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Finding curves on general spaces through quantitative topology, with applications to Sobolev and Poincaré inequalities. (English) Zbl 0870.54031
The central objective of this paper is to give geometric criteria on a metric space \((M,d)\) where there exists a (rich) family of curves, starting and ending in a prescribed pair of sets. In addition, the curves of the family are controlled by a measure \(\nu \) on the measure space \((S,\lambda )\) (=the indexing set of the family) in the sense that for each curve \(\gamma _p:[0,L(p)]\rightarrow M\) of the family, (\(p\in S\)) , one has \(\mu (\gamma _n^{-1}(A))\leq a(p)\) and \(\int_Sa(p)d\lambda (p)\leq \nu (A)\), for each Borel set \(A\subseteq M\), where \(a:S\rightarrow [0,\infty ]\) is a measurable function and \(\mu \) is the Lebesgue measure on \(\mathbb{R}\). Generally, \(S\) is replaced by the \(n\)-dimensional sphere \(S^n\) and \(\nu \) is expressed in terms of the \(n\)-dimensional Hausdorff measure on M. The conditions imposed to the space M involve asymptotic properties at infinity. The family of curves is obtained by building Lipschitz maps from M to \(S^n\) with good topological properties. To this purpose is devoted the first of the two parts of the paper. The family of curves is also used to control the values of a function in terms of its (generalized) gradient (defined on a general metric space) and to obtain Sobolev and Poincaré inequalities (in the appendix B of the paper).

MSC:
54E40 Special maps on metric spaces
57N15 Topology of the Euclidean \(n\)-space, \(n\)-manifolds (\(4 \leq n \leq \infty\)) (MSC2010)
54F45 Dimension theory in general topology
55M25 Degree, winding number
28A78 Hausdorff and packing measures
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