Rubio de Francia’s extrapolation theorem for 𝐵_{𝑝} weights

Carro, María; Lorente, María

doi:10.1090/S0002-9939-09-10040-0

Rubio de Francia’s extrapolation theorem for $B_p$ weights
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by María J. Carro and María Lorente PDF

Proc. Amer. Math. Soc. 138 (2010), 629-640 Request permission

Abstract:

In this paper, we prove some of Rubio de Francia’s extrapolation results for the class $B_p$ of weights for which the Hardy operator is bounded on $L^p(w)$ restricted to decreasing functions. Applications to the boundedness of operators on $L^p_{\textrm {dec}}(w)$ are given. We also present an extension to the $B_\infty$ case and some connections with classical $A_p$ theory.

References

Miguel A. Ariño and Benjamin Muckenhoupt, Maximal functions on classical Lorentz spaces and Hardy’s inequality with weights for nonincreasing functions, Trans. Amer. Math. Soc. 320 (1990), no. 2, 727–735. MR 989570, DOI 10.1090/S0002-9947-1990-0989570-0
Jesús Bastero, Mario Milman, and Francisco J. Ruiz, On the connection between weighted norm inequalities, commutators and real interpolation, Mem. Amer. Math. Soc. 154 (2001), no. 731, viii+80. MR 1848159, DOI 10.1090/memo/0731
María J. Carro, Amiran Gogatishvili, Joaquim Martín, and Luboš Pick, Functional properties of rearrangement invariant spaces defined in terms of oscillations, J. Funct. Anal. 229 (2005), no. 2, 375–404. MR 2182593, DOI 10.1016/j.jfa.2005.06.012
María J. Carro, José A. Raposo, and Javier Soria, Recent developments in the theory of Lorentz spaces and weighted inequalities, Mem. Amer. Math. Soc. 187 (2007), no. 877, xii+128. MR 2308059, DOI 10.1090/memo/0877
María J. Carro and Javier Soria, Boundedness of some integral operators, Canad. J. Math. 45 (1993), no. 6, 1155–1166. MR 1247539, DOI 10.4153/CJM-1993-064-2
David Cruz-Uribe, José María Martell, and Carlos Pérez, Extensions of Rubio de Francia’s extrapolation theorem, Collect. Math. Vol. Extra (2006), 195–231. MR 2264210
D. Cruz-Uribe, J. M. Martell, and C. Pérez, Extrapolation from $A_\infty$ weights and applications, J. Funct. Anal. 213 (2004), no. 2, 412–439. MR 2078632, DOI 10.1016/j.jfa.2003.09.002
Guillermo P. Curbera, José García-Cuerva, José María Martell, and Carlos Pérez, Extrapolation with weights, rearrangement-invariant function spaces, modular inequalities and applications to singular integrals, Adv. Math. 203 (2006), no. 1, 256–318. MR 2231047, DOI 10.1016/j.aim.2005.04.009
Shanzhong Lai, Weighted norm inequalities for general operators on monotone functions, Trans. Amer. Math. Soc. 340 (1993), no. 2, 811–836. MR 1132877, DOI 10.1090/S0002-9947-1993-1132877-X
José L. Rubio de Francia, Factorization theory and $A_{p}$ weights, Amer. J. Math. 106 (1984), no. 3, 533–547. MR 745140, DOI 10.2307/2374284