On Hardy, BMO and Lipschitz spaces of invariantly harmonic functions in the unit ball |
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Authors: | Bonami A; Bruna J; Grellier S |
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Institution: | Département de Mathématiques, Université d'Orléans B.P. 6759, 45067 Orléans Cedex 2, France
Departament de Matemàtiques, Universitat Autònoma de Barcelona 08193 Bellaterra (Barcelona), Spain
Departément de Mathematiques Bâtiment 425, Université Paris-Sud, 91405 Orsay Cedex, France |
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Abstract: | The invariantly harmonic functions in the unit ball Bn in Cnare those annihilated by the Bergman Laplacian . The Poisson-Szegökernel P(z, ) solves the Dirichlet problem for : if f C(Sn),the Poisson-Szegö transform of f, where d is the normalized Lebesgue measure on Sn,is the unique invariantly harmonic function u in Bn, continuousup to the boundary, such that u=f on Sn. The Poisson-Szegötransform establishes, loosely speaking, a one-to-one correspondencebetween function theory in Sn and invariantly harmonic functiontheory in Bn. When n 2, it is natural to consider on Sn functionspaces related to its natural non-isotropic metric, for theseare the spaces arising from complex analysis. In the paper,different characterizations of such spaces of smooth functionsare given in terms of their invariantly harmonic extensions,using maximal functions and area integrals, as in the correspondingEuclidean theory. Particular attention is given to characterizationin terms of purely radial or purely tangential derivatives.The smoothness is measured in two different scales: that ofSobolev spaces and that of Lipschitz spaces, including BMO andBesov spaces. 1991 Mathematics Subject Classification: 32A35,32A37, 32M15, 42B25. |
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Keywords: | Bergman laplacian invariantly harmonic admissible maximal function area integral radial derivative pluriharmonic functions |
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