MATHEMATICS

T. G. GEGELIA

ON THE PROPERTIES OF MULTIDIMENSIONAL SINGULAR INTEGRALS IN THE SPACE \(L_p(S;\rho)\)

(Presented by Academician N. I. Muskhelishvili on 6 III 1961)

1. Let \(E_{m+1}\) be Euclidean space; \(x(x_1,\ldots,x_{m+1})\), \(y(y_1,\ldots,y_{m+1})\), \(z(z_1,\ldots,z_{m+1}),\ldots\) are points of this space. Denote by \(r(x,y)\) the distance between the points \(x\) and \(y\), and by \(C(z;\delta)\) the ball with center at the point \(z\) and radius \(\delta\). Let \(l\) be an arbitrary line passing through the point \(z\), and let \(\delta\) be an arbitrary positive number. Denote by \(H(l;\delta)\) the circular cylinder of height \(2\delta\), whose axis is \(l\), whose center of symmetry is the point \(z\), and whose base radius is equal to \(\delta\).

Let \(S\) be a bounded, closed or open \(m\)-dimensional manifold from the space \(E_{m+1}\) (the set \(S\) will be regarded as closed), possessing the following property: to each point \(z\) of the manifold \(S\) one can assign a positive number \(\delta\) and a certain rectangular coordinate system \((X_1,\ldots,X_{m+1})\) with origin at the point \(z\) in such a way that the part of \(S\) contained inside \(H(X_{m+1};\delta)\) admits a representation of the form \(\eta_{m+1}=\gamma(\eta_1,\ldots,\eta_m)\), where \(\eta_1,\ldots,\eta_{m+1}\) are the coordinates of the point \(y\in S\cap H(X_{m+1};\delta)\) in the system \((X_1,\ldots,X_{m+1})\); \(\gamma\) is a single-valued function defined on \(\tau(z;\delta)\), and \(\tau(z;\delta)\) is the intersection of the ball \(C(z;\delta)\) with the hyperplane passing through \(z\) and perpendicular to the axis \(X_{m+1}\). Moreover, we shall assume that there exist continuous partial derivatives of first order of the function \(\gamma\) in the domain \(\tau(z;\delta)\), and that
\[ \gamma(0,\ldots,0)=\partial\gamma(0,\ldots,0)/\partial\eta_k=0\quad (k=1,\ldots,m) \]
and \(\omega(\partial\gamma/\partial\eta_k;t)t^{-1}\) \((k=1,\ldots,m)\) are integrable on the interval \((0,1)\), where \(\omega(\partial\gamma/\partial\eta_k;t)\) is the modulus of continuity of the function \(\partial\gamma/\partial\eta_k\) in the domain \(\tau(z;\delta)\).

Let on \(S\) be given an \((m+1)\)-dimensional symmetric matrix \(\|A_{ij}(y)\|\), possessing the following properties: the quadratic form \(\sum A_{ij}(y)t_i t_j\) is positive definite at each point \(y\) of the manifold \(S\), and the functions \(\omega(A_{ij};t)t^{-1}\) \((i,j=1,\ldots,m+1)\) are integrable on the interval \((0,1)\).

Consider the operator
\[ K_\varphi(x)=\int_S^* k(y,x)\varphi(y)\,dS_y, \]
where
\[ k(y,x)=\prod_{i=1}^{m+1}(x_i-y_i)^{\lambda_i}\,[\sigma(y,x)]^{-\lambda},\qquad \sigma^2(y,x)=\sum_{i,j=1}^{m+1} A_{ij}(y)(x_i-y_i)(x_j-y_j), \]
\(\lambda_i\) \((i=1,\ldots,m+1)\) and \(\lambda\) are arbitrary nonnegative integers;
\[ \lambda-m=\sum \lambda_i \]
is a positive odd number; \(\varphi(y)\) is the density of the integral; \(dS_y\) is the element of area of the manifold \(S\) at the point \(y\), and the integral

is understood in the sense of the principal value:

\[ \int_S^{*} k(y,x)\varphi(y)\,dS_y = \lim_{\delta\to 0}\int_{S(\delta;x)} k(y,x)\varphi(y)\,dS_y, \]

where \(S(\delta;x)=S-S(x;\delta)\); \(S(x;\delta)=S\cap H(n(x);\delta)\); \(n(x)\) is the normal to the manifold \(S\) at the point \(x\).

In the present article we consider the question of the boundedness of the operator \(K_\varphi\) in the space \(L_p(S;\rho)\), where \(p>1\), and \(\rho(x)\) is a certain nonnegative measurable function on \(S\). This question, under several other assumptions, was studied in the works \((^{1-7})\). In the article a formula is derived for changing the order of integration in repeated singular integrals of the indicated type. Analogous formulas were obtained in \((^{1,8-11})\). All these questions in the case of a singular integral of Cauchy type were studied in \((^{12,13})\).

2. Theorem 1. If \(\varphi(y)\in L_p(S)\), then \(K_\varphi\) is a bounded operator mapping \(L_p(S)\) into itself.

Proof. It is sufficient to show the boundedness of the operator

\[ T_\varphi(x)=\int_{S(z;\delta)}^{*} k(y,x)\varphi(y)\,dS_y \]

on \(L_p(S(z,\nu))\), where \(z\) is an arbitrary point of the manifold \(S\), \(0<\nu<\delta\), and \(\delta\) is the positive constant occurring in the definition of \(S\). Denote by \(\xi_1,\ldots,\xi_{m+1}\) and \(\eta_1,\ldots,\eta_{m+1}\) the coordinates of the points \(x\) and \(y\) in the system \((X_1,\ldots,X_{m+1})\), and by \(\xi\) and \(\eta\) the points \((\xi_1,\ldots,\xi_m,0)\) and \((\eta_1,\ldots,\eta_m,0)\). We shall have

\[ x_i=z_i+\sum_{j=1}^{m+1} a_{ij}\xi_j,\qquad \sigma^2(y,x)=\sum_{i,j=1}^{m+1} B_{ij}(y)(\xi_i-\eta_i)(\xi_j-\eta_j), \]

\[ B_{ij}(y)=\sum_{k,l=1}^{m+1} A_{kl}(y)a_{ki}a_{lj} \qquad (i,j=1,\ldots,m-1). \tag{1} \]

It can be shown that the operator \(T_\varphi\) is represented in the form of a finite sum of operators with certain bounded coefficients of the following two types:

\[ N_\varphi(x)=\int_{S(z;\delta)} n(x,y)\varphi(y)\,dS_y,\qquad M_\varphi(x)=\int_{\tau(z;\delta)}^{*}\frac{m(\xi,\eta)\varphi_*(\eta)}{r^m(\xi,\eta)}\,d\eta, \]

where

\[ |n(x,y)|\le r^{-m}(x,y)F(r(x,y)); \]

the function \(F(t)t^{-1}\) is integrable on \((0,\delta)\); \(\varphi_*(\eta)=\varphi(y)\); \(d\eta\) is the element of area of the tangent hyperplane at the point \(z\) of the manifold \(S\),

\[ m(\xi,\eta)= \prod_{i=1}^{m}\left[\frac{\xi_i-\eta_i}{r(\xi,\eta)}\right]^{\nu_i} \left[ \sum_{i,j=1}^{m} C_{ij}(\xi) \frac{\xi_i-\eta_i}{r(\xi,\eta)} \frac{\xi_j-\eta_j}{r(\xi,\eta)} \right]^{-\lambda/2}; \]

the constants \(\nu_i\) \((i=1,\ldots,m)\) satisfy the same conditions as the \(\lambda_i\), and

\[ C_{ij}(\xi) = B_{ij}(x) + 2B_{m+1,i}(x)\frac{\partial \gamma(\xi)}{\partial \xi_j} + B_{m+1,m+1}(x) \frac{\partial \gamma(\xi)}{\partial \xi_i} \frac{\partial \gamma(\xi)}{\partial \xi_j}. \]

The boundedness of the operator \(N_\varphi\) is obvious, and the boundedness of \(M_\varphi\) follows from the works of S. G. Mikhlin \((^{1,2})\) and A. Calderón and A. Zygmund \((^{3,4})\).

We note that Theorem 1 remains valid also in the case when

\[ k(y,x)= \sum_{k=1}^{n} C_k(y) \prod_{i=1}^{m+1} (x_i-y_i)^{\lambda_{ik}} (\sigma_{1k}+\sigma_{2k})^{-n_k} \sigma_{1k}^{-m_k} \sigma_{2k}^{-l_k}, \tag{2} \]

where \(n\) is a natural number; \(C_k(y)\) \((k=1,\ldots,n)\) are continuous functions on \(S\); \(\lambda_{lk}, n_k, m_k\), and \(l_k\) are nonnegative integers; \(n_k+m_k+l_k-m=\lambda_{1k}+\cdots+\lambda_{m+1,k}\) \((k=1,\ldots,n)\) are positive even numbers, and \(\sigma_{1k}=\sigma_{1k}(y,x)\) and \(\sigma_{2k}=\sigma_{2k}(y,x)\) \((k=1,\ldots,n)\) are expressions defined in the same way as \(\sigma=\sigma(y,x)\).

Theorem 2. If \(\varphi(y)\in L_p(S;\rho(y))\), where

\[ \rho(y)=\prod_{k=1}^{n} r^{\alpha_k}(y,z^{(k)}), \]

\[ 0<\alpha_k<m(p-1)\quad (k=0,\ldots,n_1\leq n);\qquad 0<-\alpha_k<m\quad (k=n_1+1,\ldots,n); \]

\[ z^{(k)}\in S,\quad z^{(i)}\ne z^{(j)}\ (i\ne j;\ k,i,j=1,\ldots,n);\qquad p>1, \]

then the operator \(K_\varphi\) maps the space \(L_p(S;\rho(y))\) into itself and is a bounded operator.

Proof. It is enough to prove the validity of the theorem in two cases: 1) when \(\rho(y)=r^{-\alpha}(y,z)\), where \(0<\alpha<m,\ z\in S\), and 2) when \(\rho(y)=r^\alpha(y,z)\), where \(0<\alpha<m(p-1),\ z\in S,\ p>1\).

Consider case 1). We have

\[ |K_\varphi(x)|\leq c\left(K_f^{(1)}(x)+K_f^{(2)}(x)\right)+r^{\alpha/p}(x,z)|K_f(z)|, \]

where

\[ f(x)=\varphi(x)r^{-\alpha/p}(x,z)\in L_p(S); \]

\(c\) is some positive constant;

\[ K_f^{(1)}(x)=\int_S |k(y,x)|\,r^\beta(y,x)\,r^\delta(y,z)\,|f(y)|\,dS_y, \]

\[ K_f^{(2)}(x)=r^\delta(x,z)\int_S |k(y,x)|\,r^\beta(y,x)\,|f(y)|\,dS_y, \]

\[ \beta=-\alpha/p,\quad \delta=0,\ \text{if } \alpha/p<1,\qquad \beta=-1,\quad \delta=\alpha/p-1,\ \text{if } \alpha/p>1. \]

Using the inequalities given in the notes \((^6,^ {11})\), one can show that

\[ \int_S r^{-\alpha}(y,z)\,|K_f^{(i)}(y)|^p\,dS_y < c\int_S |f(y)|^p\,dS_y \qquad (i=1,2). \]

From this inequality and from Theorem 1 the proof of Theorem 2 in case 1) follows directly. Case 2) is considered analogously.

Theorem 3. If \(\varphi(y)\in L_p(S;\mu(y))\), where \(p>1\),

\[ \mu(y)=\prod_{k=1}^{n} r^{\gamma_k}(y,z^{(k)}), \]

\[ \gamma_k=\alpha_k(p-1)\quad (k=1,\ldots,n_1\leq n),\qquad \gamma_k=-\alpha_k\quad (k=n_1+1,\ldots,n), \]

\[ 0<\alpha_k<m,\quad z^{(k)}\in S,\quad z^{(i)}\ne z^{(j)}\ (i\ne j;\ k,i,j=1,\ldots,n), \]

then the operator

\[ L_\varphi(x)=\frac{1}{\mu_*(x)}\int_S k(y,x)\mu_*(y)\varphi(y)\,dS_y, \]

where

\[ \mu_*(y)=\prod_{k=1}^{n} r^{\omega_k}(y,z^{(k)}), \]

\[ \omega_k=\alpha_k\quad (k=1,\ldots,n_1);\qquad \omega_k=-\alpha_k\quad (k=n_1+1,\ldots,n), \]

maps the space \(L_p(S;\mu(y))\) into itself and is a bounded operator.

Let us note that Theorem 2 remains valid if

\[ \rho(y)=\prod_{k=1}^{n} r^{\alpha_k}(y,z^k)\lg^p\!\left(2dr^{-1}(y,z^{(k)})\right), \tag{3} \]

where \(\alpha_k\) \((k=1,\ldots,n)\) satisfy the conditions of Theorem 2, \(d=\operatorname{div} S\), and \(p_k\) are arbitrary integers. This assertion is proved, just like Theorem 2, with the aid of the inequalities given in note \((^{14})\). Theorem 3 admits an analogous generalization.

Theorems 2 and 3, as well as their generalizations just indicated, are also valid in the case of \(k(y,x)\) defined by formula (2). Moreover, in the proofs of Theorems 2 and 3 only the fact is used that \(K_\varphi\) is a bounded operator in \(L_p(S)\) and \(k(y,x)=O(r^{-m}(x,y))\).

1. Let

\[ k^{(i)}(y,x)=[\sigma(y,x)]^{-m-1}\sum_{k=1}^{m+1} C_k^{(i)}(y)(x_k-y_k) \qquad (i=1,2), \]

where \(C_k^{(i)}(y)\) \((k=1,\ldots,m+1)\) are continuous functions on \(S\), and \(B(z)=\det\|B_{ij}(z)\|_{i,j=1}^{m}\), where \(B_{ij}(z)\) are defined by formula (1). Denote by \(b_{ij}(z)\) the ratio of the algebraic complement of the element \(B_{ij}(z)\) in the determinant \(B\) to \(B\) itself, and by \(d_j^{\,i}(z)\) the expression \(\sum C_k^{(i)}(z)a_{kj}\), where \(a_{kj}\) are defined from (1).

Theorem 4. If \(\varphi(y)\in L_p(S;\rho(y))\), where \(\rho(y)\) is a function participating in Theorem 2 or defined by formula (3), then

\[ \int_S k^{(1)}(y,z)\,dS_y\int_S k^{(2)}(x,y)\varphi(x)\,dS_x = \int_S \varphi(x)\,dS_x\int_S k^{(1)}(y,z)k^{(2)}(x,y)\,dS_y - \]

\[ -\frac{\Gamma^{m+1}\displaystyle\sum_{i,j=1}^{m} b_{ij}(z)d_i^{(1)}(z)d_j^{(2)}(z)} {m\Gamma^2\!\left(\dfrac{m+1}{2}\right)B(z)}\,\varphi(z). \]

1. The theorems formulated make it possible to study integral equations containing the indicated singular integrals extended over open manifolds.

Tbilisi Mathematical Institute
named after A. M. Razmadze
Academy of Sciences of the Georgian SSR

Received
3 III 1961

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