On the duality between $p$-modulus and probability measures

Ambrosio, Luigi, Di Marino, Simone, and Savaré, Giuseppe. "On the duality between $p$-modulus and probability measures." Journal of the European Mathematical Society 017.8 (2015): 1817-1853. <http://eudml.org/doc/277323>.

@article{Ambrosio2015,
abstract = {Motivated by recent developments on calculus in metric measure spaces $(X,\mathrm \{d\},\mathrm \{m\})$, we prove a general duality principle between Fuglede’s notion [15] of $p$-modulus for families of finite Borel measures in $(X,\mathrm \{d\})$ and probability measures with barycenter in $L^q(X,\mathrm \{m\})$, with $q$ dual exponent of $p\in (1,\infty )$. We apply this general duality principle to study null sets for families of parametric and non-parametric curves in $X$. In the final part of the paper we provide a new proof, independent of optimal transportation, of the equivalence of notions of weak upper gradient based on $p$-podulus [21, 23] and suitable probability measures in the space of curves ([6, 7]).},
author = {Ambrosio, Luigi, Di Marino, Simone, Savaré, Giuseppe},
journal = {Journal of the European Mathematical Society},
keywords = {$p$-modulus; capacity; duality; Sobolev functions; -modulus; capacity; duality; Sobolev functions},
language = {eng},
number = {8},
pages = {1817-1853},
publisher = {European Mathematical Society Publishing House},
title = {On the duality between $p$-modulus and probability measures},
url = {http://eudml.org/doc/277323},
volume = {017},
year = {2015},
}

TY - JOUR
AU - Ambrosio, Luigi
AU - Di Marino, Simone
AU - Savaré, Giuseppe
TI - On the duality between $p$-modulus and probability measures
JO - Journal of the European Mathematical Society
PY - 2015
PB - European Mathematical Society Publishing House
VL - 017
IS - 8
SP - 1817
EP - 1853
AB - Motivated by recent developments on calculus in metric measure spaces $(X,\mathrm {d},\mathrm {m})$, we prove a general duality principle between Fuglede’s notion [15] of $p$-modulus for families of finite Borel measures in $(X,\mathrm {d})$ and probability measures with barycenter in $L^q(X,\mathrm {m})$, with $q$ dual exponent of $p\in (1,\infty )$. We apply this general duality principle to study null sets for families of parametric and non-parametric curves in $X$. In the final part of the paper we provide a new proof, independent of optimal transportation, of the equivalence of notions of weak upper gradient based on $p$-podulus [21, 23] and suitable probability measures in the space of curves ([6, 7]).
LA - eng
KW - $p$-modulus; capacity; duality; Sobolev functions; -modulus; capacity; duality; Sobolev functions
UR - http://eudml.org/doc/277323
ER -