

Seminar "Complex analysis in several variables" (Vitushkin Seminar)
April 8, 2015 16:45, Moscow, MSU, auditorium 1304






On the higherdimensional harmonic analog of the Levinson loglog theorem
A. A. Logunov^{} 
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Abstract:
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 higherdimensional analog of the Levinson loglog theorem for harmonic functions.

