Sobolev Theorems for Cusp Manifolds

  • Jürgen Eichhorn

    Universität Greifswald, Germany
  • Chunpeng Wang

    Jilin University, Changchun, China


In the past, we established a module structure theorem for Sobolev spaces on open manifolds with bounded curvature and positive injectivity radius rinj(M)=infxMrinj(x)>0r_{\rm inj}(M)= \inf_{x\in M}r_{\rm inj}(x)>0. The assumption rinj(M)>0r_{\rm inj}(M)>0 was essential in the proof. But, manifolds (Mn,g)(M^n,g) with vol(Mn,g)<{\rm vol}(M^n,g)<\infty have been excluded. An extension of our former results to the case vol(Mn,g)<{\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 ξ(x)=rinj(x)n\xi(x)=r_{\rm inj}(x)^{-n} or ξ(x)=vol(B1(x))1\xi(x)={\rm vol}(B_1(x))^{-1}.

Cite this article

Jürgen Eichhorn, Chunpeng Wang, Sobolev Theorems for Cusp Manifolds. Z. Anal. Anwend. 32 (2013), no. 4, pp. 389–409

DOI 10.4171/ZAA/1491