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October, 2004 Multiplicative Square Functions
María José González, Artur Nicolau
Rev. Mat. Iberoamericana 20(3): 673-736 (October, 2004).


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.


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María José González. Artur Nicolau. "Multiplicative Square Functions." Rev. Mat. Iberoamericana 20 (3) 673 - 736, October, 2004.


Published: October, 2004
First available in Project Euclid: 27 October 2004

zbMATH: 1086.42010
MathSciNet: MR2124487

Primary: 31A20 , 42B25

Keywords: doubling measure , hyperbolic derivative , martingale , positive harmonic functions , quasiconformal mappings , square functions

Rights: Copyright © 2004 Departamento de Matemáticas, Universidad Autónoma de Madrid


Vol.20 • No. 3 • October, 2004
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