Geometric Fourier analysis
Annales de l'Institut Fourier, Volume 32 (1982) no. 3, p. 215-226

In this paper we continue the study of the Fourier transform on R n , n2, analyzing the “almost-orthogonality” of the different directions of the space with respect to the Fourier transform. We prove two theorems: the first is related to an angular Littlewood-Paley square function, and we obtain estimates in terms of powers of log(N), where N is the number of equal angles considered in R 2 . The second is an extension of the Hardy-Littlewood maximal function when one consider cylinders of R n , n2, of fixed eccentricity and direction on a given curve. We obtain sharp estimates for the L 2 -norm of such operators.

Dans ce travail nous continuons à étudier la transformée de Fourier sur R n , n2, en analysant la “presque-orthogonalité” des différentes directions de l’espace par rapport à la transformée de Fourier. Nous prouvons deux théorèmes. Dans le premier on généralise la théorie de Littlewood-Paley au cas où les angles sont égaux dans R 2 et nous obtenons des estimations de la norme L 4 de la forme (logN) a , où N est le nombre des directions. Le deuxième est une extension du théorème maximal de Hardy-Littlewood lorsqu’on considère des cylindres de R n , n2, avec excentricité fixée et direction dans une courbe donnée.

@article{AIF_1982__32_3_215_0,
     author = {Cordoba, Antonio},
     title = {Geometric Fourier analysis},
     journal = {Annales de l'Institut Fourier},
     publisher = {Imprimerie Louis-Jean},
     address = {Gap},
     volume = {32},
     number = {3},
     year = {1982},
     pages = {215-226},
     doi = {10.5802/aif.885},
     zbl = {0488.42027},
     mrnumber = {84i:42029},
     language = {en},
     url = {http://www.numdam.org/item/AIF_1982__32_3_215_0}
}
Cordoba, Antonio. Geometric Fourier analysis. Annales de l'Institut Fourier, Volume 32 (1982) no. 3, pp. 215-226. doi : 10.5802/aif.885. http://www.numdam.org/item/AIF_1982__32_3_215_0/

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