Growth problems for subharmonic functions of finite order in space

Abstract:

For a function $u(x)$ subharmonic (or ${C^2}$) in ${{\mathbf {R}}^m}$, we compare the “harmonics” (defined in §1) of u with those of a related subharmonic function whose total Riesz mass in $|x| \leqslant r$ is the same as that of u, but whose ${L^2}$ norm on $|x| = r$ is maximal, for all $0 < r < \infty$. We deduce estimates on the growth of the Riesz mass of u in $|x| \leqslant r$, as $r \to \infty$.