ENDS OF METRIC MEASURE SPACES AND SOBOLEV INEQUALITIES

Abstract. Generalizing work of Li and Wang, we prove sharp volume growth/decay rates for ends of metric measure spaces supporting a (p; p)-Sobolev inequality. A sharp result for (q; p)-Sobolev inequalities is also proved. As part of their study of complete manifolds whose spectrum of the Laplacian has a positive lower bound, Li and Wang ([LW, theorem 1.4]) prove the following result, where 1(E) is the least eigenvalue of the Laplacian on E for the Dirichlet problem, V (r) = (E\B(o; r)), and V (1) = (E). They also prove that the rate