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Autour du problème de la couronne

Abstract : In a first part, we are interested in a problem of division in Hardy spaces of the unit ball B of C^n. Given m functions g_1,...,g_m holomorphic and bounded in B, and a holomorphic function f, we give a sufficient condition, weaker than the classical corona hypothesis, so that there exist m functions f_1,...,f_m in a Hardy space of B such that f_1g_1+...+f_mg_m=f. The proof is based on the Koszul complex method, and the resolution of d"-equations with good estimates. The main new difficulty, with respect to previous works, comes from the fact that the functions g_1,...,g_m may have common zeros. In a second part we are interested in the corona problem in the Hardy spaces of the bidisc equipped with its topological boundary. We give a result about the resolution of a d"-equation in the bidisc with estimates in Lp of the bondary, when the data verify Carleson-type hypotheses. Finally, we end with a result allowing to deduce a corona theorem with values in vector spaces of finite dimension from a classical corona theorem in a Hardy space of a domain in C^n. As a consequence we obtain an operator corona theorem in Hardy spaces of the ball and of the polydisc equipped with its distinguished boundary.
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Contributor : Jessica Hergoualch <>
Submitted on : Thursday, January 6, 2005 - 4:55:32 PM
Last modification on : Thursday, January 11, 2018 - 6:12:18 AM
Long-term archiving on: : Thursday, September 13, 2012 - 2:10:10 PM


  • HAL Id : tel-00007935, version 1




Jessica Hergoualch. Autour du problème de la couronne. Mathématiques [math]. Université Sciences et Technologies - Bordeaux I, 2004. Français. ⟨tel-00007935⟩



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