Inserted: 16 nov 2012
Last Updated: 4 dec 2020
Journal: Illinois Journal of Mathematics
Links: Link to the version on arxiv.org
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 Poincar ́e inequality), we show that when $1 \leq q < p$ the Lipschitz functions are dense in those spaces; moreover, in the same setting we also show that the $p,q$-capacity is Choquet provided that $q > 1$. We provide a counterexample for the density result in the Euclidean setting when $1 < p \leq n$ and $q = \infty$.