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Weighted local estimates for singular integral operators


Authors: Jonathan Poelhuis and Alberto Torchinsky
Journal: Trans. Amer. Math. Soc. 367 (2015), 7957-7998
MSC (2010): Primary 42B20, 42B25
DOI: https://doi.org/10.1090/S0002-9947-2015-06459-9
Published electronically: February 19, 2015
MathSciNet review: 3391906
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Abstract: A local median decomposition is used to prove that a weighted mean of a function is controlled locally by the weighted mean of its local sharp maximal function. Together with the estimate $ M^{\sharp }_{0,s}(Tf)(x) \le c\,Mf(x)$ for Calderón-Zygmund singular integral operators, this allows us to express the local weighted control of $ Tf$ by $ Mf$. Similar estimates hold for $ T$ replaced by singular integrals with kernels satisfying Hörmander-type conditions or integral operators with homogeneous kernels, and $ M$ replaced by an appropriate maximal function $ M_T$. Using sharper bounds in the local median decomposition we prove two-weight, $ L^p_v-L^q_w$ estimates for the singular integral operators described above for $ 1<p\le q<\infty $ and a range of $ q$. The local nature of the estimates leads to results involving weighted generalized Orlicz-Campanato and Orlicz-Morrey spaces.


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  • [1] David R. Adams and Jie Xiao, Morrey spaces in harmonic analysis, Ark. Mat. 50 (2012), no. 2, 201-230. MR 2961318, https://doi.org/10.1007/s11512-010-0134-0
  • [2] J. Alvarez and C. Pérez, Estimates with $ A_\infty $ weights for various singular integral operators, Boll. Un. Mat. Ital. A (7) 8 (1994), no. 1, 123-133 (English, with Italian summary). MR 1273194 (95f:42027)
  • [3] T. C. Anderson and A. Vagharshakyan, A simple proof of the sharp weighted estimate for Calderón-Zygmund operators on homogenous spaces, J. Geom. Anal. DOI 10.1007/s12220-012-9372-7.
  • [4] D. L. Burkholder and R. F. Gundy, Extrapolation and interpolation of quasi-linear operators on martingales, Acta Math. 124 (1970), 249-304. MR 0440695 (55 #13567)
  • [5] A. P. Calderón, Estimates for singular integral operators in terms of maximal functions, Studia Math. 44 (1972), 563-582. Collection of articles honoring the completion by Antoni Zygmund of 50 years of scientific activity, VI. MR 0348555 (50 #1053)
  • [6] Lennart Carleson, BMO--10 years' development, 18th Scandinavian Congress of Mathematicians (Aarhus, 1980) Progr. Math., vol. 11, Birkhäuser, Boston, Mass., 1981, pp. 3-21. MR 633348 (82k:46043)
  • [7] Menita Carozza and Antonia Passarelli Di Napoli, Composition of maximal operators, Publ. Mat. 40 (1996), no. 2, 397-409. MR 1425627 (98f:42013), https://doi.org/10.5565/PUBLMAT_40296_11
  • [8] R. R. Coifman, Distribution function inequalities for singular integrals, Proc. Nat. Acad. Sci. U.S.A. 69 (1972), 2838-2839. MR 0303226 (46 #2364)
  • [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] A. Cordoba and C. Fefferman, A weighted norm inequality for singular integrals, Studia Math. 57 (1976), no. 1, 97-101. MR 0420115 (54 #8132)
  • [11] M. Cotlar, Some generalizations of the Hardy-Littlewood maximal theorem, Rev. Mat. Cuyana 1 (1955), 85-104 (1956) (English, with Spanish summary). MR 0088688 (19,564a)
  • [12] David Cruz-Uribe, José María Martell, and Carlos Pérez, Sharp two-weight inequalities for singular integrals, with applications to the Hilbert transform and the Sarason conjecture, Adv. Math. 216 (2007), no. 2, 647-676. MR 2351373 (2008k:42029), https://doi.org/10.1016/j.aim.2007.05.022
  • [13] David Cruz-Uribe, José María Martell, and Carlos Pérez, Sharp weighted estimates for classical operators, Adv. Math. 229 (2012), no. 1, 408-441. MR 2854179 (2012k:42020), https://doi.org/10.1016/j.aim.2011.08.013
  • [14] David Cruz-Uribe and Kabe Moen, A fractional Muckenhoupt-Wheeden theorem and its consequences, Integral Equations Operator Theory 76 (2013), no. 3, 421-446. MR 3065302, https://doi.org/10.1007/s00020-013-2059-z
  • [15] David Cruz-Uribe and Carlos Pérez, On the two-weight problem for singular integral operators, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 1 (2002), no. 4, 821-849. MR 1991004 (2004d:42021)
  • [16] C. Fefferman and E. M. Stein, Some maximal inequalities, Amer. J. Math. 93 (1971), 107-115. MR 0284802 (44 #2026)
  • [17] C. Fefferman and E. M. Stein, $ H^{p}$ spaces of several variables, Acta Math. 129 (1972), no. 3-4, 137-193. MR 0447953 (56 #6263)
  • [18] Nobuhiko Fujii, A proof of the Fefferman-Stein-Strömberg inequality for the sharp maximal functions, Proc. Amer. Math. Soc. 106 (1989), no. 2, 371-377. MR 946637 (91h:42022), https://doi.org/10.2307/2048815
  • [19] Nobuhiko Fujii, A condition for a two-weight norm inequality for singular integral operators, Studia Math. 98 (1991), no. 3, 175-190. MR 1115188 (92k:42022)
  • [20] José García-Cuerva and José María Martell, Two-weight norm inequalities for maximal operators and fractional integrals on non-homogeneous spaces, Indiana Univ. Math. J. 50 (2001), no. 3, 1241-1280. MR 1871355 (2003a:42023), https://doi.org/10.1512/iumj.2001.50.2100
  • [21] John B. Garnett and Peter W. Jones, BMO from dyadic BMO, Pacific J. Math. 99 (1982), no. 2, 351-371. MR 658065 (85d:42021)
  • [22] V. S. Guliyev and Sh. A.  Nazirova, Two-weighted inequalities for some sublinear operators in Lebesgue spaces, Khazar Journal of Mathematics 2 (2006), no. 1, 3-22.
  • [23] V. S. Guliyev, S. S. Aliyev, Turhan Karaman, and Parviz S. Shukurov, Boundedness of sublinear operators and commutators on generalized Morrey spaces, Integral Equations Operator Theory 71 (2011), no. 3, 327-355. MR 2852191 (2012i:42012), https://doi.org/10.1007/s00020-011-1904-1
  • [24] Guoen Hu, Dachun Yang, and Dongyong Yang, Boundedness of maximal singular integral operators on spaces of homogeneous type and its applications, J. Math. Soc. Japan 59 (2007), no. 2, 323-349. MR 2325688 (2008f:42014)
  • [25] Richard Hunt, Benjamin Muckenhoupt, and Richard Wheeden, Weighted norm inequalities for the conjugate function and Hilbert transform, Trans. Amer. Math. Soc. 176 (1973), 227-251. MR 0312139 (47 #701)
  • [26] Tuomas P. Hytönen, The sharp weighted bound for general Calderón-Zygmund operators, Ann. of Math. (2) 175 (2012), no. 3, 1473-1506. MR 2912709, https://doi.org/10.4007/annals.2012.175.3.9
  • [27] B. Jawerth and A. Torchinsky, Local sharp maximal functions, J. Approx. Theory 43 (1985), no. 3, 231-270. MR 779906 (86k:42034), https://doi.org/10.1016/0021-9045(85)90102-9
  • [28] R. A. Kerman and A. Torchinsky, Integral inequalities with weights for the Hardy maximal function, Studia Math. 71 (1981/82), no. 3, 277-284. MR 667316 (83k:42019)
  • [29] Douglas S. Kurtz and Richard L. Wheeden, Results on weighted norm inequalities for multipliers, Trans. Amer. Math. Soc. 255 (1979), 343-362. MR 542885 (81j:42021), https://doi.org/10.2307/1998180
  • [30] Michael T. Lacey, Stefanie Petermichl, and Maria Carmen Reguera, Sharp $ A_2$ inequality for Haar shift operators, Math. Ann. 348 (2010), no. 1, 127-141. MR 2657437 (2011i:42027), https://doi.org/10.1007/s00208-009-0473-y
  • [31] Andrei K. Lerner, On the John-Strömberg characterization of BMO for nondoubling measures, Real Anal. Exchange 28 (2002/03), no. 2, 649-660. MR 2010347 (2004m:42021)
  • [32] Andrei K. Lerner, A pointwise estimate for the local sharp maximal function with applications to singular integrals, Bull. Lond. Math. Soc. 42 (2010), no. 5, 843-856. MR 2721744 (2012a:42022), https://doi.org/10.1112/blms/bdq042
  • [33] Andrei K. Lerner, A ``local mean oscillation'' decomposition and some of its applications, Function Spaces, Approximation, Inequalities and Lineability, Lectures of the Spring School in Analysis, Matfyzpres, Prague (2011), pp. 71-106.
  • [34] Andrei K. Lerner, A simple proof of the $ A_2$ conjecture, Int. Math. Res. Not. (2012) DOI 10.1093/imrn/rns145.
  • [35] Andrei K. Lerner, On an estimate of Calderón-Zygmund operators by dyadic positive operators, preprint.
  • [36] M. Lorente, M. S. Riveros, and A. de la Torre, Weighted estimates for singular integral operators satisfying Hörmander's conditions of Young type, J. Fourier Anal. Appl. 11 (2005), no. 5, 497-509. MR 2182632 (2006g:42024), https://doi.org/10.1007/s00041-005-4039-4
  • [37] Benjamin Muckenhoupt, Norm inequalities relating the Hilbert transform to the Hardy-Littlewood maximal function, Functional analysis and approximation (Oberwolfach, 1980) Internat. Ser. Numer. Math., vol. 60, Birkhäuser, Basel-Boston, Mass., 1981, pp. 219-231. MR 650277 (84h:42031)
  • [38] Benjamin Muckenhoupt and Richard L. Wheeden, Two weight function norm inequalities for the Hardy-Littlewood maximal function and the Hilbert transform, Studia Math. 55 (1976), no. 3, 279-294. MR 0417671 (54 #5720)
  • [39] F. Nazarov, S. Treil, and A. Volberg, The $ Tb$-theorem on non-homogeneous spaces, Acta Math. 190 (2003), no. 2, 151-239. MR 1998349 (2005d:30053), https://doi.org/10.1007/BF02392690
  • [40] Carlos Pérez, Weighted norm inequalities for singular integral operators, J. London Math. Soc. (2) 49 (1994), no. 2, 296-308. MR 1260114 (94m:42037), https://doi.org/10.1112/jlms/49.2.296
  • [41] Carlos Pérez, Two weighted inequalities for potential and fractional type maximal operators, Indiana Univ. Math. J. 43 (1994), no. 2, 663-683. MR 1291534 (95m:42028), https://doi.org/10.1512/iumj.1994.43.43028
  • [42] Carlos Pérez, On sufficient conditions for the boundedness of the Hardy-Littlewood maximal operator between weighted $ L^p$-spaces with different weights, Proc. London Math. Soc. (3) 71 (1995), no. 1, 135-157. MR 1327936 (96k:42023), https://doi.org/10.1112/plms/s3-71.1.135
  • [43] Jonathan Poelhuis and Alberto Torchinsky, Medians, continuity, and vanishing oscillation, Studia Math. 213 (2012), no. 3, 227-242. MR 3024312, https://doi.org/10.4064/sm213-3-3
  • [44] Maria Carmen Reguera and James Scurry, On joint estimates for maximal functions and singular integrals on weighted spaces, Proc. Amer. Math. Soc. 141 (2013), no. 5, 1705-1717. MR 3020857, https://doi.org/10.1090/S0002-9939-2012-11474-1
  • [45] María Silvina Riveros and Marta Urciuolo, Weighted inequalities for integral operators with some homogeneous kernels, Czechoslovak Math. J. 55(130) (2005), no. 2, 423-432. MR 2137148 (2006a:42027), https://doi.org/10.1007/s10587-005-0032-y
  • [46] María Silvina Riveros and Marta Urciuolo, Weighted inequalities for fractional type operators with some homogeneous kernels, Acta Math. Sin. (Engl. Ser.) 29 (2013), no. 3, 449-460. MR 3019784, https://doi.org/10.1007/s10114-013-1639-9
  • [47] M. Rosenthal and H. Triebel, Calderón-Zygmund operators in Morrey spaces, Rev. Mat. Complut. (2013), DOI 10.1007/s13163-013-0125-3.
  • [48] Yoshihiro Sawano, Satoko Sugano, and Hitoshi Tanaka, Generalized fractional integral operators and fractional maximal operators in the framework of Morrey spaces, Trans. Amer. Math. Soc. 363 (2011), no. 12, 6481-6503. MR 2833565 (2012f:42037), https://doi.org/10.1090/S0002-9947-2011-05294-3
  • [49] Eric T. Sawyer, A characterization of a two-weight norm inequality for maximal operators, Studia Math. 75 (1982), no. 1, 1-11. MR 676801 (84i:42032)
  • [50] Eric T. Sawyer, Two weight norm inequalities for certain maximal and integral operators, Harmonic analysis (Minneapolis, Minn., 1981) Lecture Notes in Math., vol. 908, Springer, Berlin-New York, 1982, pp. 102-127. MR 654182 (83k:42020b)
  • [51] Eric T. Sawyer, Norm inequalities relating singular integrals and the maximal function, Studia Math. 75 (1983), no. 3, 253-263. MR 722250 (85c:42018)
  • [52] Xian Liang Shi and Alberto Torchinsky, Local sharp maximal functions in spaces of homogeneous type, Sci. Sinica Ser. A 30 (1987), no. 5, 473-480. MR 1000919 (90f:42019)
  • [53] Sven Spanne, Some function spaces defined using the mean oscillation over cubes, Ann. Scuola Norm. Sup. Pisa (3) 19 (1965), 593-608. MR 0190729 (32 #8140)
  • [54] Jan-Olov Strömberg, Bounded mean oscillation with Orlicz norms and duality of Hardy spaces, Indiana Univ. Math. J. 28 (1979), no. 3, 511-544. MR 529683 (81f:42021), https://doi.org/10.1512/iumj.1979.28.28037
  • [55] Jan-Olov Strömberg and Alberto Torchinsky, Weighted Hardy spaces, Lecture Notes in Mathematics, vol. 1381, Springer-Verlag, Berlin, 1989. MR 1011673 (90j:42053)
  • [56] Alberto Torchinsky, Interpolation of operations and Orlicz classes, Studia Math. 59 (1976/77), no. 2, 177-207. MR 0438105 (55 #11024)
  • [57] Alberto Torchinsky, Real-variable methods in harmonic analysis, Pure and Applied Mathematics, vol. 123, Academic Press, Inc., Orlando, FL, 1986. MR 869816 (88e:42001)
  • [58] Kôzô Yabuta, Sharp maximal function and $ C_p$ condition, Arch. Math. (Basel) 55 (1990), no. 2, 151-155. MR 1064382 (91i:42017), https://doi.org/10.1007/BF01189135

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Additional Information

Jonathan Poelhuis
Affiliation: Department of Mathematics, Indiana University, Bloomington, Indiana 47405
Email: jpoelhui@indiana.edu

Alberto Torchinsky
Affiliation: Department of Mathematics, Indiana University, Bloomington, Indiana 47405
Email: torchins@indiana.edu

DOI: https://doi.org/10.1090/S0002-9947-2015-06459-9
Received by editor(s): August 21, 2013
Published electronically: February 19, 2015
Dedicated: In remembrance of Björn Jawerth (1952-2013) who believed in local sharp maximal functions
Article copyright: © Copyright 2015 American Mathematical Society

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