Dimension free bounds for the Hardy–Littlewood maximal operator associated to convex sets

Deleaval, Luc; Guédon, Olivier; Maurey, Bernard

Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 27 (2018) no. 1, pp. 1-198.

Ces Notes reprennent et complètent une série d’exposés donnés par les auteurs au groupe de travail « Convexité et Probabilités » à l’UPMC Jussieu, Paris, au cours du printemps 2013. Elles sont consacrées à l’étude des fonctions maximales de type Hardy–Littlewood associées aux corps convexes symétriques dans ${ℝ}^n$. On s’intéresse tout particulièrement au comportement des constantes intervenant dans les estimations lorsque la dimension $n$ tend vers l’infini. Ce sujet a été développé principalement entre 1982 et 1990, mais a été relancé par des avancées récentes.

Le but de la série d’exposés était de prouver des inégalités maximales dans $L^p\left({ℝ}^n\right)$ avec des bornes indépendantes de la dimension $n$, pour certaines familles de corps convexes. Dans les meilleurs cas, on a pu obtenir de tels résultats pour toutes les valeurs de $p$ dans $(1,+\infty ]$. Ce thème de recherche a été initié en 1982 par Elias Stein [75], qui a démontré le premier théorème de ce genre pour la famille des boules euclidiennes en dimension arbitraire, obtenant pour tout $p\in (1,+\infty ]$ une borne dans $L^p\left({ℝ}^n\right)$ indépendante de $n$. Nous présentons ce théorème de Stein ainsi que plusieurs autres résultats dans cette direction, démontrés par Bourgain, par Carbery et par Müller dans la période 1986–1990. En 1986, Bourgain [9] obtient une borne indépendante de $n$ valable dans $L^2\left({ℝ}^n\right)$ pour tout corps convexe symétrique dans ${ℝ}^n$, puis Bourgain [10] et Carbery [21] étendent le résultat $L^p\left({ℝ}^n\right)$ de Stein aux corps convexes symétriques quelconques, mais sous la condition que $p>3/2$. Müller [59] obtient un résultat valable pour tout $p>1$ quand un certain paramètre géométrique, lié aux volumes des projections du corps convexe sur les hyperplans, reste borné. Ce paramètre ne reste pas borné pour tous les convexes, en particulier, il tend vers l’infini pour les cubes de grande dimension. Nous donnons un théorème récent (2014) dû à Bourgain [13] qui obtient pour tout $p>1$ une borne dans $L^p\left({ℝ}^n\right)$ indépendante de $n$ pour la famille des fonctions maximales associées aux cubes en dimension arbitraire. Nous complétons l’étude du cas du cube par des résultats pour la constante de type faible $(1,1)$, dus à Aldaz [1], à Aubrun [3] et à Iakovlev–Strömberg [46] entre 2009 et 2013. À l’inverse du cas $L^p\left({ℝ}^n\right)$, $1<p\le +\infty$, cette constante de type faible ne reste pas bornée quand la dimension tend vers l’infini.

This survey is based on a series of lectures given by the authors at the working seminar “Convexité et Probabilités” at UPMC Jussieu, Paris, during the spring 2013. It is devoted to maximal functions associated to symmetric convex sets in high dimensional linear spaces, a topic mainly developed between 1982 and 1990 but recently renewed by further advances.

The series focused on proving these maximal function inequalities in $L^p\left({ℝ}^n\right)$, with bounds independent of the dimension $n$ and for all $p\in (1,+\infty ]$ in the best cases. This program was initiated in 1982 by Elias Stein, who obtained the first theorem of this kind for the family of Euclidean balls in arbitrary dimension. We present several results along this line, proved by Bourgain, Carbery and Müller during the period 1986–1990, and a new one due to Bourgain (2014) for the family of cubes in arbitrary dimension. We complete the cube case with a negative answer to the possible dimensionless behavior of the weak type $(1,1)$ constant, due to Aldaz, Aubrun and Iakovlev–Strömberg between 2009 and 2013.

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