Asymptotics of $\delta$-subharmonic functions of finite order

Abstract

For $\delta$-subharmonic in $\mathbb{R}^m$, $m\geq2$, function $u=u_1-u_2$ of finite positive
order we found the asymptotical representation of the form
\[
u(x)=-I(x,u_1)+I(x,u_2) +O\left(V(|x|)\right),\ x\to\infty,
\]
where $I(x,u_i)=\int\limits_{|a-x|\leq|x|}K(x,a)d\mu_i(a)$, $K(x,a)=\ln\frac{|x|}{|x-a|}$ for $m=2$,
$K(x,a)=|x-a|^{2-m}-|x|^{2-m}$ for $m\geq3,$
$\mu_i$ is a Riesz measure of the subharmonic function $u_i,$ $V(r)=r^{\rho(r)},$ $\rho(r)$ is a proximate order of $u$.
The obtained result generalizes one theorem of I.F. Krasichkov for entire functions.