This paper studies Newtonian Sobolev-Lorentz spaces. We prove that thesespaces are Banach. We also study the global p,q-capacity and the p,q-modulus offamilies of rectifiable curves. Under some additional assumptions (that is, thespace carries a doubling measure and a weak Poincare inequality) and somerestrictions on q, we show that the Lipschitz functions are dense in thosespaces. Moreover, in the same setting we show that the p,q-capacity is Choquetprovided that q is strictly greater than 1. We also provide a counterexample tothe density result of Lipschitz functions in the Euclidean setting when q isinfinite.
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