## Revista Matemática Iberoamericana

### Multiplicative Square Functions

#### Abstract

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.

#### Article information

Source
Rev. Mat. Iberoamericana, Volume 20, Number 3 (2004), 673-736.

Dates
First available in Project Euclid: 27 October 2004

https://projecteuclid.org/euclid.rmi/1098885433

Mathematical Reviews number (MathSciNet)
MR2124487

Zentralblatt MATH identifier
1086.42010

#### Citation

González, María José; Nicolau, Artur. Multiplicative Square Functions. Rev. Mat. Iberoamericana 20 (2004), no. 3, 673--736. https://projecteuclid.org/euclid.rmi/1098885433

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