Abstract
In this paper, the boundary behaviour of a certain class of harmonic functions in Lipschitz domains is studied. It continues the work of N. Makarov on the boudary behaviour of analytic Bloch functions in the unit disk. Certain means associated with the function allow to approximate it by a martingale, and the boudary properties of the function are transferred to the asymptotic behaviour of the martingale.
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References
Anderson, J. M., Clunie, J. and Pommerenke, Ch.: 'On Bloch functions and normal functions', J. Reine Angew. Math. 270 (1974), 12–37.
Anderson, J. M. and Pitt, L. D.: 'The boundary behaviour of Bloch functions and univalent functions', Michigan Math. J. 35 (1988), 313–320.
Coifman, R. R. and Fefferman, C.: 'Weighted norm inequalities for maximal functions and singular integrals', Studia Math. 51 (1974), 241–250.
Dahlberg, B. E. J.: 'On estimates of harmonic measure', Arch. Rational Mech. Anal. 65 (1977), 272–288.
Donaire, J. J.: Conjuntos Excepcionales para las Clases de Zygmund, Tesis, Universitat Autónoma de Barcelona, 1995.
Evans, L. and Gariepy, R.: Measure Theory and Fine Properties of Functions, Studies in Advanced Mathematics, CRC, Press, 1992.
García Cuerva, J. and Rubio de Francia, J. L.: Weighted Norm Inequalities and Related Topics, Mathematical Studies 116, North-Holland, 1985.
Gehring, F.: 'The Lp-integrability of the partial derivatives of a quasi-conformal mapping', Acta Math. 130 (1973), 265–277.
Hunt, R. and Wheeden, R.: 'On the boundary values of harmonic functions', Trans. A.M.S. 132 (1968), 307–322.
Hunt, R. and Wheeden, R.: 'Positive harmonic functions on Lipschitz domains', Trans. A.M.S. 147 (1970), 507–527.
Hungerford, G. J.: Boundaries of Smooth Sets and Singular Sets of Blaschke Products in the Little Bloch Class, Thesis, California Institute of Technology, Pasadena, 1988.
Jerison, D., Kenig, C.: 'Boundary behavior of harmonic functions in nontangentially accesible domains', Adv. Math. 46 (1982), 80–147.
Korenblum, B., Rippon, P. J. and Samotij, K.: 'On integrals of harmonic functions over annuli', Ann. Acad. Sci. Fenn. A I Math. 20 (1995) 1, 3–26.
Makarov, N. G.: 'Probability methods in the theory of conformal mapping', Algebra i Analiz. (1989), 3–59 (Russian) [English transl.: Leningrad Math. J. 1 (1990), 1–56.]
Makarov, N. G.: 'Smooth measures and the law of the iterated logarithm', Izv. Akad. Nauk SSSR Ser. Mat. 53 (1989), 439–446 (Russian) [English transl.: Math. USSR Izv. 34 (1990), 455–463.]
Makarov, N. G.: 'On the radial behavior of Bloch functions', Dokl. Akad. Nauk SSSR. 309 (1989), 275–278 (Russian) [English transl.: Soviet Math. Dokl. 40 (1990), 505–508.]
Makarov, N. G.: 'Conformal mapping and Hausdorff measures', Arkiv Mat. 25 (1987), 41–89.
Makarov, N. G.: 'On the distortion of boundary sets under conformal mappings', Proc. London Math. Soc. (3) 51 (1985), 369–384.
Pommerenke, Ch.: Boundary Behaviour of Conformal Maps, Springer Verlag, 1992.
Rohde, S.: 'The boundary behaviour of Bloch functions', J. London Math. Soc. (2) 48 (1993), 488–499.
Samotij, K.: 'A representation theorem for harmonic functions in the ball of R n', Ann. Acad. Sci. Fenn. Ser A I Math. 11 (1986), 29–38.
Shiriayev, A. N.: Probability, Springer Verlag, 1984.
Stout, W. F.: Almost Sure Convergence, Academic Press, 1974.
Stout, W. F.: 'A martingale analogue of Kolmogorov's law of the iterated logarithm', Z. Wahrsch. und Verw. Gebiete. 3 (1964), 211–226.
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Llorente, J. Boundary Values of Harmonic Bloch Functions in Lipschitz Domains: A Martingale Approach. Potential Analysis 9, 229–260 (1998). https://doi.org/10.1023/A:1008655312464
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DOI: https://doi.org/10.1023/A:1008655312464