Córdoba’s differentiation theorem: revisited
[Le théorème de différentiation de Córdoba : revisité]
Annales de l'Institut Fourier, Tome 73 (2023) no. 2, pp. 559-565.

Dans cet article, on démontre un lemme de recouvrement exponentiel impliquant le cas tridimensionnel d’une conjecture bien connue formulée par A. Zygmund circa 1935 et prouvée par A. Córdoba en 1978. Notre approche évite un argument subtil qui fati appel à la série entière de la fonction exponentielle.

In this paper we prove an exponential covering lemma implying the three dimensional case of a well-known conjecture formulated by A. Zygmund circa 1935 and solved by A. Córdoba in 1978. Our approach avoids a subtle argument involving the power series of the exponential function.

Reçu le :
Révisé le :
Accepté le :
Publié le :
DOI : 10.5802/aif.3535
Classification : 42B25
Keywords: Maximal functions, Differentiation theorem, Covering lemma.
Mot clés : Fonction maximalz, théorème de différentiation, lemme de recouvrement.
Martínez, Ángel D. 1

1 Institute for Advanced Study  Fuld Hall 412, 1 Einstein Drive, Princeton, NJ 08540 (USA)
Licence : CC-BY-ND 4.0
Droits d'auteur : Les auteurs conservent leurs droits
@article{AIF_2023__73_2_559_0,
     author = {Mart{\'\i}nez, \'Angel D.},
     title = {C\'ordoba{\textquoteright}s differentiation theorem: revisited},
     journal = {Annales de l'Institut Fourier},
     pages = {559--565},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {73},
     number = {2},
     year = {2023},
     doi = {10.5802/aif.3535},
     language = {en},
     url = {https://aif.centre-mersenne.org/articles/10.5802/aif.3535/}
}
TY  - JOUR
AU  - Martínez, Ángel D.
TI  - Córdoba’s differentiation theorem: revisited
JO  - Annales de l'Institut Fourier
PY  - 2023
SP  - 559
EP  - 565
VL  - 73
IS  - 2
PB  - Association des Annales de l’institut Fourier
UR  - https://aif.centre-mersenne.org/articles/10.5802/aif.3535/
DO  - 10.5802/aif.3535
LA  - en
ID  - AIF_2023__73_2_559_0
ER  - 
%0 Journal Article
%A Martínez, Ángel D.
%T Córdoba’s differentiation theorem: revisited
%J Annales de l'Institut Fourier
%D 2023
%P 559-565
%V 73
%N 2
%I Association des Annales de l’institut Fourier
%U https://aif.centre-mersenne.org/articles/10.5802/aif.3535/
%R 10.5802/aif.3535
%G en
%F AIF_2023__73_2_559_0
Martínez, Ángel D. Córdoba’s differentiation theorem: revisited. Annales de l'Institut Fourier, Tome 73 (2023) no. 2, pp. 559-565. doi : 10.5802/aif.3535. https://aif.centre-mersenne.org/articles/10.5802/aif.3535/

[1] Córdoba, Antonio Calderón–Zygmund Theory and Beyond: Fourier Analysis, Uncertainty Principles and Modeling (Lecture notes written by E. Latorre, Fall 2014) (unpublished)

[2] Córdoba, Antonio On the Vitali covering properties of a differentiation basis, Studia Math., Volume 57 (1976) no. 1, pp. 91-95 | DOI | MR | Zbl

[3] Córdoba, Antonio s×t×Φ(s,t), Mittag-Leffler Institute Report, 9, 1978

[4] Córdoba, Antonio Maximal functions: a proof of a conjecture of A. Zygmund, Bull. Amer. Math. Soc. (N.S.), Volume 1 (1979) no. 1, pp. 255-257 | DOI | MR | Zbl

[5] Córdoba, Antonio Maximal functions: a problem of A. Zygmund, Euclidean harmonic analysis (Proc. Sem., Univ. Maryland, College Park, Md., 1979) (Lecture Notes in Math.), Volume 779, Springer, Berlin, 1980, pp. 154-161 | DOI | MR | Zbl

[6] Córdoba, Antonio La razón geométrica del Teorema Fundamental del Cálculo, Gac. R. Soc. Mat. Esp., Volume 3 (2000) no. 3, pp. 435-446 | MR

[7] Córdoba, Antonio All that Math: portraits of mathematicians as young researchers. Celebrating the Centennial of Real Sociedad Matemática Española, Revista Matemática Iberoamericana, 2011, xiv+346 pages | MR | Zbl

[8] Córdoba, Antonio; Fefferman, R. A geometric proof of the strong maximal theorem, Ann. of Math. (2), Volume 102 (1975) no. 1, pp. 95-100 | DOI | MR | Zbl

[9] Córdoba, Antonio; Fefferman, R. A geometric proof of the strong maximal theorem, Bull. Amer. Math. Soc., Volume 81 (1975) no. 5, p. 941 | DOI | MR | Zbl

[10] Fava, Norberto Angel Weak type inequalities for product operators, Studia Math., Volume 42 (1972), pp. 271-288 | DOI | MR | Zbl

[11] Fefferman, R.; Pipher, J. A covering lemma for rectangles in n , Proc. Amer. Math. Soc., Volume 133 (2005) no. 11, pp. 3235-3241 | DOI | MR | Zbl

[12] de Guzmán, Miguel An inequality for the Hardy-Littlewood maximal operator with respect to a product of differentiation bases, Studia Math., Volume 49 (1973/74), pp. 185-194 | DOI | MR | Zbl

[13] de Guzmán, Miguel Differentiation of integrals in n , Lecture Notes in Mathematics, 481, Springer-Verlag, Berlin-New York, 1975, xii+266 pages | DOI | MR | Zbl

[14] Hardy, G. H.; Littlewood, J. E. A maximal theorem with function-theoretic applications, Acta Math., Volume 54 (1930) no. 1, pp. 81-116 | DOI | MR | Zbl

[15] Jessen, B.; Marcinkiewicz, J.; Zygmund, Antoni Note on the differentiability of multiple integrals, Fundamenta Mathematicae, Volume 25 (1935) no. 1, pp. 217-234 http://eudml.org/doc/212787 | DOI | Zbl

[16] Rey, Guillermo Another counterexample to Zygmund’s conjecture, Proc. Amer. Math. Soc., Volume 148 (2020) no. 12, pp. 5269-5275 | DOI | MR | Zbl

[17] Soria, Fernando Examples and counterexamples to a conjecture in the theory of differentiation of integrals, Ann. of Math. (2), Volume 123 (1986) no. 1, pp. 1-9 | DOI | MR | Zbl

[18] Stein, Elias M. Singular integrals and differentiability properties of functions, Princeton Mathematical Series, 30, Princeton University Press, Princeton, N.J., 1970, xiv+290 pages | MR | Zbl

Cité par Sources :