We are concerned with an inequality, with an weight, for Riesz potentials in . The constant in the relevant inequality is known to depend on the constant of the weight. We find the exact form of this dependence. In particular, we exhibit the optimal exponent for the constant of the weight.
On considère une inégalité à poids , pour des potentiels de Riesz dans . La constante de l'inégalité dépend de la constante du poids. On donne la forme exacte de la dépendance, en particulier on précise l'exposant optimal de la constante du poids.
Published online:
@article{CRMATH_2009__347_21-22_1265_0,
author = {Alberico, Teresa and Cianchi, Andrea and Sbordone, Carlo},
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},
url = {http://www.numdam.org/articles/10.1016/j.crma.2009.09.001/}
}
TY - JOUR
AU - Alberico, Teresa
AU - Cianchi, Andrea
AU - Sbordone, Carlo
TI - Fractional integrals and $ {A}_{p}$-weights: A sharp estimate
JO - Comptes Rendus. Mathématique
PY - 2009
SP - 1265
EP - 1270
VL - 347
IS - 21-22
PB - Elsevier
UR - http://www.numdam.org/articles/10.1016/j.crma.2009.09.001/
DO - 10.1016/j.crma.2009.09.001
LA - en
ID - CRMATH_2009__347_21-22_1265_0
ER -
%0 Journal Article
%A Alberico, Teresa
%A Cianchi, Andrea
%A Sbordone, Carlo
%T Fractional integrals and $ {A}_{p}$-weights: A sharp estimate
%J Comptes Rendus. Mathématique
%D 2009
%P 1265-1270
%V 347
%N 21-22
%I Elsevier
%U http://www.numdam.org/articles/10.1016/j.crma.2009.09.001/
%R 10.1016/j.crma.2009.09.001
%G en
%F CRMATH_2009__347_21-22_1265_0
Alberico, Teresa; Cianchi, Andrea; Sbordone, Carlo. 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. http://www.numdam.org/articles/10.1016/j.crma.2009.09.001/
[1] Beltrami operators in the plane, Duke Math. J., Volume 107 (2001), pp. 27-56
[2] Estimates for operator norms on weighted spaces and Reverse Jensen Inequalities, Trans. Amer. Math. Soc., Volume 340 (1993), pp. 253-272
[3] Weighted norm inequalities for maximal functions and singular integrals, Studia Math., Volume 51 (1974), pp. 241-250
[4] A Note on a weighted Sobolev inequality, Proc. Amer. Math. Soc., Volume 93 (1985), pp. 703-704
[5] The local regularity of solutions of degenerate elliptic equations, Comm. Partial Differentioal Equations, Volume 7 (1982), pp. 77-116
[6] Classical and Modern Fourier Analysis, Prentice-Hall, New Jersey, 2003
[7] On certain convolution inequalities, Proc. Amer. Math. Soc., Volume 36 (1972), pp. 505-510
[8] Sobolev Spaces, Springer, Berlin, 1985
[9] Weighted norm inequalities for the Hardy maximal function, Trans. Amer. Math. Soc., Volume 165 (1972), pp. 207-226
[10] Weighted norm inequalities for fractional integrals, Trans. Amer. Math. Soc., Volume 192 (1974), pp. 261-274
[11] The sharp bound for the Hilbert transform on weighted Lebesgue spaces in terms of the classical characteristic, Amer. J. Math., Volume 129 (2007), pp. 1355-1375
[12] The sharp weighted bound for the Riesz transforms, Proc. Amer. Math. Soc., Volume 136 (2008), pp. 1237-1249
[13] Heating of the Ahlfors–Beurling operator: Weakly quasiregular maps on the plane are quasiregular, Duke Math. J., Volume 112 (2002), pp. 281-305
[14] Harmonic Analysis: Real-Variable Methods, Orthogonality and Oscillatory Integrals, Princeton University Press, Princeton, 1993
Cited by Sources:
