This article is cited in
6 papers
On means and the Laplacian of functions on Hilbert space
I. Ya. Dorfman
Abstract:
In his book Problemes concrets d'analyse fonctionnelle, Paul Levy introduced the concept of the mean
$M(f,a,\rho)$ of the function
$f$ on Hilbert space over the ball of radius
$\rho$ with center at the point
$a$, and investigated the properties of the Laplacian
$$
Lf(a)=\lim_{\rho\to0}\frac{M(f,a,\rho)-f(a)}{\rho^2},
$$
but he did not determine which functions have means. Moreover, the mean
$M(f,a,\rho)$ and the Laplacian
$Lf(a)$ are not invariant, in general, under rotation about the point
$a$.
In the present paper we give a class of functions with invariant means on Hilbert space. An example of such a class is the set of functions
$f(x)$ for which
$f(x)=\gamma(x)I+T(x)$, where the function
$\gamma(x)$ is uniformly continuous and has invariant means,
$I$ is the identity operator, and
$T(x)$ is a symmetric, completely continuous operator whose eigenvalues, arranged in decreasing order of absolute value
$\lambda_j(x)$, have the property that
$\frac1n\sum_{i=1}^n\lambda_i(x)\to0$ uniformly in
$x$ (§ 3). The invariant mean of such a function exists and is given by the formula
$$
M(f,x,r)=f(x)+\int_0^r\rho M(\gamma,x,\rho)\,d\rho,
$$
and its Laplacian is
$Lf(a)=\frac{\gamma(a)}2$. In § 4 we consider the Dirichlet problem and the Poisson problem for the ball and give sufficient conditions for the solution to be expressed by the Levy formulas.
Bibliography: 7 titles.
UDC:
513.881
MSC: 46E20,
47A70,
47L30,
15A18 Received: 23.03.1969