We study regularity properties of a positive measure in the
euclidean space in terms of two square functions which are the
multiplicative analogues of the usual martingale square function
and of the Lusin area function of a harmonic function. The size of
these square functions is related to the rate at which the measure
doubles at small scales and determines several regularity
properties of the measure. We consider the non-tangential maximal
function of the logarithm of the densities of the measure in the
dyadic setting, and of the logarithm of the harmonic extension of
the measure, in the continuous setting. We relate the size of
these maximal functions to the size of the corresponding square
functions. Fatou type results, $L^p$ estimates and versions of the
Law of the Iterated Logarithm are proved. As applications we
introduce a hyperbolic version of the Lusin Area function of an
analytic mapping from the unit disc into itself, and use it to
characterize inner functions. Another application to the theory of
quasiconformal mappings is given showing that our methods can also
be applied to prove a result by Din'kyn's on the smoothness of
quasiconformal mappings of the disc.