L\({}_ p\)-cohomology of warped cylinders
This is the second article by the same authors in which they consider the \(L_ p\)-cohomology of warped cylinders: \(C^ f_{a,b}X=X\times_ f[a,b)\) with the metric \(f^ 2(t)(dg)^ 2+(dt)^ 2,\) where \((dg)^ 2\) is a metric on X and \(a\leq t<b\). In the first article [Sib. Math. J. 31, No.5, 716-727 (1990); translation from Sib. Mat. Zh. 31, No.5(183), 10-23 (1990; Zbl 0722.53034)] they considered the \(\bar H^ k_ p\)-reduced \(L_ p\)-cohomology of warped cylinders. In the article under review they study the \(L_ p\)-cohomology \(H^ k_ p\). Among some other results using the methods described previously, it is proved, that \(H^ k_ p (C^ f_{a,b}X,X_ a)=0\) if \[ B(a,b,f^ m)=\sup_{a<t<b}(\int^{b}_{t}f^{pm}(\tau)d\tau)^{1/p}(\int^{t}\;sb{a}f^{-p'm}(\tau)d\tau)^{1/p'}<\infty \] where \(p'=p/(p-1)\), \(m=n/p- k+1\); and \(H^ k_ p(C^ f_{a,b}X)=0\) if in addition \(\int^{b}_{a}f^{(m-1)p}(t)dt=\infty\).
scientific article; zbMATH DE number 4210976