# Contribution à l'étude des opérateurs multilinéaires et des espaces de Hardy

Abstract : This thesis contains two independent parts. In the first one, we are interested in the study of bilinear operators. We dedicate the two first chapters, to describe "time-frequency" arguments aiming to get local estimates about these operators. Using these "off-diagonal" estimates, we mainly get the continuities of these bilinear operators on Lebesgue spaces and Sobolev spaces. At the end of the second chapter, we study a bilinear pseudo-differential calculus. The third chapter is about a geometrical study of these bilinear operators. To complete this work, in the fourth chapter, we study some various results, for example, we try to generalize our results to multi-dimensional variables. The second part is about the concept of "Hardy spaces". We define an abstract construction of new Hardy spaces. Then, comparing with the already known and studied Hardy spaces, we try to clear up the minimal conditions to keep the main properties of these spaces. So we also get a criterion in order to prove the $H^1-L^1$ continuity of some operators. Then we take an interest in the study of intermediate spaces, got by interpolation between these new $H^1$ spaces and Lebesgue spaces. Finally, we use our abstract theory to solve the problem of maximal $L^p$ regularity on evolution differential equations.
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Cited literature [80 references]

https://tel.archives-ouvertes.fr/tel-00199735
Contributor : Frederic Bernicot <>
Submitted on : Wednesday, December 19, 2007 - 2:24:58 PM
Last modification on : Wednesday, September 16, 2020 - 4:04:54 PM
Long-term archiving on: : Thursday, September 27, 2012 - 11:55:43 AM

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• HAL Id : tel-00199735, version 1

### Citation

Frederic Bernicot. Contribution à l'étude des opérateurs multilinéaires et des espaces de Hardy. domain_other. Université Paris Sud - Paris XI, 2007. Français. ⟨tel-00199735⟩

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