Illinois Journal of Mathematics

Newtonian Lorentz metric spaces

Şerban Costea and Michele Miranda Jr.

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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 $q=\infty$.

Article information

Source
Illinois J. Math., Volume 56, Number 2 (2012), 579-616.

Dates
First available in Project Euclid: 22 November 2013

Permanent link to this document
https://projecteuclid.org/euclid.ijm/1385129966

Digital Object Identifier
doi:10.1215/ijm/1385129966

Mathematical Reviews number (MathSciNet)
MR3161342

Zentralblatt MATH identifier
1319.31015

Subjects
Primary: 31C15: Potentials and capacities 46E35: Sobolev spaces and other spaces of "smooth" functions, embedding theorems, trace theorems

Citation

Costea, Şerban; Miranda Jr., Michele. Newtonian Lorentz metric spaces. Illinois J. Math. 56 (2012), no. 2, 579--616. doi:10.1215/ijm/1385129966. https://projecteuclid.org/euclid.ijm/1385129966


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