Newtonian Lorentz metric spaces

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, $X$ carries a doubling measure and a weak Poincaré inequality), we show that when $1\le q<p$ 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 $1<p\le n$ and $q=\infty$.