UDC 517.516

MATHEMATICS

B. K. PCHELİN

ON THE GENERAL THEORY OF POLYHARMONIC FUNCTIONS

(Presented by Academician P. S. Novikov, 27 XII 1968)

1°. A generalization of the ordinary differential Laplace operator \(\Delta^{(p)}U(P)\), which a priori requires the existence of partial derivatives up to order \(2p\) inclusive, is the generalized polyharmonic operator \(\Delta^{*(p)}U(P)\), introduced by I. I. Privalov and B. K. Pchelin in the work \((^1)\). We shall call \(\Delta^{*(p)}U(P)\) the generalized spherical polyharmonic operator of the first kind of order \(p\). It is obvious that the operator \(\Delta^{*(p)}U(P)\) does not exist for all functions integrable in the sense of Lebesgue. Thus, for example, consider the function \(U(P)\) defined by the conditions: \(U(P)=U(x_1,x_2,\ldots,x_n)\) at those points \(P(x_1,x_2,\ldots,x_n)\) of the domain \(D\) at which at least one of the coordinates \(x_1,x_2,\ldots,x_n\) is irrational; \(U(P)=0\) at those points \(P\) of the domain \(D\) all of whose coordinates \(x_1,x_2,\ldots,x_n\) are rational. It is easy to prove that for such a function \(U(P)\) the operator \(\Delta^{*(p)}U(P)\) exists at those points \(P\) of the domain \(D\) at which at least one of its coordinates is irrational, and does not exist at the remaining points of the domain \(D\). In connection with this there arises the question of a broader definition of the generalized polyharmonic operator.

2°. In the present paper the following notation is adopted: \(D_{ih}\) \((i=1,2,\ldots,p)\) is the domain lying inside the domain \(D\) and whose boundary is removed from the boundary of the domain \(D\) by a distance \(>ih\); \(\omega(P_i,h)\) and \(\sigma(P_i,h)\) are respectively the hypersphere and its surface with center at the point \(P_i\) of radius \(h\); \(\omega(h)\) and \(\sigma(h)\) are the volume of the hypersphere and the area of its surface; \(P=P_p\subset \overline{D}_{ph}\), \(P_{p-1}\subset \overline{D}_{(p-1)h},\ldots, P_1\subset \overline{D}_{h}\), \(M\subset D\); \(C_p^{(k)}\) is the number of combinations of \(p\) elements taken \(k\) at a time.

3°. Let \(U(P)\) be a single-valued function in a certain domain \(D\) of \(n\)-dimensional space, Lebesgue integrable over every hypersphere and over its surface lying inside the domain \(D\). Consider the expression

\[ \delta_h U(P_1)= \frac{1}{\sigma(h)}\int_{\sigma(P_1,h)} U(M)\,d\sigma - \frac{1}{\omega(h)}\int_{\omega(P_1,h)} U(M)\,d\omega. \tag{1} \]

This expression represents a function of the point \(P_1\), Lebesgue integrable in the domain \(\overline{D}_h\subset D\). Let the point \(P_2\subset \overline{D}_{2h}\). Repeating operation (1) on the function \(\delta_h U(P_1)\), we obtain

\[ \delta_h^{(2)}U(P_2)=\delta_h\delta_h U(P_2)= \]

\[ = \frac{1}{\sigma(h)} \int_{\sigma(P_2,h)} \left[ \frac{1}{\sigma(h)} \int_{\sigma(P_1,h)} U(M)\,d\sigma - \frac{1}{\omega(h)} \int_{\omega(P_1,h)} U(M)\,d\omega \right] d\sigma_1 - \]

\[ - \frac{1}{\omega(h)} \int_{\omega(P_2,h)} \left[ \frac{1}{\sigma(h)} \int_{\sigma(P_1,h)} U(M)\,d\sigma - \frac{1}{\omega(h)} \int_{\omega(P_1,h)} U(M)\,d\omega \right] d\omega_1 . \]

Repeating this process \(p\) times, we arrive at the expression \(\delta_h\delta_h\cdots\delta_h U(P)=\delta_h^{(p)}U(P)\), defined at the points \(P=P_p\subset \overline{D}_{ph}\). The expressions

\[ \overline{\delta^{*(p)}U}(P)= \varlimsup_{h\to 0} \left[ \delta_h^{(p)}U(P): \frac{h^{2p}}{n^p(n+2)^p} \right], \]

\[ \underline{\delta^{*(p)}}\,U(P)= \varliminf_{h\to 0} \left[ \delta_h^{(p)}U(P): \frac{h^{2p}}{n^p(n+2)^p} \right] \]

we shall call, respectively, the upper and lower generalized polyharmonic operators of the second kind of order \(p\) for the function \(U(P)\). We then arrive at the following definition of a generalized polyharmonic operator of the second kind of order \(p\):

Definition. The function \(U(P)\) has at the point \(P\) a generalized polyharmonic operator \(\delta^{*(p)}U(P)\) of the second kind of order \(p\), if \(\overline{\delta}^{*(p)}U(P)=\underline{\delta}^{*(p)}U(P)\), and we set

\[ \delta^{*(p)}U(P)=\lim_{h\to 0}\left[\delta_h^{(p)}U(P):\frac{h^{2p}}{n^p(n+2)^p}\right]. \]

The operator \(\delta^{*(p)}U(P)\) exists, for example, for the function \(U(P)\) indicated above. Consequently, it is an operator existing for a broader class of Lebesgue-integrable functions.

To justify this definition, the following is proved.

Theorem 1. Let \(U(P)\) be a single-valued function of the point \(P\) in some domain \(D\) of \(n\)-dimensional space, Lebesgue-integrable over every hypersphere and over its surface lying inside the domain \(D\). If for the function \(U(P)\) at the points \(P\) of the domain \(D\) there exists the operator \(\Delta^{*(p)}U(P)\), then for the function \(U(P)\) at the same points \(P\) of the domain \(D\) there exists the operator \(\delta^{*(p)}U(P)\), equal to \(\Delta^{*(p)}U(P)\).

For the proof of this theorem, one first introduces the generalized polyharmonic spherical operator of the first kind of order \(p\), denoted by \(\nabla^{*(p)}U(P)\), by means of the equality

\[ \nabla^{*(p)}U(P)=\lim_{h\to 0}\left[\Delta_h^{(p)}U(P):\frac{h^{2p}}{2^p n^p}\right], \]

where

\[ \nabla_h U(P)=\frac{1}{\sigma(h)}\int_{\sigma(P,h)} U(M)\,d\sigma-U(P), \]

and the existence of this operator and its equality to the operator \(\Delta^{*(p)}U(P)\) is proved, provided the latter exist.

\(3^\circ\). In order to arrive at a new definition of a polyharmonic function, Theorems A and B are first proved.

Theorem A. Let the function \(U(P)\) be a polyharmonic function of order \(p\) in the domain \(D\). Then, whatever the domain \(D'\), \(D'\subset D\), for all points \(P\) of this domain the function \(\delta_h U(P)\), defined by expression (1), where \(h\) is any sufficiently small positive number, is a polyharmonic function of order \(p-1\).

Theorem B. If a function \(U(P)\), continuous in the domain \(D\), is such that the function \(\delta_h U(P)\), defined by expression (1), where \(h\) is any sufficiently small positive number, is a polyharmonic function of order \(p-1\), then the function \(U(P)\) is a polyharmonic function of order \(p\) in the domain \(D\).

Proceeding from the proved Theorems A and B, we arrive at the following definition of a polyharmonic function:

Definition. A function \(U(P)\), continuous in the domain \(D\), is called a polyharmonic function of order \(p\) in this domain if it satisfies the condition: whatever the domain \(D'\), \(D'\subset D\), for all points \(P\) of this domain the expression

\[ \delta_h U(P)=\frac{1}{\sigma(h)}\int_{\sigma(P,h)} U(M)\,d\sigma-\frac{1}{\omega(h)}\int_{\omega(P,h)} U(M)\,d\omega, \]

where \(h\) is any sufficiently small positive number, is a polyharmonic function of order \(p-1\).

Gorky Polytechnic Institute
named after A. A. Zhdanov

Received
5 III 1968

REFERENCES

1. I. I. Privalov, B. K. Pchelin, Matem. sborn., 2 (44), 745 (1937).