$\bar\partial$-problem in fiber bundles for decreasing $(0,1)$-forms

TL;DR

This paper studies the $ar ext{ extdegree}$-problem in fiber bundles with non-compactly supported forms that decrease along fibers, establishing existence of solutions with controlled decay, relevant to complex analysis and extension phenomena.

Contribution

It proves the existence of solutions to the $ar ext{ extdegree}$-problem in fiber bundles for forms decreasing along fibers without assuming compact support, extending previous results.

Findings

Solutions exist for forms decreasing along fibers with order slightly more than one.

Solutions also decrease along fibers, but not necessarily with the same order as the original forms.

Results have applications in complex analysis, especially in the Hartogs extension phenomenon.

Abstract

In this paper we consider the $\bar\partial$-problem in fiber bundles (fibers biholomorphic to $\mathbb C^k$, $k\geq 1$), namely the equation $\bar\partial\sigma =\omega$ for $(0,1)$-forms $\omega$ which decrease along the fibers. The order of decrease is slightly more than one. The important fact is that we do not assume that $\omega$ has compact support. The main theorem says that the equation has a solution which also decreases along fibers, however, not necessarily with the order as the original form. Existence of solution of the above mentioned $\bar\partial$-problem can be applied in various situations in Complex Analysis, in particular, to the Hartogs extension phenomenon.