Hölder and L^p estimates for solutions of ∂u = f in strongly pseudoconvex domains

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

“…La partie a) du théorème 1.1 découle du lemme 2.6.1 de [13], (qui reste valable si on remplace la mesure de Lebesgue de D par des mesures positives sur D), et du résultat suivant :…”

Approximation to bounded holomorphic functions on strictly pseudoconvex domains

“…We also obtain the following theorem, which, for E -3D, contains the bounded pointwise approximation theorem noted by N. Kerzman [6]. THEOREM …”

“…A recent result of I. Lieb [7] and N. Kerzman [6] states that any continuous function on the closure of a strictly pseudoconvex domain D with smooth boundary which is holomorphic on D, can be approximated uniformly on D by functions holomorphic in a neighborhood of D. 1 Here we prove the following theorem, which contains the above mentioned result in case E = 3D. H°°(D) denotes the Banach algebra of bounded holomorphic functions on D.…”

Singular integrals and estimates for the Cauchy-Riemann equations

“…Recent integral representations for u and estimates are due to Grauert and Lieb [2], Henkin [3], and Kerzman [4]. Basing ourselves on these integral representations, then for the resulting linear mapping ƒ -» u we can prove THEOREM 9 and trivially A x c T 1/2 <= f 1/2 .…”

“…(2) The singular integrals that occur in the solution of the du = ƒ problem, as given by Grauert and Lieb [2], Henkin [3], and Kerzman [4]. From the qualitative point of view what is common to these areas (and distinguishes them from the more classical integrals and estimates) is the splitting of directions at each point, together with the nonisotropic way that singularities behave and estimates are made.…”