On a theorem of M. Cartwright in higher dimensions

TL;DR

This paper investigates conditions on radial weights that ensure harmonic functions in higher dimensions can be estimated from below by these weights, extending Cartwright's results from two dimensions to higher dimensions.

Contribution

It generalizes Cartwright's reverse estimate for harmonic functions from two dimensions to higher dimensions under specific weight conditions.

Findings

Established conditions on weights guaranteeing lower bounds for harmonic functions

Extended Cartwright's theorem from 2D to higher dimensions

Identified classes of weights for which reverse estimates hold

Abstract

We consider harmonic functions in the unit ball of $\mathbb{R}^{n+1}$ that are unbounded near the boundary but can be estimated from above by some (rapidly increasing) radial weight $w$. Our main result gives some conditions on $w$ that guarantee the estimate from below on the harmonic function by a multiple of this weight. In dimension two this reverse estimate was first obtained by M. Cartwright for the case of the power weights, $w_p(z)=(1-|z|)^{-p}$ for $p>1$, and then generalized to a wide class of regular weights by a number of authors.