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

In this paper we consider the $\bar{\mathrm{\partial }}$-problem in fiber bundles (fibers biholomorphic to ${\mathbb{C}}^k$, $k\ge 1$), namely the equation $\bar{\mathrm{\partial }}\sigma =\omega$ 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 $\bar{\mathrm{\partial }}$-problem can be applied in various situations in Complex Analysis, in particular, to the Hartogs extension phenomenon.