Motivated by recent developments on calculus in metric measure spaces $(X,mathsf d,mathfrak m)$, we prove a general duality principle between Fuglede's notion of $p$-modulus for families of finite Borel measures in $(X,mathsf d)$ and probability measures with barycenter in $L^q(X,mathfrak m)$, with $q$ dual exponent of $pin (1,infty)$. 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 $p$-Modulus (Koskela-MacManus '98, Shanmugalingam '00) and suitable probability measures in the space of curves (Ambrosio-Gigli-Savare '11)

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

Ambrosio, Luigi;Di Marino, Simone;
2015

Abstract

Motivated by recent developments on calculus in metric measure spaces $(X,mathsf d,mathfrak m)$, we prove a general duality principle between Fuglede's notion of $p$-modulus for families of finite Borel measures in $(X,mathsf d)$ and probability measures with barycenter in $L^q(X,mathfrak m)$, with $q$ dual exponent of $pin (1,infty)$. 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 $p$-Modulus (Koskela-MacManus '98, Shanmugalingam '00) and suitable probability measures in the space of curves (Ambrosio-Gigli-Savare '11)
Settore MAT/05 - Analisi Matematica
Mathematics - Functional Analysis; Mathematics - Functional Analysis; Mathematics - Metric Geometry; Mathematics - Probability; 28A12, 31C15, 26A46
Fondi MUR
Calculus of Variations
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11384/60286
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