An andreotti-grauert theorem with l^r estimates

2012/03/04 by Éric Amar, Eric Amar, Amar, Eric · 1 citation
Mathematics · #Advanced Harmonic Analysis Research #Advanced Mathematical Physics Problems #Complex Variables (math.CV) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometry and complex manifolds #math.CV

paper · pdf · doi:10.48550/arxiv.1203.0759

Thanks to the referee, the presentation is highly enhanced and some typos are fixed. This will appear in:Annali de la Scuola Norm. Sup. di Pisa

openalex publication_date 2012/03/04 · arxiv created 2019/10/11 · arxiv updated 2019/10/14 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/04

Abstract

By a theorem of Andreotti and Grauert if ω is a (p,q) current, q < n, in a Stein manifold Ω, ∂ closed and with compact support, then there is a solution u to ∂ u=ω still with compact support in Ω. The main result of this work is to show that if moreover ω∈ L^r(m), where m is a suitable Lebesgue measure on the Stein manifold, then we have a solution u with compact support \sl and in L^s(m), (1)/(s)=(1)/(r)-(1)/(2(n+1)). We prove it by estimates in L^r spaces with weights.

Cited by

• Serre duality and Hörmander's solution of the ∂ -equation

Related

• Solutions of the ∂ -equation on Stein and on Kähler manifold with compact support
• Levi Problem in Complex Manifolds
• The raising steps method. Applications to the ∂ equation in Stein manifolds
• Théorie L^p pour l'équation de Cauchy-Riemann
• Grauert's theorem for subanalytic open sets in real analytic manifolds