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

Luigi Ambrosio; Simone Di Marino; Giuseppe Savaré

Journal of the European Mathematical Society (2015)

- Volume: 017, Issue: 8, page 1817-1853
- ISSN: 1435-9855

## Access Full Article

top## Abstract

top## How to cite

topAmbrosio, 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 -

## NotesEmbed ?

topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.