Subharmonic functions with a Bloch type growth
The article compares various types of growth for subharmonic functions in the unit ball $B_N\subsetR^N$. Given a decreasing non-negative function $\omega$ on $[0,1)$, let $SH(\omega)$ be the collection of all non-negative subharmonic functions in $B_N$ satisfying $$\sup_{a\in B_N}\int_{B_N}[u(x)]^{N/2}\omega(|\varphi_a(x)|)\,dx<\infty,$$ where $\varphi_a(x)$ are Möbius transformations of $B_N$ with $\varphi_a(0)=a$. It is shown that if $$\int_0^1 r^{N-1}(1-r^2)^{-(N+1)/2}\omega(r)\,dr<\infty,$$ then ${\scr B}_1\subset SH(\omega)\subset {\scr B}_2$, where the ${\scr B}_k$ ($k=1,2$) stand for the sets of non-negative subharmonic functions in $B_N$ satisfying $$\sup_{x\in B_N}(1-|x|^2)^ku(x)<\infty.$$