$L^2$ Extension of $\bar{\partial }$-closed forms from a hypersurface

Abstract

We establish $L^2$ extension theorems for $\bar{\partial }$-closed $(0,q)$-forms with values in a holomorphic line bundle with smooth Hermitian metric, from a smooth hypersurface on a Stein manifold. Our result extends (and gives a new, perhaps more classical, proof of) a theorem of Berndtsson on compact Kähler manifolds, which itself is a sharpening of the theorem of Koziarz. The proof makes use of the Kohn solution, which is the solution of an (interior) elliptic problem, to handle the well-known regularity issues. As such, our methods require the line bundle to be equipped with a smooth metric.

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Cited by in corpus (2)

• Extension of L^2, di-bar-closed, forms
• Extension of L^2 holomorphic functions