The Sobolev--Poincar\'e Inequality and the $L_{q,p}$-Cohomology of Twisted Cylinders

Abstract

We establish a vanishing result for the $L_{q,p}$-cohomology ($q\ge p$) of a twisted cylinder, which is a generalization of a warped cylinder. The result is new even for warped cylinders. We base on the methods for proving the $(p,q)$ Sobolev--Poincar\'e inequality developed by L.~Shartser.