Newtonian Lorentz Metric Spaces

TL;DR

This paper investigates the properties of Newtonian Sobolev-Lorentz spaces, establishing their Banach space structure, analyzing capacities and moduli, and exploring the density of Lipschitz functions under various conditions.

Contribution

It proves that Newtonian Sobolev-Lorentz spaces are Banach and examines conditions for Lipschitz density and capacity properties, extending existing theory.

Findings

Newtonian Sobolev-Lorentz spaces are Banach spaces.

Lipschitz functions are dense under doubling measure and Poincare inequality with certain q.

p,q-capacity is Choquet when q > 1.

Abstract

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.