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 $r\_{\rm inj}(M)= \inf\_{x\in M}r\_{\rm inj}(x)>0$. The assumption $r\_{\rm inj}(M)>0$ was essential in the proof. But, manifolds $(M^n,g)$ with ${\rm vol}(M^n,g)<\infty$ have been excluded. An extension of our former results to the case ${\rm vol}(M^n,g)<\infty$ 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 $\xi(x)=r\_{\rm inj}(x)^{-n}$ or $\xi(x)={\rm vol}(B\_1(x))^{-1}$.