no. 5
Sparse bounds for maximal rough singular integrals via the Fourier transform
[Un contrôle épars à travers la transformée de Fourier]
Di Plinio, Francesco1 ; Hytönen, Tuomas P.2 ; Li, Kangwei3
1 Washington University in St. Louis One Brookings Drive St. Louis, MO 63130-4899 (USA)
2 (TPH) Department of Mathematics and Statistics, P.O.B. 68 FI-00014 University of Helsinki (Finland)
3 (KL) BCAM Basque Center for Applied Mathematics Bilbao (Spain)
Annales de l'Institut Fourier, Tome 70 (2020) no. 5, pp. 1871-1902.

Nous démontrons un contrôle épars qualitatif pour la troncature maximale des noyaux de convolution dont la transformée de Fourier satisfait des conditions de décroissance appropriées. Notre résultat étend le principe de contrôle épars de Conde-Alonso, Culiuc, Ou et du premier auteur au cas de la troncature maximale, et inclut le cas des intégrales singulières homogènes T Ω sur ℝ d dont la composante angulaire Ω est bornée et a une moyenne nulle. Parmi les diverses conséquences, nous obtenons de nouvelles inégalités quantitatives pondérées pour la troncature maximale de T Ω , l’extension d’un résultat de Roncal, Tapiola et du second auteur. De plus, une extension appropriée aux valeurs vectorielles du contrôle épars implique de nouvelles estimations à poids matriciels pour les troncatures maximales de T Ω . Notre résultat est quantitatif, mais il est nouveau même d’un point de vue qualitatif. L’approche actuelle basée sur le contrôle épars est la seule actuellement connue pour les estimations à poids matriciels de cette classe d’opérateurs.

We prove a quantified sparse bound for the maximal truncations of convolution-type singular integrals with suitable Fourier decay of the kernel. Our result extends the sparse domination principle by Conde-Alonso, Culiuc, Ou and the first author to the maximally truncated case, and covers the rough homogeneous singular integrals T Ω on ℝ d with bounded angular part Ω having vanishing integral on the sphere. Among several consequences, we obtain new quantitative weighted norm inequalities for the maximal truncation of T Ω , extending a result by Roncal, Tapiola and the second author.

A convex-body valued version of the sparse bound is also deduced and employed towards novel matrix-weighted norm inequalities for the maximal truncations of T Ω . Our result is quantitative, but even the qualitative statement is new, and the present approach via sparse domination is the only one currently known for the matrix weighted bounds of this class of operators.

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DOI : 10.5802/aif.3354
Classification : 42B20, 42B25
Keywords: Sparse domination, rough singular integrals, weighted norm inequalities
Mot clés : contrôle épars, intégrales singulières de finesse limitée, estimations á poids
Di Plinio, Francesco 1 ; Hytönen, Tuomas P. 2 ; Li, Kangwei 3

1 Washington University in St. Louis One Brookings Drive St. Louis, MO 63130-4899 (USA)
2 (TPH) Department of Mathematics and Statistics, P.O.B. 68 FI-00014 University of Helsinki (Finland)
3 (KL) BCAM Basque Center for Applied Mathematics Bilbao (Spain)
Licence : CC-BY-ND 4.0
Droits d'auteur : Les auteurs conservent leurs droits 
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     title = {Sparse bounds for maximal rough singular integrals via the {Fourier} transform},
     journal = {Annales de l'Institut Fourier},
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Di Plinio, Francesco; Hytönen, Tuomas P.; Li, Kangwei. Sparse bounds for maximal rough singular integrals via the Fourier transform. Annales de l'Institut Fourier, Tome 70 (2020) no. 5, pp. 1871-1902. doi : 10.5802/aif.3354. https://aif.centre-mersenne.org/articles/10.5802/aif.3354/

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