On the higher-dimensional harmonic analog of the Levinson loglog theorem

Let $P$ be a rectangle $(-a,a)\times(-b,b)$ in $\mathbb{R}^2$ and let $M:(0,b)\to [e,+\infty)$ be a decreasing function. Consider the set $F_M$ of all functions $f$ holomorphic in $P$ such that $|f(x,y)| \leq M(|y|)$, $(x,y)\in P$. The classical Levinson theorem asserts that $F_M$ is a normal family in $P$ if $\int_{0}^{b}\log\log M(y)dy<+\infty$.
One can replace holomorphic functions by harmonic functions in the statement above and it will remain true.
We are going to prove the higher-dimensional analog of the Levinson loglog theorem for harmonic functions.