Formules explicites pour les solutions minimales de l'équation $\bar\partial u=f$ dans la boule et dans le polydisque de ${\Bbb C}^n$

Extension dans des classes de Hardy de fonctions holomorphes et estimations de type «mesures de Carleson» pour l'équation $\bar\partial $

“…Les résultats de a) ne sont pas nouveaux (voir [7], et lorsque q = 1 , D boule de C 2 , [5] ; par contre le noyau utilisé dans [7] ne semble pas satisfaire toutes les estimations de b).…”

Henkin-Ramirez formulas with weight factors

“…(21) on the boundary is the minimal solution in L 2 . This solution has been found earlier by Charpentier [2], in the interior and on the boundary already in [9]. Intuitively (22) (N = 1) could be viewed as an approximate Bergman kernel, in the same way as the CauchyLeray kernel is an approximate Szego kernel.…”

“…The logarithm actually occurs only when a is an integer, and the forms 9z\ log(l + |z| 2 ) show that the logarithm cannot be dispensed with. When n > 1 and / is a (0,l)-form a small additional argument shows that Theorem 9' holds for all a G R .…”

“…One type of such kernels has been given by Dautov and Henkin [4]. They use weights which behave like a power of the distance to the boundary (see also [2] and [ 1 ] for the case of the ball).…”

Integral formulas with weight factors

“…If D c C is a bounded domain in the complex plane with smooth boundary dD then for / e C ' ( 5 ) and z e D w e have where F(£, z) in any C -function which is holomorphic in z and F = 1 if f = z (see [2, p. 93]). If F = 1 then (3) becomes (2).…”