Fractional integrals and $ {A}_{p}$-weights: A sharp estimate

Teresa Alberico; Andrea Cianchi; Carlo Sbordone

doi:10.1016/j.crma.2009.09.001

Harmonic Analysis/Calculus of Variations

Fractional integrals and

A_{p}

-weights: A sharp estimate
[Intégrales fractionnaires et

A_{p}

-poids : une estimation optimale]

Présenté par : Haïm Brezis

Teresa Alberico ¹ ; Andrea Cianchi ² ; Carlo Sbordone ¹

¹ Dipartimento di Matematica e Applicazioni “R. Caccioppoli”, Università di Napoli “Federico II”, Via Cintia, 80126 Napoli, Italy
² Dipartimento di Matematica e Applicazioni per l'Architettura, Università di Firenze, Piazza Ghiberti 27, 50122 Firenze, Italy

Comptes Rendus. Mathématique, Volume 347 (2009) no. 21-22, pp. 1265-1270.

Résumé
Abstract

On considère une inégalité à poids $A_{p}$ , pour des potentiels de Riesz dans $R^{n}$ . La constante de l'inégalité dépend de la constante $A_{p}$ du poids. On donne la forme exacte de la dépendance, en particulier on précise l'exposant optimal de la constante $A_{p}$ du poids.

We are concerned with an inequality, with an $A_{p}$ weight, for Riesz potentials in $R^{n}$ . The constant in the relevant inequality is known to depend on the $A_{p}$ constant of the weight. We find the exact form of this dependence. In particular, we exhibit the optimal exponent for the $A_{p}$ constant of the weight.

Accepté le : 2009-07-12
Publié le : 2009-10-28

DOI : 10.1016/j.crma.2009.09.001

Affiliations des auteurs :

Teresa Alberico ¹ ; Andrea Cianchi ² ; Carlo Sbordone ¹

¹ Dipartimento di Matematica e Applicazioni “R. Caccioppoli”, Università di Napoli “Federico II”, Via Cintia, 80126 Napoli, Italy
² Dipartimento di Matematica e Applicazioni per l'Architettura, Università di Firenze, Piazza Ghiberti 27, 50122 Firenze, Italy

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     author = {Teresa Alberico and Andrea Cianchi and Carlo Sbordone},
     title = {Fractional integrals and $ {A}_{p}$-weights: {A} sharp estimate},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {1265--1270},
     publisher = {Elsevier},
     volume = {347},
     number = {21-22},
     year = {2009},
     doi = {10.1016/j.crma.2009.09.001},
     language = {en},
}

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Teresa Alberico; Andrea Cianchi; Carlo Sbordone. Fractional integrals and $ {A}_{p}$-weights: A sharp estimate. Comptes Rendus. Mathématique, Volume 347 (2009) no. 21-22, pp. 1265-1270. doi : 10.1016/j.crma.2009.09.001. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2009.09.001/

Bibliographie
Cité par

[1] K. Astala; T. Iwaniec; E. Saksman Beltrami operators in the plane, Duke Math. J., Volume 107 (2001), pp. 27-56

[2] S.M. Buckley Estimates for operator norms on weighted spaces and Reverse Jensen Inequalities, Trans. Amer. Math. Soc., Volume 340 (1993), pp. 253-272

[3] R. Coifman; C. Fefferman Weighted norm inequalities for maximal functions and singular integrals, Studia Math., Volume 51 (1974), pp. 241-250

[4] F. Chiarenza; M. Frasca A Note on a weighted Sobolev inequality, Proc. Amer. Math. Soc., Volume 93 (1985), pp. 703-704

[5] E.B. Fabes; C.E. Kenig; R.P. Serapioni The local regularity of solutions of degenerate elliptic equations, Comm. Partial Differentioal Equations, Volume 7 (1982), pp. 77-116

[6] L. Grafakos Classical and Modern Fourier Analysis, Prentice-Hall, New Jersey, 2003

[7] L.I. Hedberg On certain convolution inequalities, Proc. Amer. Math. Soc., Volume 36 (1972), pp. 505-510

[8] V.G. Maz'ya Sobolev Spaces, Springer, Berlin, 1985

[9] B. Muckenhoupt Weighted norm inequalities for the Hardy maximal function, Trans. Amer. Math. Soc., Volume 165 (1972), pp. 207-226

[10] B. Muckenhoupt; R. Wheeden Weighted norm inequalities for fractional integrals, Trans. Amer. Math. Soc., Volume 192 (1974), pp. 261-274

[11] S. Petermichl The sharp bound for the Hilbert transform on weighted Lebesgue spaces in terms of the classical $A_{p}$ characteristic, Amer. J. Math., Volume 129 (2007), pp. 1355-1375

[12] S. Petermichl The sharp weighted bound for the Riesz transforms, Proc. Amer. Math. Soc., Volume 136 (2008), pp. 1237-1249

[13] S. Petermichl; A. Volberg Heating of the Ahlfors–Beurling operator: Weakly quasiregular maps on the plane are quasiregular, Duke Math. J., Volume 112 (2002), pp. 281-305

[14] E.M. Stein Harmonic Analysis: Real-Variable Methods, Orthogonality and Oscillatory Integrals, Princeton University Press, Princeton, 1993

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