A Dolbeault Lemma for Temperate Currents
Annales Henri Lebesgue, Volume 4 (2021) , pp. 879-896.

Metadata

KeywordsStein open subset of n or of a Stein manifold, L 2 estimates, ¯-operator, Dolbeault ¯-complex, temperate distributions and currents, temperate cohomology, Sobolev spaces

Abstract

We consider a bounded open Stein subset Ω of a complex Stein manifold X of dimension n. We prove that if f is a current on X of bidegree (p,q+1), ¯-closed on Ω, we can find a current u on X of bidegree (p,q) which is a solution of the equation ¯u=f in Ω. In other words, we prove that the Dolbeault complex of temperate currents on Ω (i.e. currents on Ω which extend to currents on X) is concentrated in degree 0. Moreover if f is a current on X= n of order k, then we can find a solution u which is a current on n of order k+2n+1.


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