Espaces fonctionnels associés au calcul de Weyl-Hörmander (d'après un travail avec J.-M. Bony)
We consider generalized square function norms of holomorphic functions with values in a Banach space. One of the main results is a characterization of embeddings of the form , in terms of the type p and cotype q of the Banach space X. As an application we prove -estimates for vector-valued Littlewood-Paley-Stein g-functions and derive an embedding result for real and complex interpolation spaces under type and cotype conditions.
We investigate the Fourier transforms of functions in the Sobolev spaces . It is proved that for any function the Fourier transform f̂ belongs to the Lorentz space , where . Furthermore, we derive from this result that for any mixed derivative the weighted norm can be estimated by the sum of -norms of all pure derivatives of the same order. This gives an answer to a question posed by A. Pełczyński and M. Wojciechowski.
In this work we study the problem in , in , on , in , is a bounded regular domain such that , , , , and are positive functions such...
We prove an existence and uniqueness theorem for the elliptic Dirichlet problem for the equation div a(x,∇u) = f in a planar domain Ω. Here and the solution belongs to the so-called grand Sobolev space . This is the proper space when the right hand side is assumed to be only -integrable. In particular, we obtain the exponential integrability of the solution, which in the linear case was previously proved by Brezis-Merle and Chanillo-Li.