Half-spherical means and boundary behaviour of subharmonic functions in half-spaces
\textit{L. Naïm} [Ann. Inst. Fourier 7, 183-281 (1958; Zbl 0086.306)] has shown how the minimal fine topology and Martin boundary may be used to obtain elegant generalizations of the maximum principle and the solution to the Dirichlet problem. The paper under review shows that analogous results can be obtained for half-spherical means in the Euclidean half-space \(\Omega =\{X=(x_ 1,...,x_ n):\) \(x_ n>0\}\). The half-spherical mean of a function f is defined by \[ N(f:P,r)=r^{-n- 1}\int_{\{X\in \Omega: | X-P| =r\}}x_ n\quad f(X)\quad d\sigma (X) \] provided the integral exists (\(\sigma\) denotes surface area measure). The maximum principle type of result in the paper has the following corollary. Let s be a subharmonic function in \(\Omega\) such that \(s^+\) has a harmonic majorant in \(\Omega\). If \(\lim_{r\to 0+}\inf r N(s:P,r)\) is less than \(+\infty\) for all \(P\in \partial \Omega\) and non positive for a.e. (\(\sigma)\)P\(\in \partial \Omega\), and \(\lim_{r\to +\infty}\inf N(s:O,r)\leq 0,\) then \(s\leq 0\) in \(\Omega\). (This also generalizes a result of \textit{D. H. Armitage} [Ann. Acad. Sci. Fenn., Ser. A I 6, 161-171 (1981; Zbl 0441.31003)]). The Dirichlet problem is reformulated in terms of quotients of half- spherical means, and various types of regularity are defined. These are shown to be equivalent and are simply characterized, leading to the deduction that almost every boundary point is regular. Finally, the reader is cautioned that the notation \(G\nu\) is used in a non-standard way in the paper.
scientific article; zbMATH DE number 3910744