Calculus of Variations and Geometric Measure Theory
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S. Costea - M. Miranda Jr

Newtonian Lorentz Metric Spaces

created by miranda on 16 Nov 2012
modified by costea on 04 Dec 2020


Published Paper

Inserted: 16 nov 2012
Last Updated: 4 dec 2020

Journal: Illinois Journal of Mathematics
Volume: 56
Number: 2
Pages: 579-616
Year: 2012
Doi: 10.1215/ijm/1385129966
Links: Link to the version on


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 $q = \infty$.


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