Displaying similar documents to “The Cauchy-Riemann equations in infinite dimensions”

Semi-global solutions of ∂ with L (1 ≤ p ≤ ∞) bounds on strongly pseudoconvex real hypersurfaces in C (n ≥ 3).

C. H. Chang, H. P. Lee (1999)

Publicacions Matemàtiques

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Let M be an open subset of a compact strongly pseudoconvex hypersurface {ρ = 0} defined by M = D × C ∩ {ρ = 0}, where 1 ≤ m ≤ n-2, D = {σ(z, ..., z) < 0} ⊂ C is strongly pseudoconvex in C. For ∂ closed (0, q) forms f on M, we prove the semi-global existence theorem for ∂ if 1 ≤ q ≤ n-m-2, or if q = n - m - 1 and f satisfies an additional “moment condition”. Most importantly, the solution operator satisfies L estimates for 1 ≤ p ≤ ∞ with p = 1 and ∞ included.

On ∂̅-problems on (pseudo)-convex domains

R. Range (1995)

Banach Center Publications

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In this survey we shall tour the area of multidimensional complex analysis which centers around ∂̅-problems (i.e., the Cauchy-Riemann equations) on pseudoconvex domains. Along the way we shall highlight some of the classical milestones as well as more recent landmarks, and we shall discuss some of the major open problems and conjectures. For the sake of simplicity we will only consider domains in ℂ n ; intriguing phenomena occur already in the simple setting of (Euclidean) convex domains....

Hilbert-valued forms and barriers on weakly pseudoconvex domains.

Vincent Thilliez (1998)

Publicacions Matemàtiques

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We introduce an alternative proof of the existence of certain C barrier maps, with polynomial explosion of the derivatives, on weakly pseudoconvex domains in C. Barriers of this sort have been constructed very recently by J. Michel and M.-C. Shaw, and have various applications. In our paper, the adaptation of Hörmander's L techniques to suitable vector-valued functions allows us to give a very simple approach of the problem and to improve some aspects of the result of Michel and Shaw,...

Hölder and L estimates for the solutions of the ∂-equation in non-smooth strictly pseudoconvex domains.

Josep M. Burgués Badía (1990)

Publicacions Matemàtiques

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Let D be a bounded strict pseudoconvex non-smooth domain in C. In this paper we prove that the estimates in L and Lipschitz classes for the solutions of the ∂-equation with L-data in regular strictly pseudoconvex domains (see [2]) are also valid for D. We also give estimates of the same type for the ∂ in the regular part of the boundary of these domains.