^1 \frac{\partial f(\bar x+t(\bar y-\bar x))}{\partial x_j}\,dt \\ &\quad + A_{mm}(\bar y,f(\bar y)) \int_0^1 \frac{\partial f(\bar x+t(\bar y-\bar x))}{\partial x_i}\,dt \int_0^1 \frac{\partial f(\bar x+t(\bar y-\bar x))}{\partial x_j}\,dt. \end{aligned} \tag{3.10} \]

After the substitution (2.4),

\[ \rho^2=r^2\sum_{ij=1}^{m-1}B_{ij}(\bar x;r,\varphi)=r^2B(\bar x;r,\varphi), \tag{3.11} \]

where

\[ B_{ij}(\bar x;r,\varphi)=B_{ij}(\bar x,\bar y)\big|_{\bar y=\bar x+r\varphi}. \]

Thus,

\[ F(\bar x;r,\varphi)=[B(\bar x;r,\varphi)]^{1-\frac m2}. \]

From the uniform ellipticity of the operator (1.1) it follows that for all \((r,\varphi)\) and \(\bar x\in\sigma_2\),

\[ B(\bar x;r,\varphi)\ge a_0. \tag{3.12} \]

By virtue of condition \(A\), from (3.10) it is clear that \(B(\bar x;r,\varphi)\) belongs to the class \(C^{(n+1)}\) in \(\bar x\) in \(\sigma_2\), and \(\|B\|_{C^{(n+1)}(\sigma_2)}\) is estimated in terms of \(\|a_{ij}\|_{C^{(n+1)}}\) and \(\|f\|_{C^{(n+2)}}\), uniformly with respect to all \(r\) and \(\varphi\).

The derivatives of order \(s\) of \(F(\bar x;r,\varphi)\) with respect to \(x_i\) \((i=1,\ldots,m-1)\) are finite sums of the form

\[ \frac{\left[B^{(i_1)}\right]^{k_1},\ldots,\left[B^{(i_s)}\right]^{k_s}} {[B]^{\frac m2+p-1}}, \qquad s\le n+1, \tag{3.13} \]

where

\[ 0 \leqslant i_j \leqslant s,\quad 0 \leqslant k_j \leqslant s,\quad j=1,\ldots,s,\quad 1 \leqslant p \leqslant s, \]

and

\[ B^{(i_s)}=B^{(i_s)}(\bar{x};r,\varphi)=D_x^{(i_s)}(\bar{x};r,\varphi) \]

is some derivative of order \(i_s\) of \(B(\bar{x};r,\varphi)\) with respect to \(\bar{x}\).

We note that, by the conditions \(K\), each factor in the numerator of (3.13) belongs, with respect to \(\bar{x}\), to the class \(C^{(0,\lambda)}(\sigma_2)\) for all \(r\) and \(\varphi\); therefore, by (3.12), (3.13) as a whole belongs to this same class.

\(5^\circ\). Thus (3.8) may be differentiated \(n\) times under the integral sign. The derivatives of order \(n\) of \(\varphi(x)\) with respect to \(x_1,\ldots,x_{m-1}\) are finite sums of the form

\[ \varphi_l(\bar{x})=\int_{\vartheta} d\theta \int_0^\infty \bar{z}^{(l)}(\bar{x}+r\varphi)F^{(n-l)}(\bar{x};r,\varphi)\,dr,\quad l=0,1,\ldots,n. \tag{3.14} \]

It follows directly from this that the norm \(\|\varphi\|_{C^{(l)}(\sigma_0)}\) is estimated in terms of \(\|\bar{z}\|_{C^{(l)}}\).

Thus, it remains to prove that \(\varphi_l(\bar{x})\) belongs to the class \(C^{(1,\lambda')}(\sigma_0)\) for any \(0\leqslant l\leqslant n\), and to estimate its norm in \(C^{(1,\lambda')}(\sigma_0)\) in terms of \(\|\bar{z}\|_{C^{n,\lambda}_{(l)}}\).

\(6^\circ\). To this end, first consider the case when \(\bar{z}^{(l)}=1\), i.e., consider the integral

\[ \bar{\varphi}_l(\bar{x})=\int_{\vartheta} d\theta \int_0^\infty F^{(n-l)}(\bar{x};r,\varphi)\,dr. \]

By the arguments in item \(4^\circ\), \(\bar{\varphi}_l(\bar{x})\) belongs to the class \(C^{(1,\lambda)}(\sigma_0)\), and its norm \(\|\bar{\varphi}_l\|_{C^{(1,\lambda)}(\sigma_0)}\) can be estimated independently of the density \(z(x)\).

\(7^\circ\). Returning to the original local Cartesian coordinate system, write (3.14) in the form

\[ \varphi_l(\bar{x})=\int_{\pi} \bar{z}^{(l)}(\bar{y})F^{(n-l)}(\bar{x},\bar{y})\,r_{\bar{x}\bar{y}}^{\,2-m}\,d\bar{y}, \]

where

\[ F^{(n-l)}(\bar{x},\bar{y})= F^{(n-l)}(\bar{x};r,\varphi)\big|_{(r,\varphi)=\bar{y}-\bar{x}}, \]

\[ r_{\bar{x}\bar{y}}^2=\sum_{i=1}^{m-1}(y_i-x_i)^2. \]

Let \(\bar{x}=(x_1,\ldots,x_{m-1})\), \(\bar{x}_2=(x_1,\ldots,x_{i-1},x_i+h,x_{i+1},\ldots,x_{m-1})\) be two arbitrary points belonging to \(\bar{\sigma}_0\). Form the expression

\[ \frac{1}{h}\,[\varphi_l(\bar{x})-\varphi_l(\bar{x}_2)] = \frac{\bar{z}^{(l)}(\bar{x})}{h} \left[ \int_{\pi}\frac{F^{(n-l)}(\bar{x},\bar{y})}{r_{\bar{x}\bar{y}}^{\,m-2}}\,d\bar{y} - \int_{\pi}\frac{F^{(n-l)}(\bar{x}_2,\bar{y})}{r_{\bar{x}_2\bar{y}}^{\,m-2}}\,d\bar{y} \right] +\frac{1}{h} \left\{ \int_{\pi} \bigl(\bar{z}^{(l)}(\bar{y})-\bar{z}^{(l)}(\bar{x})\bigr) \frac{F^{(n-l)}(\bar{x},\bar{y})}{r_{\bar{x}\bar{y}}^{\,m-2}}\,d\bar{y} - \]

\[ -\int_{\pi}\left[z^{(l)}(\bar y)-z^{(l)}(\bar x)\right] \frac{F^{(n-l)}(\bar x_2,\bar y)}{r_{\bar x_2 y}^{m-2}}\,d\bar y \Bigg\}, \tag{3.15} \]

where

\[ r_{\bar x_2 y}^{2}=\sum_{\substack{j=1\\ j\ne i}}^{m-1}(y_j-x_j)^2+(y_i-x_i-h)^2. \]

If, as \(h\to 0\), (3.15) has a limit, then it will be the derivative of \(\varphi_l(\bar x)\) with respect to \(x_i\). It is necessary to prove the existence of this limit, that it belongs to the class \(C^{(0,\lambda')}(\sigma_0)\), and then to estimate its norm in \(C^{(0,\lambda')}(\sigma_0)\) in terms of \(\|z^{(l)}\|_{C^{(0,\lambda)}}\).

This is easy to do for the first term on the right-hand side of (3.15), by virtue of the results of item \(6^\circ\). Let us consider the second term in (3.15). For this purpose we shall prove that in the expression

\[ \begin{aligned} &\int_{\pi}\left[z_l(\bar y)-z_l(\bar x)\right] \left\{ \frac{1}{h}\left[ \frac{F^{(n-l)}(\bar x\bar y)}{r_{\bar x y}^{m-2}} - \frac{F^{(n-l)}(\bar x_2,\bar y)}{r_{\bar x_2 y}^{m-2}} \right] - \frac{\partial}{\partial x_i} \left[ \frac{F^{(n-l)}(\bar x\bar y)}{r_{\bar x y}^{m-2}} \right] \right\}\,d\bar y \\ &= \frac{1}{h}\int_{r_{\bar x y}\le 2|h|} \left[z_l(\bar y)-z_l(\bar x)\right] \frac{F^{(n-l)}(\bar x\bar y)}{r_{\bar x y}^{m-2}}\,d\bar y - \frac{1}{h}\int_{r_{\bar x y}\le 2|h|} \left[z_l(\bar y)-z_l(\bar x)\right] \frac{F^{(n-l)}(\bar x_2,\bar y)}{r_{\bar x_2 y}^{m-2}}\,d\bar y \\ &\quad -\int_{r_{\bar x y}\le 2|h|} \left[z_l(\bar y)-z_l(\bar x)\right] \frac{\partial}{\partial x_i} \left[ \frac{F^{(n-l)}(\bar x\bar y)}{r_{\bar x y}^{m-2}} \right]\,d\bar y \\ &\quad +\int_{r_{\bar x y}\ge 2|h|} \left[z_l(\bar y)-z_l(\bar x)\right] \left\{ \frac{1}{h}\left[ \frac{F^{(n-l)}(\bar x\bar y)}{r_{\bar x y}^{m-2}} - \frac{F^{(n-l)}(\bar x_2,\bar y)}{r_{\bar x_2 y}^{m-2}} \right] - \frac{\partial}{\partial x_i} \left[ \frac{F^{(n-l)}(\bar x\bar y)}{r_{\bar x y}^{m-2}} \right] \right\}\,d\bar y \\ &=I_1+I_2+I_3+I_4 \end{aligned} \]

as \(h\to 0\) each term tends to zero, where \(z_l(\bar x)=z^{(l)}(\bar x)\). Then, instead of the second term on the right-hand side of (3.15), one may consider the integral

\[ \chi(\bar x)= \int_{\pi}\left[z_l(\bar y)-z_l(\bar x)\right] \frac{\partial}{\partial x_i} \left[ \frac{F^{(n-l)}(\bar x\bar y)}{r_{\bar x y}^{m-2}} \right]\,d\bar y . \tag{3.16} \]

\(8^\circ\). We shall prove that, as \(h\to 0\), \(I_k\to 0\), \(k=1,2,3,4\). We have

\[ I_1=O\left(\frac{1}{|h|}\int_{0}^{2|h|} r^\lambda\,dr\right)\to 0 \quad\text{as } h\to 0, \]

since \(z_l\in C^{(0,\lambda)}\) and \(F^{(n-l)}(\bar x\bar y)=O(1)\).

Similarly,

\[ I_2=O\left(\frac{1}{|h|}\int_0^{3|h|}(r_2^\lambda+|h|^\lambda)\,dr_2\right)\to 0 \quad \text{as } h\to 0, \]

since

\[ |z_1(\bar y)-z_1(\bar x)|\leq |z_1(\bar y)-z_1(\bar x_2)|+ |z_1(\bar x_2)-z_1(\bar x)| \tag{3.17} \]

and the ball \(r_{\bar x y}\leq 2|h|\) is contained in the ball
\(r_{\bar x_2 y}\leq 3|h|\).

To prove that \(I_3\to 0\) as \(h\to 0\), it is enough to establish the estimate

\[ \frac{\partial}{\partial x_i}F^{(n-1)}(\overline{x y})=O(r_{\bar x y}^{-1}). \tag{3.18} \]

Indeed, if (3.18) were true, then

\[ I_3=O\left(\int_0^{2|h|} r^{\lambda-1}\,dr\right)\to 0 \quad \text{as } h\to 0. \]

Finally, consider \(I_4\). We note that

\[ \frac{1}{h}\left[ \frac{F^{(n-1)}(\overline{x y})}{r_{\bar x y}^{m-2}} - \frac{F^{(n-1)}(\overline{x_2 y})}{r_{\bar x_2 y}^{m-2}} \right] \]

is the derivative with respect to \(x_i\) of the function

\[ \frac{F^{(n-1)}(\overline{x y})}{r_{\bar x y}^{m-2}} \]

at some point
\(\bar x'=(x_1,\ldots,x_{i-1},x_i+\theta h,x_{i+1},\ldots,x_{m-1})\),
\(0\leq\theta\leq 1\). We further note that, by the property of a Lyapunov surface, outside
\(r_{\bar x y}>2|h|\) the following holds [2]:

\[ \left\{ \begin{aligned} \frac{1}{r_{\bar x y}}&=O\left(\frac{1}{r_{\bar x_2 y}}\right),\\ \frac{1}{r_{\bar x_2 y}}&=O\left(\frac{1}{r_{\bar x y}}\right), \end{aligned} \right. \tag{3.19} \]

where \(O\) does not depend on \(h\).

Therefore, in order to prove that \(I_4\to 0\) as \(h\to 0\), it is enough to prove that

\[ \left[ \frac{\partial}{\partial x_i} \left( \frac{F^{(n-1)}(\overline{x y})}{r_{\bar x y}^{m-2}} \right) \right]_{x=x'} - \frac{\partial}{\partial x_i} \left( \frac{F^{(n-1)}(\overline{x y})}{r_{\bar x y}^{m-2}} \right) = O\left(\frac{|h|}{r_{\bar x y}^{m}}\right). \tag{3.20} \]

Indeed, if this were proved, then

\[ \begin{aligned} I_4 &= \int_{r_{\bar x y}>2|h|} [z_1(\bar y)-z_1(\bar x)] \left\{ \left[ \frac{\partial}{\partial x_i} \left( \frac{F^{(n-1)}(\overline{x y})}{r_{\bar x y}^{m-2}} \right) \right]_{x=x'} \right.\\ &\qquad\left. - \frac{\partial}{\partial x_i} \left( \frac{F^{(n-1)}(\overline{x y})}{r_{\bar x y}^{m-2}} \right) \right\}\,dy = O\left(|h|\int_{2|h|}^{a} r^{\lambda-2}\,dr\right)\to 0 \quad \text{as } h\to 0, \end{aligned} \tag{3.21} \]

where \(a\) is some sufficiently large number.

\(9^\circ\). In this subsection we shall establish the estimates (3.18) and (3.20). For simplicity we consider only the case when \(l=0\).

Let us note that

\[ r_{\bar x\bar y}=O(|h|), \]

\[ \frac{\partial}{\partial x_i}\left[\frac{F^{(n)}(\bar x\bar y)}{r_{\bar x\bar y}^{\,m-2}}\right] = \frac{\dfrac{\partial}{\partial x_i}F^{(n)}(\bar x\bar y)}{r_{\bar x\bar y}^{\,m-2}} -(m-2)\frac{\partial r_{\bar x\bar y}}{\partial x_i}\, \frac{F^{(n)}(\bar x\bar y)}{r_{\bar x\bar y}^{\,m-1}}. \]

Therefore, in order to verify (3.18) and (3.20), it suffices to prove that

\[ \frac{\partial}{\partial x_i}F^{(n)}(\bar x\bar y)=O(r_{\bar x\bar y}^{-1}), \tag{3.22} \]

\[ \frac{\partial^2}{\partial x_i\partial x_j}F^{(n)}(\bar x\bar y)=O(r_{\bar x\bar y}^{-2}). \tag{3.23} \]

From (3.13) it is clear that it suffices to prove (3.22) and (3.23) for all

\[ D^{(l)}B(\bar x\bar y)=B^{(l)}(\bar x;r,\varphi)\big|_{(r\varphi)=\bar y-\bar x},\quad l=0,1,\ldots,n. \]

Introduce the following notation.

Let \(f(\bar x)\) be any function of \(\bar x\). By \(\Delta f\) we denote the difference

\[ f(\bar y)-f(\bar x)=\Delta f. \]

The function obtained from \(\Delta f\) by replacing \(\bar y\) by \(\bar x+r\varphi\) according to formula (2.4) will be denoted by \(\overline{\Delta f}\). Then

\[ D^{(s)}(\overline{\Delta f})=\overline{\Delta(D^{(s)}f)}, \tag{3.24} \]

where the differentiation is performed with respect to \(x_i\) \((i=1,\ldots,m-1)\).

We have

\[ B(\bar x;r,\varphi)= \sum_{ij=1}^{m-1} A_{ij}(\bar x+r\varphi)g_i(\varphi)g_j(\varphi)+ \]

\[ +2\overline{\Delta f}\sum_{i=1}^{m-1}A_{im}(\bar x+r\varphi)g_i(\varphi)\frac{1}{r} +\frac{1}{r^2}A_{mm}(\bar x+r\varphi)(\overline{\Delta f})^2 \]

\[ =B_1+B_2+B_3, \]

\[ B_1^{(l)}(\bar x;r,\varphi)= \sum_{ij=1}^{m-1}A_{ij}^{(l)}(\bar x+r\varphi)g_i(\varphi)g_j(\varphi), \]

\[ B_2^{(l)}(\bar x;r,\varphi)\sim \frac{1}{r}\sum_{i=1}^{m-1} A_{im}^{(s)}(\bar x+r\varphi)g_i(\varphi)D^{(l-s)}(\overline{\Delta f}) \]

\[ =\psi D^{(l-s)}(\overline{\Delta f}),\quad 0\le s\le l, \]

where the symbol \(\sim\) means that \(B_2^{(l)}\) is a finite sum of terms of the form

\[ \psi D^{(l-s)}(\overline{\Delta f}), \]

\[ B_3^{(l)}(\bar x;r,\varphi)\sim \frac{1}{r^2}A_{mm}^{(s)}(\bar x+r\varphi)D^{(l-s)}\bigl[(\overline{\Delta f})^2\bigr], \quad 0\le s\le l. \]

HEIGHTENING PROPERTY OF DIRECT VALUES OF POTENTIALS

In view of \(f\in C^{(n+2,\lambda)}\), it is easy to verify that \(\dfrac{\partial}{\partial x_i}B_k^{(l)}(\overline{xy})\) and \(\dfrac{\partial^2}{\partial x_i\partial x_j}B_k^{(l)}(\overline{xy})\) exist and

\[ B_k^{(l)}(\overline{xy}) = \bigl[B_k^{(l)}(\overline{x};r,\varphi)\bigr]_{(r\varphi)=\overline{y}-\overline{x}} =O(1), \]

\[ \frac{\partial}{\partial x_i}B_k^{(l)}(\overline{xy})=O(r_{\overline{xy}}^{-1}), \tag{3.25} \]

\[ \frac{\partial^2}{\partial x_i\partial x_j}B_k^{(l)}(\overline{xy}) = O(r_{\overline{xy}}^{-2}), \qquad k=1,2,3,\quad 0\leq l\leq n. \]

Thus, (3.22) and (3.23) are proved.

\(10^\circ\). To estimate \(\omega_1(x)\) in \(C^{(n+1,\lambda')}(\sigma_0)\) (see (3.3)), it remains to prove that the integral (3.16) belongs to the class \(C^{(0,\lambda')}(\sigma_0)\) and can be estimated in \(C^{(0,\lambda')}(\sigma_0)\) in terms of \(\|z_l\|_{C^{(0,\lambda)}}\).

We have

\[ \chi(\overline{x}_1)-\chi(\overline{x}_2) = \int_{r_1\leq 2\delta} [z_l(\overline{y})-z_l(\overline{x})] \left\{ \frac{\partial}{\partial x_i} \left[ \frac{F^{(n-l)}(\overline{xy})}{r_{\overline{xy}}^{m-2}} \right] \right\}_{\overline{x}=\overline{x}_1} \,d\overline{y} - \]

\[ - \int_{r_1\leq 2\delta} [z_l(\overline{y})-z_l(\overline{x}_2)] \left\{ \frac{\partial}{\partial x_i} \left[ \frac{F^{(n-l)}(\overline{xy})}{r_{\overline{xy}}^{m-2}} \right] \right\}_{\overline{x}=\overline{x}_2} \,d\overline{y} + \]

\[ + \int_{r_1\geq 2\delta} \left\{ [z_l(\overline{y})-z_l(\overline{x}_1)] \left[ \frac{\partial}{\partial x_i} \left( \frac{F^{(n-l)}(\overline{xy})}{r_{\overline{xy}}^{m-2}} \right) \right]_{\overline{x}=\overline{x}_1} \right. \]

\[ \left. - [z_l(\overline{y})-z_l(\overline{x}_2)] \left[ \frac{\partial}{\partial x_i} \left( \frac{F^{(n-l)}(\overline{xy})}{r_{\overline{xy}}^{m-2}} \right) \right]_{\overline{x}=\overline{x}_2} \right\} \,d\overline{y} = \]

\[ =I_5+I_6+I_7, \]

where \(\overline{x}_1,\overline{x}_2\) are any two points in \(\sigma_0\), the distance between them is \(\delta\), \(r_1=r_{\overline{x}_1\overline{y}}\), \(r_2=r_{\overline{x}_2\overline{y}}\).

By virtue of (3.23),

\[ I_5 = O\left( \|z_l\|_{C^{(0,\lambda)}} \int_0^{2\delta} r_1^{\lambda-1}\,dr_1 \right) = O\left(\delta^\lambda\|z_l\|_{C^{(0,\lambda)}}\right), \]

\[ I_6 = O\left( \|z_l\|_{C^{(0,\lambda)}} \int_0^{3\delta} r_2^{\lambda-1}\,dr_2 \right) = O\left(\delta^\lambda\|z_l\|_{C^{(0,\lambda)}}\right), \]

where \(O\) does not depend on \(z_l\).

Let us rewrite \(I_7\) in the form \(I_7=I_7'+I_7''\), where

\[ I_7' = \int_{r_1\geq 2\delta} [z_l(\overline{x}_2)-z_l(\overline{x}_1)] \left[ \frac{\partial}{\partial x_i} \left( \frac{F^{(n-l)}(\overline{xy})}{r_{\overline{xy}}^{m-2}} \right) \right]_{\overline{x}=\overline{x}_2} \,d\overline{y} = \]

\[ = O\left( \|z_l\|_{C^{(0,\lambda)}}\delta^\lambda \right) \left( \int_{2\delta}^{a} \left| \frac{F^{(n+1-l)}(\overline{xy})}{r_2^{m-2}} - (m-2)\frac{\partial r_2}{\partial x_i} \frac{F^{(n-l)}(\overline{xy})}{r_2^{m-1}} \right| \,d\overline{y} \right) = \]

\[ =O\left(\delta^\lambda \|z_1\|_{C(0,\lambda)} \left(1+\log \frac{\alpha}{2\delta}\right)\right) =O\left(\delta^{\lambda'}\|z_1\|_{C(0,\lambda)}\right), \]

where \(0<\lambda'<\lambda\) is arbitrary,

\[ \begin{aligned} I_7''&=\int_{r_1\ge 2\delta} [z_1(\bar y)-z_1(\bar x_1)] \left\{ \left[ \frac{\partial}{\partial x_i} \left( \frac{F^{(n-l)}(\overline{xy})}{r_{\overline{xy}}^{m-2}} \right) \right]_{\bar x=\bar x_1} - \left[ \frac{\partial}{\partial x_i} \left( \frac{F^{(n-l)}(\overline{xy})}{r_{\overline{xy}}^{m-2}} \right) \right]_{\bar x=\bar x_2} \right\}\,d\bar y \\ &= O\left(\|z_1\|_{C(0,\lambda)}\, \delta\int_{2\delta}^{\alpha} r_1^{\lambda-2}\,dr_1\right) = O\left(\delta^\lambda\|z_1\|_{C(0,\lambda)}\right). \end{aligned} \]

The estimates for \(I_5\), \(I_6\), and \(I_7\) give the proof of our assertion concerning the integral (3.16).

\(11^\circ\). Let us now consider \(\omega_2(x)\) (see (3.2)). Since \(a_{ij}\in C^{(n+1,\lambda)}\), we transfer

\[ \frac{\partial A_{sl}(y)}{\partial y_i}\bigg/ \sqrt{A(y)} \]

into the density and, instead of \(\omega_2(x)\), consider only

\[ \omega_2^{(sl)}(x)= \int_{\Sigma} z(y)\rho^{-m}(y_s-x_s)(y_l-x_l)\,d s_y,\qquad 1\le s,l\le m. \tag{3.26} \]

This integral differs from the integral (3.3) only in that in (3.26) the kernel is

\[ F^{(sl)}(\overline{xy})=\rho^{-m}(x_s-y_s)(x_l-y_l), \]

whereas in (3.3) it is \(\rho^{2-m}\). Therefore all the arguments carried out above with respect to (3.3) are also applicable to (3.26).

Indeed, after passing to polar coordinates in (3.26), instead of the former kernel \(\bar F(\bar x;r,\varphi)\) there will be the new integral kernel

\[ F^{(sl)}(\bar x;r,\varphi) = T(\bar x;r,\varphi)\,[B(\bar x;r,\varphi)]^{-\frac{m}{2}}, \]

where

\[ T(\bar x;r,\varphi)= \begin{cases} g_s(\varphi)g_l(\varphi), & \text{if } 1\le s,l\le m-1,\\[1.2ex] g_l(\varphi)\displaystyle\sum_{i=1}^{m-1}g_i(\varphi) \int_0^1 \frac{\partial f(\bar x+tr\varphi)}{\partial x_i}\,dt, & \text{if } s=m,\ 1\le l\le m-1,\\[2.2ex] g_s(\varphi)\displaystyle\sum_{i=1}^{m-1}g_i(\varphi) \int_0^1 \frac{\partial f(\bar x+tr\varphi)}{\partial x_i}\,dt, & \text{if } l=m,\ 1\le s\le m-1,\\[2.2ex] \left[ \displaystyle\sum_{i=1}^{m-1}g_i(\varphi) \int_0^1 \frac{\partial f(\bar x+tr\varphi)}{\partial x_i}\,dt \right]^2, & \text{if } s=l=m. \end{cases} \]

It is seen from this that all the results of items \(5^\circ\)—\(10^\circ\) are also valid for \(F^{(s)}(x; r,\varphi)\), since the results were obtained only from the properties of the function \(B(\bar x; r,\varphi)\).

\(12^\circ\). We now pass to the integral

\[ \omega_3(x)\int_{\Sigma}\frac{z(y)}{a(y)} \sum_{i=1}^{m}\frac{x_i-y_i}{\omega_m\sqrt{A(y)}\rho^m}\cos(n x_i)\,ds_y . \]

It is known that

\[ \cos(n x_i)=-f'_{x_i}(\bar x)\cos(n x_m),\qquad 1\le i\le m-1. \]

Let us write \(\omega_3(x)\) in the form

\[ \omega_3(x)=\int_{\Sigma} z_1(y)\rho^{-m} \left[f(\bar x)-f(\bar y)+\sum_{i=1}^{m-1}(y_i-x_i)f'_{x_i}(\bar y)\right]\cos(n y_m)\,ds_4, \]

where

\[ z_1(y)=z(y)/\omega_m a(y)\sqrt{A(y)}. \]

Passing to the integral over \(\pi\) in polar coordinates, we rewrite \(\omega_3(\bar x)\) in the form

\[ \omega_3(\bar x)=\int_{\theta} a\theta\int_{0}^{\infty} z_1(\bar x+r\varphi)K(\bar x;r,\varphi)\,dr, \tag{3.27} \]

where

\[ z_1(\bar x+r\varphi)=z_1(\bar y,f(\bar y))\big|_{\bar y=\bar x+r\varphi}; \]

\[ K(\bar x;r,\varphi)=E(\bar x;r,\varphi)/r^2\,B^{-\frac{m}{2}}(\bar x;r,\varphi); \]

\[ E(\bar x;r,\varphi)=r\sum_{i=1}^{m-1} g_i(\varphi)f'_{x_i}(\bar x+r\varphi) -f(\bar x+r\varphi)+f(\bar x). \]

Our immediate task consists in proving that \(\omega_3(\bar x)\) belongs to the class \(C^{(n+1,\lambda')}(\bar\sigma_0)\) and in obtaining for \(\omega_3(\bar x)\) an estimate in \(C^{(n+1,\lambda')}(\bar\sigma_0)\) in terms of \(\|z_1\|_{C(n,\lambda)}\).

\(13^\circ\). Consider the differentiability of the kernel \(K(\bar x;r,\varphi)\).

Let \(\Phi(\bar x)\) be an arbitrary function. By \(\nabla\Phi\) we denote

\[ \Phi(\bar x)-\Phi(\bar y)+\sum_{i=1}^{m-1}(y_i-x_i)\Phi'_{x_i}(\bar y)=\nabla\Phi. \]

Replacing \(\bar y\) by \(\bar x+r\varphi\) according to formula (2.4), from \(\nabla\Phi\) we obtain a new function, which we denote by the symbol \(\nabla\Phi\). It is easy to verify that if \(\Phi\in C^{(2)}\), then

\[ \nabla\Phi= \sum_{ij=1}^{m-1}(y_i-x_i)(y_j-x_j) \int_{0}^{1} t\, \frac{\partial^2\Phi\bigl(\bar x+t(\bar y-\bar x)\bigr)} {\partial x_i\,\partial x_j}\,dt. \tag{3.28} \]

Obviously,

\[ D^{(s)}[\overline{\nabla \Phi}] = \overline{\nabla [D^{(s)}\Phi]}, \]

where \(D^{(s)}\) is any derivative with respect to \(\bar{x}\) of order \(s\). We have

\[ D^{(s)}E(\bar{x};r,\varphi) = D^{(s)}[\overline{\nabla f}] = \overline{\nabla [D^{(s)}f]} = \]

\[ = r^2\sum_{ij=1}^{m-1} g_i(\varphi)g_j(\varphi) \int_0^1 t\,\frac{\partial^2[D^{(s)}f(\bar{x}+tr\varphi)]}{\partial x_i\,\partial x_j}\,dt. \]

It follows from this that \(E(\bar{x};r,\varphi)\) and, consequently, \(K(\bar{x};r,\varphi)\) have continuous derivatives with respect to \(\bar{x}\) up to order \(n\) inclusive, for all \(r\) and \(\varphi\).

Consider the case when \(s=n+1\). We have

\[ D^{(n+1)}E(\bar{x};r,\varphi) = D^{(n+1)}f(\bar{x}) - D^{(n+1)}f(\bar{x}+r\varphi) + \]

\[ + \sum_{i=1}^{m-1} rg_i(\varphi)D^{(n+2)}f(\bar{x}+r\varphi) = -r\sum_{i=1}^{m-1}g_i(\varphi) \left[ D^{(n+2)}f(\bar{x}+\theta r\varphi) -\right. \]

\[ \left. - D^{(n+2)}f(\bar{x}+r\varphi) \right] = O(r^{1+\lambda}), \tag{3.29} \]

for \(f\in C^{(n+2,\lambda)}\). This estimate holds uniformly with respect to all \(\bar{x}\in\sigma_2\). Therefore, in \(\sigma_2\), \(K(\bar{x};r,\varphi)\) has continuous derivatives with respect to \(\bar{x}\) up to order \(n+1\) inclusive for \(r>0\), and for derivatives of order \(n+1\) the following estimate holds:

\[ K^{(n+1)}(\bar{x};r,\varphi)=O(r^{\lambda-1}) \tag{3.30} \]

uniformly for all \(\bar{x}\in\sigma_2\).

The derivative of order \(s\) of \(K(\bar{x};r,\varphi)\) is a finite sum of terms of the form

\[ K^{(s)}(\bar{x};r,\varphi)\sim \frac{1}{r^2}\, \overline{\nabla(D^{(p)}f)}\, D^{(s-p)} \left[ B^{-\frac{m}{2}}(\bar{x};r,\varphi) \right], \tag{3.31} \]

where \(0\le s\le n+1,\quad 0\le p\le s\).

\(14^\circ\). Thus, differentiation with respect to \(\bar{x}\) under the integral sign in (3.27) is legitimate. Hence we may assert that \(\omega_3(\bar{x})\) has continuous derivatives with respect to \(\bar{x}\) in \(\sigma_0\) up to order \(n\) inclusive; moreover, one can estimate \(\|\omega_3\|_{C^{(l)}(\sigma_0)}\) in terms of \(\|z_1\|_{C^{(n)}}\), \(l=0,1,\ldots,n\).

The derivative of order \(n\) of \(\omega_3(\bar{x})\) is a finite sum of terms of the form

\[ D^{(n)}\omega_3(\bar{x})\sim \psi_l(\bar{x}) = \int_\theta d\theta \int_0^\infty z_1^{(l)}(\bar{x}+r\varphi) K^{(n-l)}(\bar{x};r,\varphi)\,dr, \qquad 0\le l\le n. \]

For simplicity of notation, in what follows we shall consider only the case \(l=0\). In this case

\[ \psi_0(\bar{x}) \sim \int_\theta d\theta \int_0^\infty z_1(\bar{x}+r\varphi) \frac{\overline{\nabla(D^{(s)}f)}}{r^2} D^{(n-s)} \left[ B^{-\frac{m}{2}}(\bar{x};r,\varphi) \right]dr, \qquad 0\le s\le n. \]

Without loss of generality, we shall restrict ourselves only to the case when \(s=n\). In this case we write

\[ \psi_0(\bar x)\sim \int_{\theta} d\theta \int_0^\infty z_1(\bar x+r\varphi)\, \frac{\overline{\nabla\left(D^{(n)}f\right)}}{r^2} \left[B^{-\frac m2}(\bar x;r,\varphi)\right]. \tag{3.32} \]

Thus, it remains to prove that the integral (3.32) belongs to the class
\(C^{(1,\lambda')}(\bar\sigma_0)\), and to estimate it in
\(C^{(1,\lambda')}(\bar\sigma_0)\) in terms of \(\|z_1\|_{C^{(0,\lambda)}}\).

\(15^\circ\). For this purpose, first of all, let us prove that

\[ \widetilde\psi_0(\bar x)= \int_{\theta} d\theta \int_0^\infty \frac{\overline{\nabla\left(D^{(n)}f\right)}}{r^2} \left[B^{-\frac m2}(\bar x;r,\varphi)\right]\,dr \]

belongs to the class \(C^{(1,\lambda')}(\bar\sigma_0)\) and has a finite norm in
\(C^{(1,\lambda')}(\bar\sigma_0)\).

By virtue of (3.29), the integral

\[ \int_0^\alpha \frac1{r^2}\, \frac{\partial}{\partial x_i} \left[\overline{\nabla\left(D^{(n)}f\right)}\right]\,dr \]

converges uniformly with respect to all \(\bar x\in \bar\sigma_2\). Therefore
\(\widetilde\psi_0(\bar x)\) may be differentiated under the integral sign.

Thus, it remains to prove that

\[ \frac{\partial\widetilde\psi_0(\bar x)}{\partial x_i} = \int_{\theta} d\theta \int_0^\infty \frac1{r^2}\, \frac{\partial}{\partial x_i} \left\{ \overline{\Delta\left(D^{(n)}f\right)} \left[B^{-\frac m2}(\bar x;r,\varphi)\right] \right\}\,dr = \]

\[ = \int_{\theta} d\theta \int_0^\infty Q(\bar x;r,\varphi)\,dr \]

belongs to the class \(C^{(0,\lambda')}(\bar\sigma_0)\).

By virtue of (3.9) we have

\[ \overline{\nabla\left[D^{(n+1)}f(\bar x_1;r,\varphi)\right]} - \overline{\nabla\left[D^{(n+1)}f(\bar x_2;r,\varphi)\right]} = O\left(r|\bar x_1-\bar x_2|^\lambda\right), \]

where \(\bar x_1,\bar x_2\) are two arbitrary points in \(\bar\sigma_0\), and \(O\) does not depend on \(r\) or on \(\bar x_1,\bar x_2\). Starting from this estimate, it is not difficult to verify that

\[ |Q(\bar x_1;r,\varphi)-Q(\bar x_2;r,\varphi)| = O\left(\frac{|\bar x_1-\bar x_2|^\lambda}{r}\right) \tag{3.33} \]

holds for all \(\bar x_1,\bar x_2\in\bar\sigma_2\). By virtue of this, it is easy to prove that
\(\partial\widetilde\psi_0(\bar x)/\partial x_i\) belongs to the class
\(C^{(0,\lambda')}(\bar\sigma_0)\), where \(0<\lambda'<\lambda\) is arbitrary.

\(16^\circ\). In conclusion of this paragraph we shall complete the proof of Theorem 1. It remains only to show that the integral (3.32) belongs to the class
\(C^{(1,\lambda')}(\bar\sigma_0)\) and can be estimated in
\(C^{(1,\lambda')}(\bar\sigma_0)\) in terms of \(\|z_1\|_{C^{(0,\lambda)}}\).

Here we shall not repeat all the arguments of points \(7^\circ\)—\(10^\circ\). Let us note only that all the investigations in points \(7^\circ\), \(8^\circ\), and \(10^\circ\) also apply to the integral (3.32).

To verify the estimates (3.22) and (3.23) for \(K(\bar{x}; r, \varphi)\), it is enough only to verify these estimates for the function

\[ \frac{\nabla(D^{(p)} f)}{r_{\bar{x}y}^{2}} = \frac{\nabla(D^{(p)} f)} {\displaystyle\sum_{i=1}^{m-1}(x_i-y_i)^2}, \qquad 0\le p\le n . \]

This is very easy to do if one takes into account that

\[ \nabla(D^{(p)} f)=O(r_{\bar{x}y}^{2}), \]

\[ \frac{\partial}{\partial x_i}\,[\nabla(D^{(p)} f)]=O(r_{\bar{x}y}), \]

\[ \frac{\partial^2}{\partial x_i\partial x_j}\,[\nabla(D^{(p)} f)]=O(1). \]

Thus, Theorem 1 is proved.

§ 4. DIRECT VALUE OF THE CONORMAL DERIVATIVE OF THE SIMPLE-LAYER POTENTIAL FOR THE COMPARISON FUNCTION

In this section Theorem 2 is proved for the case \(m>3\). The case \(m=2\) will be considered in the next section.

In this section the same computational technique is used as in the preceding section. In this connection we shall not repeat what has already been done in § 3.

1°. By definition,

\[ V_{\operatorname{pr}}(x) = \int_{\Gamma} v(y)\,[P_x H(xy)]_{x\in\Gamma}\,ds_y, \]

where

\[ P_x H(xy)= I(xy)/\omega_m\sqrt{A(y)}\,\rho^m, \]

\[ I(xy)=I_1(xy)\cos(nx_m)+I_2(xy), \]

\[ I_1(xy)\cos(nx_m) = \sum_{i,j,l=1}^{m} a_{ij}(y)A_{il}(y)(y_l-x_l)\cos(nx_j) = \]

\[ = \left[ \sum_{j=1}^{m-1}(y_j-x_j)f'_{x_j}(\bar{x}) + f(\bar{x})-f(\bar{y}) \right]\cos(nx_m), \]

\[ I_2(xy) = \sum_{i,j,l=1}^{m} [a_{ij}(x)-a_{ij}(y)]A_{il}(y)(y_l-x_l)\cos(nx_j). \]

Then

\[ V_{\operatorname{pr}}(x) = \int_{\Gamma} v(y) \left[ \frac{I_1(xy)\cos(xy)} {\omega_m\sqrt{A(y)}\,\rho^m} \right]_{x\in\Gamma} ds_y + \]

\[ + \int_{\Gamma} v(y) \left[ \frac{I_2(xy)} {\omega_m\sqrt{A(y)}\,\rho^m} \right]_{x\in\Gamma} ds_y = W_{\operatorname{pr}}^{1}(x)+W_{\operatorname{pr}}^{2}(x); \]

\(V_{\mathrm{pr}}^{1}(x)\) and \(V_{\mathrm{pr}}^{2}(x)\) are considered separately. Items \(2^\circ\)—\(3^\circ\) are devoted to \(V_{\mathrm{pr}}^{1}(x)\), and items \(4^\circ\) and \(5^\circ\) to \(V_{\mathrm{pr}}^{2}(x)\).

\(2^\circ\). It is known that \(\cos(nx_m)\in C^{(n+1,\lambda)}(\Gamma)\). Therefore, instead of \(V_{\mathrm{pr}}^{1}(x)\), one may consider only the integral

\[ S_1(\bar x)=\int_{\sigma} v(\bar y) \left[\frac{I_1(xy)}{\rho^m}\right]_{x\in\sigma_0}\,d\bar y . \]

We must prove that if \(v(\bar y)\in C^{(n,\lambda)}(\Gamma)\), then \(S_1(\bar x)\in C^{(n+1,\lambda')}(\sigma_0)\) and
\(\|S_1\|_{C^{(n+1,\lambda')}(\sigma_0)}\) is estimated in terms of
\(\|v\|_{C^{(n,\lambda)}}\).

We have

\[ I_1(\bar x\bar y)=E(\bar x\bar y)+E_1(\bar x\bar y), \]

where

\[ E_1(\bar x\bar y)=\sum_{i=1}^{m-1}(y_i-x_i)\bigl[f'_{x_i}(\bar x)-f'_{x_i}(\bar y)\bigr]= \]

\[ =-\sum_{i,j=1}^{m-1}(y_i-x_i)(y_j-x_j) \int_{0}^{1} \frac{\partial^2 f\bigl(\bar x+t(\bar y-\bar x)\bigr)} {\partial x_i\,\partial x_j}\,dt . \]

For \(E_1(\bar x;r,\varphi)=E_1(\bar x\bar y)|_{\bar y=\bar x+r\varphi}\) we establish a number of estimates which held for \(E(\bar x;r,\varphi)\).

Obviously,

\[ D^{(s)}E_1(\bar x;r,\varphi) = r\sum_{i=1}^{m-1}g_i(\varphi) \bigl[\Delta(D^{(s+1)}f)\bigr] = \]

\[ = r^2\sum_{i,j=1}^{m-1}g_i(\varphi)g_j(\varphi) \int_{0}^{1} \frac{\partial^2\bigl[D^{(s)}f(\bar x+tr\varphi)\bigr]} {\partial x_i\,\partial x_j}\,dt, \qquad 0\le s\le n, \]

\[ D^{(n+1)}E_1(\bar x;r,\varphi)=O(r^{1+\lambda}). \tag{4.1} \]

Hence it is clear that estimate (3.29) is fulfilled for \(E_1\).

Further, estimate (3.33) is also valid in the case when, instead of \(E\), \(E_1\) enters the kernel \(K(\bar x;r,\varphi)\) (see (3.27)), since

\[ \Delta(D^{(n+2)}f)\in C^{(0,\lambda)} \quad \text{with respect to } \bar x \text{ in } \bar\sigma_2 . \tag{4.2} \]

To establish for \(E_1\) the estimate corresponding to (3.22), it is enough to show that

\[ \frac{\partial}{\partial x_i} \left[ \frac{ \Delta(D^{(n+1)}f)\displaystyle\sum_{j=1}^{m-1}(y_j-x_j) }{ \displaystyle\sum_{j=1}^{m-1}(y_j-x_j)^2 } \right] = O(r_{\bar x\bar y}^{-1}). \tag{4.3} \]

This follows immediately from the fact that

\[ \Delta(D^{(n+1)}f)=O(r_{\bar x\bar y}), \]

\[ \frac{\partial}{\partial x_i}\bigl[\Delta(D^{(n+1)}f)\bigr]=O(1). \tag{4.4} \]

It is no longer possible to prove the estimate corresponding to (3.23) for \(E_1\), because second derivatives of \(\Delta(D^{n+1}f)\), generally speaking, do not exist. To avoid this, let us recall that the purpose of establishing (3.22) and (3.23) is to prove (3.30) and, consequently, (3.21). However, in order that (3.21) be valid, it is sufficient to require that, instead of (3.20), the following estimate be satisfied:

\[ \left\{ \frac{\partial}{\partial x_i} \left[ \frac{F^{(n-l)}(\bar x\bar y)}{r_{\bar x\bar y}^{m-2}} \right] \right\}_{x=\bar x'} - \frac{\partial}{\partial x_i} \left[ \frac{F^{(n-l)}(\bar x\bar y)}{r_{\bar x\bar y}^{m-2}} \right] = O\left( \frac{|h|}{r_{\bar x\bar y}^{m}} + \frac{|h|^\lambda}{r_{\bar x\bar y}^{m-1}} \right). \tag{4.5} \]

Thus, it remains only to prove (4.5) for our new kernel \(E_1(x;r,\varphi)/r^2\rho^2\).

\(3^\circ\). For this it is sufficient to verify that

\[ \left[ \frac{ \dfrac{\partial}{\partial x_i}R^{(n)}(\bar x\bar y) }{ r_{\bar x\bar y}^{m-2} } \right]_{x=\bar x'} - \frac{ \dfrac{\partial}{\partial x_i}R^{(n)}(\bar x\bar y) }{ r_{\bar x\bar y}^{m-2} } = O\left( \frac{|h|}{r_{\bar x\bar y}^{m}} + \frac{|h|^\lambda}{r_{\bar x\bar y}^{m-1}} \right), \tag{4.6} \]

where

\[ R^{(n)}(\bar x\bar y) = \left[ R^{(n)}(\bar x;r,\varphi) \right]_{(r\varphi)=\bar y-\bar x} = \]

\[ = \left\{ \frac{ \displaystyle \sum_{i=1}^{m-1} g_i(\varphi)\,[\Delta(D^{(s+1)}f)] }{r} D^{(n-s)} \left[ B^{-\frac m2}(\bar x;r,\varphi) \right] \right\}_{(r\varphi)=\bar y-\bar x} = \]

\[ = g(\bar x\bar y)\,k(\bar x\bar y), \]

\[ k(\bar x\bar y) = \frac{ \Delta(D^{(s+1)}f)\displaystyle\sum_{i=1}^{m-1}(y_i-x_i) }{ \displaystyle\sum_{i=1}^{m-1}(y_i-x_i)^2 }, \qquad s\le n. \]

\[ \bar x=(x_1,\ldots,x_{m-1}),\quad \bar x^{\,1}=(x_1,\ldots,x_{i-1},\,x_i+h,\,x_{i+1},\ldots,x_{m-1}). \]

It was proved that

\[ g(\bar x\bar y)=O(1); \qquad \frac{\partial}{\partial x_i}g(\bar x\bar y)=O(r_{\bar x\bar y}^{-1}); \qquad \frac{\partial^2}{\partial x_i\partial x_j}g(\bar x\bar y)=O(r_{\bar x\bar y}^{-2}) \tag{4.7} \]

(see (3.22), (3.23)). Obviously

\[ k(\bar x\bar y)=O(1), \qquad \frac{\partial}{\partial x_i}k(\bar x\bar y)=O(r_{\bar x\bar y}^{-1}). \tag{4.8} \]

Therefore, in order to prove (4.6), it is sufficient to show that

\[ J= \frac{\partial}{\partial x_i}k(\bar x\bar y) - \frac{\partial}{\partial x_i}k(\bar x'\bar y) = O\left( \frac{|h|}{r_{\bar x\bar y}^{2}} + \frac{|h|^\lambda}{r_{\bar x\bar y}} \right). \tag{4.9} \]

Suppose

\[ k(\bar x\bar y) = z(\bar x\bar y)\,[\Delta(D^{(s+1)}f)], \qquad 0\le s\le n . \]

and write

\[ J=J_1+J_2+J_3+J_4, \]

where

\[ J_1=\bigl(\Delta(D^{(s+1)}f)\bigr)_1 \left[\left(\frac{\partial z}{\partial x_i}\right)_1- \left(\frac{\partial z}{\partial x_i}\right)_2\right]; \]

by \((\Phi)_i\) \((i=1,2)\) we denote the value of the function \(\Phi(\bar x \bar y)\) at the point \(\bar x_i\),

\[ (\bar x_1=\bar x,\ \bar x_2=\bar x'), \]

\[ J_2=\left(\frac{\partial z}{\partial x_i}\right)_2 \left\{\bigl(\Delta(D^{(s+1)}f)\bigr)_1- \bigl(\Delta(D^{(s+1)}f)\bigr)_2\right\}, \]

\[ J_3=\left(\frac{\partial}{\partial x_i}[\Delta(D^{(s+1)}f)]\right)_2 \bigl[(z)_1-(z)_2\bigr], \]

\[ J_4=(z)_1\left\{ \left(\frac{\partial}{\partial x_i}[\Delta(D^{(s+1)}f)]\right)_1- \left(\frac{\partial}{\partial x_i}[\Delta(D^{(s+1)}f)]\right)_2 \right\}. \]

It is easy to verify that

\[ J_k=O\left(\frac{|h|}{r_{\bar x\bar y}^{\,2}}\right),\quad k=1,2,3, \]

\[ J_4=O\left(\frac{|h|^\lambda}{r_{\bar x\bar y}}\right). \]

Thus, (4.6) is proved.

\(4^\circ\). In view of \(\Gamma\subset A^{(n+2,\lambda)}\), instead of \(V_{\mathrm{pr}}^2(x)\) it suffices to consider only the integral

\[ S_2(\bar x)=\int_\sigma \nu(y)\,[\bar I_2(\bar x\bar y)\rho^{-m}]_{x\in\sigma_0}\,d y, \]

where

\[ \bar I_2(\bar x\bar y)=I_2^1(\bar x\bar y)+I_2^2(\bar x\bar y), \]

\[ I_2^1(\bar x\bar y)= \sum_{i,j,l=1}^{m-1} \bigl[a_{ij}(\bar x)-a_{ij}(\bar y)\bigr]A_{ij}(\bar y)(y_l-x_l), \]

\[ I_2^2(\bar x\bar y)= \bigl(f(\bar y)-f(\bar x)\bigr) \sum_{ij=1}^{m} \bigl[a_{ij}(\bar x)-a_{ij}(\bar y)\bigr]A_{ij}(\bar y). \]

Obviously,

\[ I_2^k(\bar x;r,\varphi)= I_2^k(\bar x\bar y)\big|_{\bar y=\bar x+r\varphi} \quad (k=1,2) \]

belongs to the class \(C^{(n,\lambda)}(\sigma_2)\) with respect to \(\bar x\) for all \(r\) and \(\varphi\). We shall prove next the estimates corresponding to (4.1) for our kernels \(I_2^1\) and \(I_2^2\).

Assume

\[ L_k(\bar x;r,\varphi)=\frac{1}{n^2}I_2^k(\bar x;r,\varphi),\quad k=1,2. \]

Then

\[ D^{(q)}L_1(\bar x;r,\varphi)\sim \frac{1}{r} \sum_{l=1}^{m-1} g_l(\varphi) \sum_{ij=1}^{m} D^{(s)}A_{il}(\bar x+r\varphi)\, \bigl[\Delta(D^{(q-p)}a_{ij})\bigr]. \]

\[ D^{(q)} L_2(\bar{x}; r,\varphi)\sim \frac{1}{r^2}\,[\Delta(D^{(s)}f)]\sum_{ij=1}^{m}D^{(p)}A_{im}(\bar{x}+r\varphi)[\Delta(D^{(q-s-p)}a_{ij})], \]

\[ 0\leq q\leq n+1,\quad 0\leq s\leq q,\quad 0\leq p\leq q. \]

Hence it follows easily that

\[ D^{(n+1)}L_k(\bar{x};r,\varphi)=O(r^{\lambda-1}),\quad k=1,2. \]

The property corresponding to (4.2) is also present in our case:

\[ D^{(s)}A_{il}\in C^{(0,\lambda)},\quad \Delta(D^{(s)}f)\in C^{(0,\lambda)},\quad \Delta(D^{(s)}a_{ij})\in C^{(0,\lambda)} \]

in \(\sigma_2\) with respect to \(\bar{x}\), for any \(0\leq s\leq n+1\).

Next, it is necessary to verify the estimate corresponding to (4.3) for \(L_k\), i.e., it is necessary to prove that

\[ \frac{\partial}{\partial x_i}\,[D^{(n)}L_k(\bar{x}\bar{y})]=O(r_{\bar{x}\bar{y}}^{-1}),\quad k=1,2, \tag{4.10} \]

where

\[ D^{(n)}L_k(\bar{x}\bar{y})=(D^{(n)}L_k(\bar{x};r,\varphi))_{(r\varphi)=\bar{y}-\bar{x}}. \]

Obviously,

\[ D^{(n)}L_1(\bar{x}\bar{y})\sim \frac{\displaystyle\sum_{l=1}^{m-1}(y_l-x_l)} {\displaystyle\sum_{l=1}^{m-1}(y_l-x_l)^2} \sum_{ij=1}^{m}D^{(s)}A_{il}(\bar{y})[\Delta D^{(n-s)}a_{ij}], \]

\[ D^{(n)}L_2(\bar{x}\bar{y})\sim \frac{1}{\displaystyle\sum_{l=1}^{m-1}(y_l-x_l)^2}\, \Delta(D^{(s)}f)\sum_{ij=1}^{m}D^{(p)}A_{im}(\bar{y})[\Delta(D^{(n-s-p)}a_{ij})]. \]

Then estimate (4.8) can be obtained from (4.4) and from the fact that

\[ \Delta(D^{(n-s)}a_{ij})=O(r_{\bar{x}\bar{y}}), \]

\[ \frac{\partial}{\partial x_i}\,[\Delta(D^{(n-s)}a_{ij})]=O(1). \]

\(5^\circ\). In general, it is no longer possible to prove the inequality corresponding to (3.23) for \(I_k^2\). We shall prove, as before, the estimate corresponding to (4.5), for two kernels:

\[ Q_k^{(n)}(\bar{x};r,\varphi) = D^{(s)}\left[\frac{1}{r^2}L_k(\bar{x};r,\varphi)\right] D^{(n-s)}\left[B^{-\frac{m}{2}}(\bar{x};r,\varphi)\right], \]

\[ 0\leq s\leq n,\quad k=1,2. \]

Let

\[ Q_k^{(n)}(\bar{x}\bar{y}) = Q_k^{(n)}(\bar{x};r,\varphi)\big|_{(r\varphi)=\bar{y}-\bar{x}} = g(\bar{x}\bar{y})P_k(\bar{x}\bar{y}),\quad k=1,2, \]

where

\[ P_1(\bar x \bar y) = \frac{\displaystyle \sum_{l=1}^{m-1}(y_l-x_l)} {\displaystyle \sum_{l=1}^{m-1}(y_l-x_l)^2} \sum_{ij=1}^{m} D^{(s)} A_{il}(\bar y)\,[\Delta(D^{(n-s)}a_{ij})], \]

\[ P_2(\bar x \bar y) = \frac{1}{\displaystyle \sum_{l=1}^{m-1}(y_l-x_l)^2} \Delta(D^{(s)}f)\sum_{lj=1}^{m}D^{(p)}A_{im}(\bar y)[\Delta(D^{(n-s-p)}a_{ij})]. \]

The function \(g(\bar x\bar y)\) has been well studied. In order to establish (4.5) for \(Q_k^{(n)}\), in view of (4.7) it is enough to show that

\[ \frac{\partial}{\partial x_i}P_k(\bar x\bar y) - \frac{\partial}{\partial x_i}P_k(\bar x'\bar y) = O\left( \frac{|h|}{r_{\bar x y}^{\,2}} + \frac{|h|^\lambda}{r_{\bar x y}} \right), \qquad k=1,2. \]

These estimates can be proved in the same way as (4.9). Thus, Theorem 2 is proved.

§ 5. THE TWO-DIMENSIONAL CASE. ANALYSIS OF THE RESULTS

We now consider the two-dimensional case.

\(1^\circ\). We begin with the double-layer potential. We have

\[ \frac{\partial H(xy)}{\partial y_i} = \frac{\partial A(y)/\partial y_i}{4\pi[A(y)]^{3/2}}\ln\rho - \frac{\partial \rho/\partial y_i}{2\pi\rho\sqrt{A(y)}}. \tag{5.1} \]

The last term in (5.1) has already been studied by us (see § 3, \(2^\circ\)). Therefore it remains to consider only the first term, or, for simplicity, the following integral:

\[ T_1(x)=\int_{\Gamma} z(y)\ln\rho\,ds_y. \]

In the case \(m=2\), the substitution (2.4) takes the form

\[ y_1=x_1+r, \]

and formula (3.11) is written in the form

\[ \rho^2=r^2 B(x_1,r), \]

where

\[ B(x_1,r)=A_{11}(x_1,r)+2A_{12}(x_1,r)\int_0^1 \frac{\partial f(x_1+tr)}{\partial x_1}\,dt+ \]

\[ {}+A_{22}(x_1,r)\left[\int_0^1 \frac{\partial f(x_1+tr)}{\partial x_1}\,dt\right], \]

\[ A_{ij}(x_1,r)=A_{ij}(x_1+r,f(x_1+r)),\qquad i,j=1,2. \]

Then

\[ \ln \rho=\frac{1}{2}\ln B+\ln |r|. \]

All the arguments in § 3 go through without change for the function \(\ln B\). Thus, after smoothing it remains only to consider the integral

\[ T_2(x_1)=\int_{-\infty}^{\infty} z(x_1+r)\ln |r|\,dr . \tag{5.3} \]

It is necessary to prove that \(T_2(x_1)\in C^{(n+1,\lambda')}(\sigma_0)\) when \(z\in C^{(n,\lambda)}\), and to estimate \(T_2\) in \(C^{(n+1,\lambda')}(\sigma_0)\) in terms of \(\|z\|_{C^{(n,\lambda)}}\), where \(\sigma_0\) is some neighborhood of zero.

Let us note that (5.3) can be differentiated \(n\) times under the integral sign. After this it suffices to show that, if the finite density \(\mu(x_1)\in C^{(0,\lambda)}(-\infty,\infty)\), then

\[ \varphi(x_1)=\int_{-\infty}^{\infty}\mu(y_1)\ln |y_1-x_1|\,dy_1 \]

belongs to \(C^{(1,\lambda')}(\sigma_0)\), and \(\|\varphi\|_{C^{(1,\lambda')}(\tau_0)}\) is estimated in terms of \(\|\mu\|_{C^{(0,\lambda)}}\).

This can be done by the usual device, i.e., by considering first the case \(\mu\equiv 1\), and then the general case.

For \(\mu\equiv 1\),

\[ \varphi(x_1)=(y_1-x_1)[\ln |y_1-x_1|-1]\big|_{y_1=-\alpha}^{y_1=\alpha}, \]

where \(\alpha\) is some sufficiently large number. Hence it follows at once that, for \(\mu\equiv 1\), \(\varphi(x_1)\in C^{(1,\lambda')}(\sigma_0)\), where \(0<\lambda'<1\) is arbitrary.

For the general case one must establish the estimate corresponding to (3.20). This estimate, in the case \(m=2\), has the form

\[ \frac{1}{y_1-x'_1}-\frac{1}{y_1-x_1} = O\left(\frac{|x'_1-x_1|}{(y_1-x_1)^2}\right), \tag{5.4} \]

where \(x_1,x'_1\) are any two points in \(\sigma_0\). Let us observe that estimate (3.20) was obtained under condition (3.11), which in our case takes the form

\[ \frac{1}{y_1-x_1}=O\left(\frac{1}{y_1-x_1}\right). \tag{5.5} \]

From (5.5), (5.4) follows elementarily.

As for the single-layer potential, the case \(m=2\) leads to no difficulties in comparison with the cases \(m>3\), since in the case \(m=2\)

\[ \frac{\partial H(xy)}{\partial x_1} = -\frac{\partial \rho/\partial x_i}{2\pi \rho\sqrt{A(y)}}, \]

which was already considered in § 3.

\(2^\circ\). We shall now briefly analyze the results obtained. To this end, consider the following example.

Suppose an elliptic operator is given by

\[ Mu=\frac{\partial^2 u}{\partial x_1^2}+ \frac{1}{b^2(x_1x_2)}\frac{\partial^2 u}{\partial x_2^2}, \tag{5.6} \]

where \(A_0 \geq b(x_1x_2) \geq A_0^{-1}\), \(A_0>0\) is some positive constant. The comparison function for (5.6) will be

\[ H(xy)=\frac{b(y_1y_2)}{2\pi}\ln\frac{1}{\rho}, \]

where

\[ \rho^2=(y_1-x_1)^2+b(y_1y_2)(y_2-x_2)^2 . \]

Assume that a piece of the boundary of the domain \(g\) is a segment of the \(x_2\)-axis situated near the origin. In view of the local character of our results, one may consider the integral extended only over this segment.

On the \(x_2\)-axis,

\[ \left[\frac{\partial H}{\partial \nu_y}\right]_{x_1=0} = \left[ \frac{1}{2\pi}\frac{\partial b(y_1y_2)}{\partial y_1}\ln\frac{1}{\rho} - \frac{1}{2\pi}\frac{\partial b(y_1y_2)}{\partial y_1} \right]_{x_1=0}. \]

Discarding the constant term, after smoothing we write the direct value of the double-layer potential in the form

\[ W_{\mathrm{pr}}(x_2)= \int_{-\infty}^{\infty} \mu(y_2)\beta(y_2)\ln|y_2-x_2|\,dy_2, \]

where \(\mu(y_2)\) is the density,

\[ \beta(y_2)= \left[ \frac{\partial b(y_1y_2)}{\partial y_1} \right]_{y_1=0}. \]

Let \(\mu=1\) and

\[ b(x_1x_2)=|x_1+x_2|^{1+\lambda}+K, \]

where \(K>0\) is some number. Then \(b(x_1x_2)\) and, consequently, \(1/b^2(x_1x_2)\), in any bounded domain containing the origin, belong only to the class \(C^{(1,\lambda)}\) and do not belong to the class \(C^{(1,\lambda+\delta)}\), where \(\delta>0\) is arbitrary.

It is easy to see that, for \(x_2>0\),

\[ \frac{d}{dx_2}\bigl[W_{\mathrm{pr}}(x_2)\bigr] = (1+\lambda)x_2^\lambda \left[ \int_0^\alpha \ln\left|\frac{y_2-x_2}{y_2+x_2}\right|\,dy_2 \right]' x_2 + \]

\[ + (1+\lambda) \int_0^\alpha (y_2^\lambda-x_2^\lambda) \frac{\partial}{\partial x_2} \left[ \ln\left|\frac{y_2-x_2}{y_2+x_2}\right| \right]dy_2 . \tag{5.7} \]

This equality shows the connection between the smoothness of the direct value of the double-layer potential and the smoothness of the coefficients of the operator \(M\).

Let us note that the first term on the right-hand side of (5.7) belongs only to the class \(C^{(0,\lambda)}\) near the origin. It is not difficult to verify that the second term in (5.7), near the origin, belongs only to the class \(C^{(0,\lambda')}\) and does not belong to the class \(C^{(0,\lambda)}\), where \(0<\lambda'<\lambda\) is arbitrary. Hence the following assertion follows.

Suppose that the coefficients of the operator belong to the class \(C^{(n+1,\lambda)}\). Then, although the boundary of the domain \(g\) and the density \(\mu(x)\) are infinitely smooth, the direct value \(W_{\mathrm{pr}}(x_2)\) of the double-layer potential belongs only to a class no better than \(C^{(n+1,\lambda')}\), where \(0<\lambda'<\lambda\) is arbitrary. This shows that our results (Theorem 1) cannot be improved.

On the other hand, it is also clear that in order that the direct value \(W_{\mathrm{pr}}(x_2)\) belong to the class \(C^{(n+1,\lambda')}\), where \(0<\lambda'<\lambda\) is arbitrary, it is necessary, even in the case when the boundary of the domain \(g\) and the density \(\mu(x)\) are infinitely smooth, to require that the coefficients of the operator \(M\) be at least of the class \(C^{(n+1,\lambda')}\). This shows that the conditions of Theorem 1 imposed on the smoothness of the coefficients of the operator \(M\) are minimal, up to an arbitrarily small \(\varepsilon>0\), with respect to the Hölder exponent.

In an entirely similar way one can construct examples showing the minimality of the conditions of Theorem 1 imposed on the smoothness of the boundary and of the density.

For the simple-layer potential, examples of the type indicated above are also easy to give.

§ 6. VOLUME POTENTIAL FOR THE COMPARISON FUNCTION

In this paragraph we shall prove Theorem 3. Let \(g'\) be any interior subdomain of the domain \(g\). For smoothing we take still another subdomain \(g''\subset A^{(n,\lambda)}\) of the domain \(g\) such that \(g\supset \overline{g''}\supset \overline{g'}\).

After smoothing, it remains to consider only the integral (see § 2)

\[ \int_{\omega} d\omega \int_0^\infty \mu_1(x+r\varphi)F(x;r,\varphi)\,dr, \tag{6.1} \]

where

\[ \mu_1(x+r\varphi) = \left. \frac{\mu(y)}{\omega_m(m-2)\sqrt{A(y)}} \right|_{y=x+r\varphi}, \]

\[ F(x;r,\varphi) = \frac{\xi r^{m-1}}{\rho^m} = r[A(x;r,\varphi)]^{1-\frac{m}{2}}, \]

\[ A(x;r,\varphi) = \sum_{i,j=1}^{m} A_{ij}(x+r\varphi)\,g_i(\varphi)g_j(\varphi). \]

Obviously, \(F(x;r,\varphi)\) has continuous derivatives with respect to \(x\) in \(g''\) up to order \(n\) inclusive, and moreover

\[ F^{(l)}(x;r,\varphi) \sim r[A(x;r,\varphi)]^{-\frac{m}{2}-p} \sum_{i=1}^{l} \bigl[D^{(q_i)}A(x;r,\varphi)\bigr]^{s_i}, \]

\[ 0\le l\le n,\qquad 0\le p\le l,\qquad 0<q_i\le l,\qquad 0\le s_i\le l,\qquad q_i s_i\le l. \]

Therefore differentiation under the integral sign in (6.1) is legitimate. The derivative of order \(n\) of (6.1) is a finite sum of terms of the form

\[ \int_{\omega} d\omega \int_0^\infty \mu_1^{(s)}(x;r,\varphi)F^{(n-s)}(x;r,\varphi)\,dr,\qquad 0\le s\le n. \]

Having returned to the original Cartesian coordinate system, we write this integral in the form

\[ \int_{R_m}\frac{\mu_2(y)\,F^{(n-s)}(xy)}{r_{xy}^{m-1}}\,dy=U_1(x), \]

where

\[ \mu_2(y)=\mu_1^{(s)}(y), \]

\[ F^{(n-s)}(xy)=F^{(n-s)}(x;r,\varphi)\big|_{(r\varphi)=y-x}. \]

Thus, it remains to prove that \(U_1(x)\in C^{(2,\lambda')}(g')\), if \(\mu_2\in C^{(0,\lambda)}(g+\Gamma)\), and to estimate \(\|U_1\|C^{(2,\lambda')}(g')\) in terms of \(\|\mu_2\|C^{(0,\lambda)}(g+\Gamma)\). For simplicity, we consider only the case when \(s=0\).

We have

\[ F^{(n)}(xy)\sim \left[\sum_{i=1}^{m}(x_i-y_i)^2\right]^{1/2} [A(xy)]^{1-\frac{m}{2}-p} \prod_{i=1}^{n}\left[D^{(q_i)}A(xy)\right]^{s_i} \]

\[ 0\leq p,\ q_i,\ s_i\leq n,\qquad s_iq_i\leq n, \]

where

\[ A(xy)= \frac{\sum_{i=1}^{m} A_{ij}(y)(x_i-y_i)(x_j-y_j)} {\sum_{i=1}^{m}(x_i-y_i)^2}, \]

\[ D^{(q_k)}A(xy)= \sum_{ij=1}^{m} \frac{\partial^{q_k}A(y)} {\partial y_{1}^{q_{k1}},\ldots,\partial y_{m}^{q_{km}}} \times \]

\[ \times \frac{(x_i-y_i)(x_j-y_j)} {\sum_{i=1}^{m}(x_i-y_i)^2}, \qquad q_k=\sum_{l=1}^{m}q_{kl}. \]

Obviously,

\[ D^{(q_i)}A(xy)=O(1);\qquad F^{(n)}(xy)=O(r_{xy});\qquad \frac{\partial}{\partial x_i}F^{(n)}(xy)=O(1). \]

Then \(U_1(x)\) can once again be differentiated under the integral sign.

Let

\[ \Phi(xy)=\frac{F^{(n)}(xy)}{r_{xy}^{m-1}}, \]

then

\[ \frac{\partial}{\partial x_i}U_1(x) = \int_{\overset{\circ}{R}_m} \mu_2(y)\, \frac{\partial}{\partial x_i}\Phi(xy)\,dy. \tag{6.3} \]

To obtain the derivatives of \(\dfrac{\partial}{\partial x_i} U_1(x)\), note that in (6.3) as the domain of integration one may take a certain ball \(T\) of sufficiently large radius containing \(g\) in its interior, i.e.

\[ \frac{\partial}{\partial x_i} U_1(x) = \int_T \mu_2(y)\,\frac{\partial}{\partial x_i}\Phi(xy)\,dy. \]

The computation of the derivatives of \(\dfrac{\partial}{\partial x_i} U_1(x)\) may be carried out as indicated on p. 37 of the book [1], i.e.

\[ \frac{\partial^2}{\partial x_i\partial x_j}U_1(x) = \int_T \left[ \mu_2(y)\frac{\partial^2}{\partial x_i\partial x_j}\Phi(xy) - \mu_2(x)\frac{\partial^2}{\partial y_i\partial y_j}\Phi(yx) \right]dy + \mu_2(x)\int_{\partial T} \frac{\partial}{\partial y_i}\Phi(yx)\cos(ny_i)\,ds_y, \]

where \(\partial T\) is the boundary of \(T\), and \(n\) is the normal to \(\partial T\). After this it is not difficult to prove Theorem 3, since

\[ \left[ \mu_2(y)\frac{\partial^2}{\partial x_i\partial x_j}\Phi(xy) - \mu_2(x)\frac{\partial^2}{\partial y_i\partial y_j}\Phi(yx) \right]\in N^{(\lambda,\lambda')}, \tag{6.4} \]

\[ \frac{\partial}{\partial y_i}\Phi(yx)\in C^{(0,\lambda)}(g') \quad\text{with respect to }x\text{ for }y\in\partial T, \]

and (6.4) can be verified as indicated on p. 37 of the book [1].

Thus, Theorem 3 is proved.

Taking this opportunity, the author expresses his deep gratitude to his scientific adviser V. A. Il’in for posing the problem, for guidance, and for valuable advice; to A. N. Tikhonov for discussing the results. The author thanks I. A. Shishmarev for reading the manuscript of the present article.

References

1. C. Miranda. Partial Differential Equations of Elliptic Type. Moscow, Foreign Literature Publishing House, 1957.

2. N. M. Günter. Potential Theory and Its Application to the Basic Problems of Mathematical Physics. Moscow, Gostekhizdat, 1953.

3. Kh. L. Smolitskii. The Boundary-Value Problem for the Wave Equation. Doctoral dissertation. Leningrad, 1950.

4. G. M. Fichtenholz. A Course of Differential and Integral Calculus, vol. 3. Moscow, Fizmatgiz, 1963.

5. M. Gevrey. Les quasi fonctions de Green et les systèmes d’équations aux dérivées partielles du type elliptique. Ann. Ec., N. Sup. 52, 39—108, 1935.

6. Van Tung. Dokl. Akad. Nauk SSSR, 152 No. 6, 1288—1291, 1963.

Received by the editors
March 12, 1965

Nankai University,
Tianjin, PRC