On a theorem of Cartwright in higher dimensions

We consider harmonic functions in the unit ball of Rⁿ⁺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 2, this reverse estimate was first obtained by Cartwright for the case of the power weights, w_p(z) = (1 - |z|)^-p, p>1, and then generalized to a wide class of regular weights by a number of authors.