Sobolev Theorems for Cusp Manifolds

Abstract

In the past, we established a module structure theorem for Sobolev spaces on open manifolds with bounded curvature and positive injectivity radius $r_{\mathrm{inj}}(M)={\inf }_{x\in M}{r}_{\mathrm{inj}}(x)>0$. The assumption $r_{\mathrm{inj}}(M)>0$ was essential in the proof. But, manifolds $(M^n,g)$ with $\mathrm{vol}(M^n,g)<\infty$ have been excluded. An extension of our former results to the case $\mathrm{vol}(M^n,g)<\infty$ seems to be hopeless. In this paper, we show that certain Sobolev embedding theorems and a (generalized) module structure theorem are valid in weighted spaces with the weight $\xi (x)=r_{\mathrm{inj}}(x{)}^{-n}$ or $\xi (x)=\mathrm{vol}(B_1(x){)}^{-1}$.