NEWTONIAN LORENTZ METRIC SPACES

摘要

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 rec-tifiable curves. Under some additional assumptions (that is, X carries a doubling measure and a weak Poincare inequality), we show that when 1 ≤ q ＜ p 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 ＞ 1. We also provide a counterexample to the density result in the Euclidean setting when 1 ＜ p ≤ n and q = ∞.