${\ell }^p$-improving for discrete spherical averages

Hughes, Kevin

Annales Henri Lebesgue, Tome 3 (2020), pp. 959-980.

Nous initions la théorie sur les inégalités de gain d’intégrabilité ${\ell }^p$ pour des moyennes arithmétiques sur des hypersurfaces ainsi que pour leurs fonctions maximales. En particulier, nous prouvons un gain d’intégrabilité pour les moyennes sphériques discrètes et certaines généralisations. Comme application de notre résultat, nous obtenons une nouvelle estimation pour la fonction maximale sphérique discrète en dimension 4. Nos preuves sont analogues à celle de Littman sur les moyennes sphériques Euclidiennes. Un aspect important de la preuve est une décomposition de Littlewood–Paley reposant simultanément sur des aspects arithmétiques et analytiques. L’aspect arithmétique est encodé par une méthode de décomposition du cercle en arcs mineurs et majeurs.

We initiate the theory of ${\ell }^p$-improving inequalities for arithmetic averages over hypersurfaces and their maximal functions. In particular, we prove ${\ell }^p$-improving estimates for the discrete spherical averages and some of their generalizations. As an application of our ${\ell }^p$-improving inequalities for the dyadic discrete spherical maximal function, we give a new estimate for the full discrete spherical maximal function in four dimensions. Our proofs are analogous to Littman’s result on Euclidean spherical averages. One key aspect of our proof is a Littlewood–Paley decomposition in both the arithmetic and analytic aspects. In the arithmetic aspect this is a major arc-minor arc decomposition of the circle method.

Classification : 37A45, 42B25, 11P05, 11P55, 11L07
Mots-clés : $L^p$-improving, discrete averages, discrete maximal functions, circle method, Littlewood–Paley theory

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