Newtonian Lorentz Metric Spaces

Details Newtonian Lorentz Metric Spaces Newtonian Lorentz Metric Spaces

2011-04-18

arxiv.org/abs/1104.3475v2

Mathematics

Metric Geometry

v2: 32 pages. Formula on page 23 corrected; typos removed

Scientific paper

This paper studies Newtonian Sobolev-Lorentz spaces. We prove that these spaces are Banach. We also study the global p,q-capacity and the p,q-modulus of families of rectifiable curves. Under some additional assumptions (that is, the space carries a doubling measure and a weak Poincare inequality) and some restrictions on q, we show that the Lipschitz functions are dense in those spaces. Moreover, in the same setting we show that the p,q-capacity is Choquet provided that q is strictly greater than 1. We also provide a counterexample to the density result of Lipschitz functions in the Euclidean setting when q is infinite.

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