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Zeitschrift für Analysis und ihre Anwendungen

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Volume 32, Issue 4, 2013, pp. 389–409
DOI: 10.4171/ZAA/1491

Published online: 2013-09-19

Sobolev Theorems for Cusp Manifolds

Jürgen Eichhorn[1] and Chunpeng Wang[2]

(1) Universität Greifswald, Germany
(2) 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 $r_{\rm inj}(M)= \inf_{x\in M}r_{\rm inj}(x)>0$. 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 $\xi(x)={\rm vol}(B_1(x))^{-1}$.

Keywords: Weighted Sobolev spaces, open manifolds, injectivity radius, finite volume, embedding theorems

Eichhorn Jürgen, Wang Chunpeng: Sobolev Theorems for Cusp Manifolds. Z. Anal. Anwend. 32 (2013), 389-409. doi: 10.4171/ZAA/1491