Revista Matemática Iberoamericana

Multiplicative Square Functions

María José González and Artur Nicolau

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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

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

First available in Project Euclid: 27 October 2004

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Zentralblatt MATH identifier

Primary: 42B25: Maximal functions, Littlewood-Paley theory 31A20: Boundary behavior (theorems of Fatou type, etc.)

Doubling measure martingale positive harmonic functions square functions hyperbolic derivative quasiconformal mappings


González, María José; Nicolau, Artur. Multiplicative Square Functions. Rev. Mat. Iberoamericana 20 (2004), no. 3, 673--736.

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  • Aleksandrov, A. B., Anderson, J. M. and Nicolau, A: Inner functions, Bloch spaces and symmetric measures. Proc. London Math. Soc. (3) 79 (1999), 318-352.
  • Anderson, J. M., Fernández, J. L. and Shields, A. L.: Inner functions and cyclic vectors in the Bloch space. Trans. Amer. Math. Soc. 323 (1991), 429-448.
  • Bañuelos, R., Klemeš, I. and Moore, C. N.: An analogue for harmonic functions of Kolmogorov's law of the iterated logarithm. Duke Math. J. 57 (1988), 37-68.
  • Bañuelos, R., Klemeš, I. and Moore, C. N.: The lower bound in the law of the iterated logarithm for harmonic functions. Duke Math. J. 60 (1990), 689-715.
  • Bañuelos, R. and Moore, C. N.: Laws of the iterated logarithm, sharp good-$\lambda$-inequalities and $L^p $-estimates for caloric and harmonic functions. Indiana Univ. Math. J. 38 (1989), 315-344.
  • Bañuelos, R. and Moore, C. N.: Probabilistic behavior of harmonic functions. Progress in Mathematics 175. Birkhäuser Verlag, Basel, 1999.
  • Beurling, A. and Ahlfors, L. V.: The boundary correspondence under quasiconformal mappings. Acta Math. 96 (1956), 125-142.
  • Brossard, J.: Intégrale d'aire et support d'une mesure positive. C. R. Acad. Sci. Paris Sér I. Math. 296 (1983), 231-232.
  • Burkholder, D. L. and Gundy, R. F.: Extrapolation and interpolation of quasi-linear operators on martingales. Acta Math. 124 (1970), 249-304.
  • Burkholder, D. L. and Gundy, R. F.: Distribution function inequalities for the area integral. Studia Math. 44 (1972), 527-544.
  • Buckley, S. M.: Estimates for operator norms on weighted spaces and reverse Jensen inequalities. Trans. Amer. Math. Soc. 340 (1993), 253-272.
  • Cantón, A.: Singular measures and the little Bloch space. Publ. Mat. 42 (1998), 211-222.
  • Carleson, L.: On mappings, conformal at the boundary. J. Analyse Math. 19 (1967), 1-13.
  • Chang, S.-Y. A., Wilson, J. M. and Wolff, T. H.: Some weighted norm inequalities concerning the Schrödinger operator. Comment. Math. Helv. 60 (1985), 217-246.
  • Dyn'kin, E.: Estimates for asymptotically conformal mappings. Ann. Acad. Sci. Fenn. Math. 22 (1997), 275-304.
  • Doubtsov, E. and Nicolau, A.: Symmetric and Zygmund measures in several variables. Ann. Inst. Fourier (Grenoble) 52 (2002), no. 1, 153-177.
  • Fefferman, R. A. Kenig, C. E. and Pipher, J.: The theory of weights and the Dirichlet problem for elliptic equations. Ann. of Math. (2) 134 (1991), 65-124.
  • Fefferman, C. and Stein, E. M.: $H^p$ spaces of several variables. Acta Math. 129 (1972), 137-193.
  • Fernández, J. L. and Nicolau, A.: Boundary behaviour of inner functions and holomorphic mappings. Math. Ann. 310 (1998), 423-445.
  • Garnett, J. B. and Jones, P. W.: The distance in BMO to $L^\infty$. Ann. of Math. (2) 108 (1978), 373-393.
  • Garnett, J. B. and Jones, P. W.: BMO from dyadic BMO. Pacific J. Math. 99 (1982), 351-371.
  • Heurteaux, Y.: Sur la comparaison des mesures avec les mesures de Hausdorff. C. R. Acad. Sci. Paris Sér I Math. 321 (1995), 61-65.
  • Jones, P. W.: Square functions, Cauchy integrals, analytic capacity and harmonic measure. In Harmonic analysis and partial differential equations (El Escorial, 1987), 24-68. Lecture Notes in Math. 1384. Springer, Berlin, 1989.
  • Kahane, J. P.: Trois notes sur les ensembles parfaits linéaires. Enseignement Math. (2) 15 (1969), 185-192.
  • Kaufman, R. and Wu, J. M.: Two problems on doubling measures. Rev. Mat. Iberoamericana 11 (1995), no. 3, 527-545.
  • Llorente, J. G.: Boundary values of harmonic Bloch functions in Lipschitz domains: a martingale approach. Potential Anal. 9 (1998), no. 3, 229-260.
  • Makarov, N. G.: Probability methods in the theory of conformal mappings. (Russian) Algebra i Analiz 1 (1989), no. 1, 3-59. Translation in Leningrad Math. J. 1 (1990), no. 1, 1-56.
  • Makarov, N. G.: Smooth measures and the law of the iterated logarithm. (Russian) Izv. Akad. Nauk SSSR Ser. Mat. 53 (1989), 439-446. Translation in Math. USSR-Izv. 34 (1990), 455-463.
  • MacManus, P.: Quasiconformal mappings and Ahlfors-David curves. Trans. Amer. Math. Soc. 343 (1994), 853-881.
  • McConnell, T. R.: Area integrals and subharmonic functions. Indiana Univ. Math. J. 33 (1984), 289-303.
  • Murai, T. and Uchiyama, A.: Good $\lambda$ inequalities for the area integral and the nontangential maximal function. Studia Math. 83 (1986), 251-262.
  • Nevanlinna, R.: Analytic functions. Springer-Verlag, New York, Berlin, 1970.
  • Piranian, G.: Two monotonic, singular, uniformly almost smooth functions. Duke Math. J. 33 (1966), 255-262.
  • Smith, W.: Inner functions in the hyperbolic little Bloch class. Michigan Math. J. 45 (1998), 103-114.
  • Stein, E. M.: Singular integrals and differentiability properties of functions. Princeton Mathematical Series 30. Princeton University Press, Princeton, N. J., 1970.
  • Sweezy, C.: $L$-harmonic functions and the exponential square class. Pacific J. Math. 147 (1991), 187-200.
  • Treil, S. R., Volberg, A. and Zheng, D.: Hilbert transform, Toeplitz operators and Hankel operators, and invariant $A_\infty$ weights. Rev. Mat. Iberoamericana 13 (1997), no. 2, 319-360.
  • Uchiyama, A.: On McConnell's inequality for functionals of subharmonic functions. Pacific J. Math. 128 (1987), 367-377.
  • Wu, J.-M.: Doubling measures with different basis. Colloq. Math. 76 (1998), no. 1, 49-55.
  • Wu, J.-M.: Hausdorff dimension and doubling measures on metric spaces. Proc. Amer. Math. Soc. 126 (1998), no. 5, 1453-1459.