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Extrapolation and weighted norm inequalities in the variable Lebesgue spaces


Authors: David Cruz-Uribe, SFO and Li-An Daniel Wang
Journal: Trans. Amer. Math. Soc. 369 (2017), 1205-1235
MSC (2010): Primary 42B25, 42B35
DOI: https://doi.org/10.1090/tran/6730
Published electronically: April 8, 2016
MathSciNet review: 3572271
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Abstract: We extend the theory of Rubio de Francia extrapolation, including off-diagonal, limited range and $ A_\infty $ extrapolation, to the weighted variable Lebesgue spaces $ L^{p(\cdot )}(w)$. As a consequence we are able to show that a number of different operators from harmonic analysis are bounded on these spaces. The proofs of our extrapolation results are developed in a way that outlines a general approach to proving extrapolation theorems on other Banach function spaces.


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  • [1] Kenneth F. Andersen and Russel T. John, Weighted inequalities for vector-valued maximal functions and singular integrals, Studia Math. 69 (1980/81), no. 1, 19-31. MR 604351 (82b:42015)
  • [2] Pascal Auscher, On necessary and sufficient conditions for $ L^p$-estimates of Riesz transforms associated to elliptic operators on $ \mathbb{R}^n$ and related estimates, Mem. Amer. Math. Soc. 186 (2007), no. 871, xviii+75. MR 2292385 (2007k:42025), https://doi.org/10.1090/memo/0871
  • [3] Pascal Auscher, Steve Hofmann, Michael Lacey, Alan McIntosh, and Ph. Tchamitchian, The solution of the Kato square root problem for second order elliptic operators on $ {\mathbb{R}}^n$, Ann. of Math. (2) 156 (2002), no. 2, 633-654. MR 1933726 (2004c:47096c), https://doi.org/10.2307/3597201
  • [4] Pascal Auscher and José María Martell, Weighted norm inequalities, off-diagonal estimates and elliptic operators. III. Harmonic analysis of elliptic operators, J. Funct. Anal. 241 (2006), no. 2, 703-746. MR 2271934 (2007g:42022), https://doi.org/10.1016/j.jfa.2006.07.008
  • [5] Pascal Auscher and José María Martell, Weighted norm inequalities, off-diagonal estimates and elliptic operators. I. General operator theory and weights, Adv. Math. 212 (2007), no. 1, 225-276. MR 2319768 (2008m:42015), https://doi.org/10.1016/j.aim.2006.10.002
  • [6] Ana Bernardis, Estefanía Dalmasso, and Gladis Pradolini, Generalized maximal functions and related operators on weighted Musielak-Orlicz spaces, Ann. Acad. Sci. Fenn. Math. 39 (2014), no. 1, 23-50. MR 3186804, https://doi.org/10.5186/aasfm.2014.3904
  • [7] B. Bongioanni, A. Cabral, and E. Harboure, Extrapolation for classes of weights related to a family of operators and applications, Potential Anal. 38 (2013), no. 4, 1207-1232. MR 3042701, https://doi.org/10.1007/s11118-012-9313-x
  • [8] B. Bongioanni, E. Harboure, and O. Salinas, Classes of weights related to Schrödinger operators, J. Math. Anal. Appl. 373 (2011), no. 2, 563-579. MR 2720705 (2011i:35034), https://doi.org/10.1016/j.jmaa.2010.08.008
  • [9] R. R. Coifman and C. Fefferman, Weighted norm inequalities for maximal functions and singular integrals, Studia Math. 51 (1974), 241-250. MR 0358205 (50 #10670)
  • [10] Michael Cowling, José García-Cuerva, and Hendra Gunawan, Weighted estimates for fractional maximal functions related to spherical means, Bull. Austral. Math. Soc. 66 (2002), no. 1, 75-90. MR 1922609 (2003e:42025), https://doi.org/10.1017/S0004972700020694
  • [11] D. Cruz-Uribe, L. Diening, and A. Fiorenza, A new proof of the boundedness of maximal operators on variable Lebesgue spaces, Boll. Unione Mat. Ital. (9) 2 (2009), no. 1, 151-173. MR 2493649 (2010i:42023)
  • [12] David Cruz-Uribe, Lars Diening, and Peter Hästö, The maximal operator on weighted variable Lebesgue spaces, Fract. Calc. Appl. Anal. 14 (2011), no. 3, 361-374. MR 2837636 (2012j:42039), https://doi.org/10.2478/s13540-011-0023-7
  • [13] David V. Cruz-Uribe and Alberto Fiorenza, Variable Lebesgue spaces, Applied and Numerical Harmonic Analysis, Birkhäuser/Springer, Heidelberg, 2013. Foundations and harmonic analysis. MR 3026953
  • [14] David Cruz-Uribe and Alberto Fiorenza, Introduction to the variable Lebesgue spaces, Variable Lebesgue spaces and hyperbolic systems, Adv. Courses Math. CRM Barcelona, Birkhäuser/Springer, Basel, 2014, pp. 1-90. MR 3364251
  • [15] D. Cruz-Uribe, A. Fiorenza, J. M. Martell, and C. Pérez, The boundedness of classical operators on variable $ L^p$ spaces, Ann. Acad. Sci. Fenn. Math. 31 (2006), no. 1, 239-264. MR 2210118 (2006m:42029)
  • [16] D. Cruz-Uribe, A. Fiorenza, and C. J. Neugebauer, Weighted norm inequalities for the maximal operator on variable Lebesgue spaces, J. Math. Anal. Appl. 394 (2012), no. 2, 744-760. MR 2927495, https://doi.org/10.1016/j.jmaa.2012.04.044
  • [17] David Cruz-Uribe, Eugenio Hernández, and José María Martell, Greedy bases in variable Lebesgue spaces, Monatsh. Math. 179 (2016), no. 3, 355-378. MR 3460475, https://doi.org/10.1007/s00605-015-0862-0
  • [18] 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 (2005g:42029), https://doi.org/10.1016/j.jfa.2003.09.002
  • [19] David V. Cruz-Uribe, José Maria Martell, and Carlos Pérez, Weights, extrapolation and the theory of Rubio de Francia, Operator Theory: Advances and Applications, vol. 215, Birkhäuser/Springer Basel AG, Basel, 2011. MR 2797562 (2012f:42001)
  • [20] David Cruz-Uribe and C. J. Neugebauer, The structure of the reverse Hölder classes, Trans. Amer. Math. Soc. 347 (1995), no. 8, 2941-2960. MR 1308005 (95m:42026), https://doi.org/10.2307/2154763
  • [21] D. Cruz-Uribe and L.-A. Wang, The structure of Muckenhoupt weights in the variable Lebesgue spaces,
    preprint.
  • [22] David Cruz-Uribe and Li-An Daniel Wang, Variable Hardy spaces, Indiana Univ. Math. J. 63 (2014), no. 2, 447-493. MR 3233216, https://doi.org/10.1512/iumj.2014.63.5232
  • [23] Lars Diening, Maximal function on Musielak-Orlicz spaces and generalized Lebesgue spaces, Bull. Sci. Math. 129 (2005), no. 8, 657-700 (English, with English and French summaries). MR 2166733 (2006e:46032), https://doi.org/10.1016/j.bulsci.2003.10.003
  • [24] Lars Diening, Petteri Harjulehto, Peter Hästö, and Michael R $ \overset {\circ }{\textup {u}}$žička, Lebesgue and Sobolev spaces with variable exponents, Lecture Notes in Mathematics, vol. 2017, Springer, Heidelberg, 2011. MR 2790542
  • [25] L. Diening and P. Hästö, Muckenhoupt weights in variable exponent spaces,
    preprint, 2010.
  • [26] L. Diening, P. Hästö, and S. Roudenko, Function spaces of variable smoothness and integrability, J. Funct. Anal. 256 (2009), no. 6, 1731-1768. MR 2498558 (2010b:46067), https://doi.org/10.1016/j.jfa.2009.01.017
  • [27] Javier Duoandikoetxea, Weighted norm inequalities for homogeneous singular integrals, Trans. Amer. Math. Soc. 336 (1993), no. 2, 869-880. MR 1089418 (93f:42030), https://doi.org/10.2307/2154381
  • [28] Javier Duoandikoetxea, Fourier analysis, Graduate Studies in Mathematics, vol. 29, American Mathematical Society, Providence, RI, 2001. Translated and revised from the 1995 Spanish original by David Cruz-Uribe. MR 1800316 (2001k:42001)
  • [29] Javier Duoandikoetxea, Adela Moyua, Osane Oruetxebarria, and Edurne Seijo, Radial $ A_p$ weights with applications to the disc multiplier and the Bochner-Riesz operators, Indiana Univ. Math. J. 57 (2008), no. 3, 1261-1281. MR 2429092 (2009d:42034), https://doi.org/10.1512/iumj.2008.57.3282
  • [30] Alberto Fiorenza, Amiran Gogatishvili, and Tengiz Kopaliani, Boundedness of Stein's spherical maximal function in variable Lebesgue spaces and application to the wave equation, Arch. Math. (Basel) 100 (2013), no. 5, 465-472. MR 3057132, https://doi.org/10.1007/s00013-013-0509-0
  • [31] A. Gogatishvili and T. Kopaliani, Extensions of Rubio de Francia's extrapolation theorem in variable Lebesgue space and application,
    preprint, 2014.
    arXiv 1407.5216v1.
  • [32] Loukas Grafakos, Modern Fourier analysis, 3rd ed., Graduate Texts in Mathematics, vol. 250, Springer, New York, 2014. MR 3243741
  • [33] Eleonor Harboure, Roberto A. Macías, and Carlos Segovia, Extrapolation results for classes of weights, Amer. J. Math. 110 (1988), no. 3, 383-397. MR 944321 (89i:47055), https://doi.org/10.2307/2374616
  • [34] R. Johnson and C. J. Neugebauer, Change of variable results for $ A_p$- and reverse Hölder $ {\rm RH}_r$-classes, Trans. Amer. Math. Soc. 328 (1991), no. 2, 639-666. MR 1018575 (92c:42019), https://doi.org/10.2307/2001798
  • [35] A. Lerner, On a dual property of the maximal operator on weighted variable $ L^p$ spaces, 2015, preprint. arXiv:1509.07664v2.
  • [36] Joaquín Motos, María Jesús Planells, and César F. Talavera, On variable exponent Lebesgue spaces of entire analytic functions, J. Math. Anal. Appl. 388 (2012), no. 2, 775-787. MR 2869787 (2012j:46039), https://doi.org/10.1016/j.jmaa.2011.09.069
  • [37] Benjamin Muckenhoupt, Weighted norm inequalities for the Hardy maximal function, Trans. Amer. Math. Soc. 165 (1972), 207-226. MR 0293384 (45 #2461)
  • [38] Benjamin Muckenhoupt and Richard Wheeden, Weighted norm inequalities for fractional integrals, Trans. Amer. Math. Soc. 192 (1974), 261-274. MR 0340523 (49 #5275)
  • [39] Elias M. Stein, Maximal functions. I. Spherical means, Proc. Nat. Acad. Sci. U.S.A. 73 (1976), no. 7, 2174-2175. MR 0420116 (54 #8133a)
  • [40] David K. Watson, Weighted estimates for singular integrals via Fourier transform estimates, Duke Math. J. 60 (1990), no. 2, 389-399. MR 1047758 (91b:42035), https://doi.org/10.1215/S0012-7094-90-06015-6

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Additional Information

David Cruz-Uribe, SFO
Affiliation: Department of Mathematics, Trinity College, Hartford, Connecticut 06106
Address at time of publication: Department of Mathematics, University of Alabama, Tuscaloosa, Alabama 35487
Email: dcruzuribe@ua.edu

Li-An Daniel Wang
Affiliation: Department of Mathematics and Statistics, Sam Houston State University, Huntsville, Texas 77341
Email: ldw@shsu.edu

DOI: https://doi.org/10.1090/tran/6730
Keywords: Variable Lebesgue spaces, weights, Muckenhoupt weights, maximal operator, singular integrals, fractional integrals, Rubio de Francia extrapolation
Received by editor(s): August 19, 2014
Received by editor(s) in revised form: March 26, 2015
Published electronically: April 8, 2016
Additional Notes: Both authors were supported by the Stewart-Dorwart faculty development fund at Trinity College, and the first author was also supported by NSF grant 1362425
Article copyright: © Copyright 2016 American Mathematical Society

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