Conservation de certaines propri\'et\'es \`a travers un contr\^ole \'epars d'un op\'erateur et applications au projecteur de Leray-Hopf

TL;DR

This paper investigates how sparse controls can preserve certain properties of singular operators, with applications to Riesz transforms and Leray projectors, offering new insights into sparse domination and atomic decompositions.

Contribution

It introduces methods to maintain operator properties via sparse controls and links sparse domination with atomic decomposition in Hardy spaces.

Findings

Boundedness of the adjoint of Riesz transform established

Leray projector boundedness demonstrated

New connection between sparse domination and atomic decomposition revealed

Abstract

Nous poursuivons l'\'etude d'un contr\^ole \'epars d'un op\'erateur singulier. Plus pr\'ecis\'ement nous expliquons comment on peut conserver certaines propri\'et\'es de l'op\'erateur initial \`a travers un tel contr\^ole et d\'ecrivons quelques applications: bornitude de l'adjoint de la transform\'ee de Riesz et du projecteur de Leray. De plus, nous nous int\'eresserons \`a donner un regard nouveau sur les dominations \'eparses \`a travers les oscillations et les fonctions carr\'ees localis\'ees. Aussi, nous d\'evoilerons une connexion entre les bons intervalles de la d\'ecomposition \'eparse et une d\'ecomposition atomique. We pursue the study of a sparse control for a singular operator. More precisely, we describe how one can track some properties of the initial operator, through such a control and describe also some applications: boundedness of the adjoint of a Riesz transform and of the Leray projector. Moreover, we will be interested in giving a new insight on the sparse domination through the oscillations and the localized square functions. Also, we will reveal a connection between the good intervals of the sparse domination and the atomic decomposition for a function in a Hardy space.