Calculus of Variations and Geometric Measure Theory

G. Cavagnari - G. Savaré - G. E. Sodini

Extension of monotone operators and Lipschitz maps invariant for a group of isometries

created by sodini on 10 May 2023
modified on 09 Jan 2024


Published Paper

Inserted: 10 may 2023
Last Updated: 9 jan 2024

Journal: Canadian Journal of Mathematics
Year: 2023
Doi: 10.4153/S0008414X23000846

ArXiv: 2305.04678 PDF


We study monotone operators in reflexive Banach spaces that are invariant with respect to a group of suitable isometric isomorphisms and we show that they always admit a maximal extension which preserves the same invariance. A similar result applies to Lipschitz maps in Hilbert spaces, thus providing an invariant version of Kirzsbraun-Valentine extension Theorem. We then provide a relevant application to the case of monotone operators in $L^p$-spaces of random variables which are invariant with respect to measure-preserving isomorphisms, proving that they always admit maximal dissipative extensions which are still invariant by measure-preserving isomorphisms. We also show that such operators are law invariant, a much stronger property which is also inherited by their resolvents, the Moreau-Yosida approximations, and the associated semigroup of contractions. These results combine explicit representation formulae for the maximal extension of a monotone operator based on selfdual lagrangians and a refined study of measure-preserving maps in standard Borel spaces endowed with a nonatomic measure, with applications to the approximation of arbitrary couplings between measures by sequences of maps.