Weighted local estimates for singular integral operators

Poelhuis, Jonathan; Torchinsky, Alberto

doi:10.1090/S0002-9947-2015-06459-9

Weighted local estimates for singular integral operators
HTML articles powered by AMS MathViewer

by Jonathan Poelhuis and Alberto Torchinsky PDF

Trans. Amer. Math. Soc. 367 (2015), 7957-7998 Request permission

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.

References

David R. Adams and Jie Xiao, Morrey spaces in harmonic analysis, Ark. Mat. 50 (2012), no. 2, 201–230. MR 2961318, DOI 10.1007/s11512-010-0134-0
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
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.
D. L. Burkholder and R. F. Gundy, Extrapolation and interpolation of quasi-linear operators on martingales, Acta Math. 124 (1970), 249–304. MR 440695, DOI 10.1007/BF02394573
A. P. Calderón, Estimates for singular integral operators in terms of maximal functions, Studia Math. 44 (1972), 563–582. MR 348555, DOI 10.4064/sm-44-6-563-582
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
Menita Carozza and Antonia Passarelli Di Napoli, Composition of maximal operators, Publ. Mat. 40 (1996), no. 2, 397–409. MR 1425627, DOI 10.5565/PUBLMAT_{4}0296_{1}1
R. R. Coifman, Distribution function inequalities for singular integrals, Proc. Nat. Acad. Sci. U.S.A. 69 (1972), 2838–2839. MR 303226, DOI 10.1073/pnas.69.10.2838
R. R. Coifman and C. Fefferman, Weighted norm inequalities for maximal functions and singular integrals, Studia Math. 51 (1974), 241–250. MR 358205, DOI 10.4064/sm-51-3-241-250
A. Cordoba and C. Fefferman, A weighted norm inequality for singular integrals, Studia Math. 57 (1976), no. 1, 97–101. MR 420115, DOI 10.4064/sm-57-1-97-101
M. Cotlar, Some generalizations of the Hardy-Littlewood maximal theorem, Rev. Mat. Cuyana 1 (1955), 85–104 (1956) (English, with Spanish summary). MR 88688
D. Cruz-Uribe, J. M. Martell, and C. 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, DOI 10.1016/j.aim.2007.05.022
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, DOI 10.1016/j.aim.2011.08.013
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, DOI 10.1007/s00020-013-2059-z
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
C. Fefferman and E. M. Stein, Some maximal inequalities, Amer. J. Math. 93 (1971), 107–115. MR 284802, DOI 10.2307/2373450
C. Fefferman and E. M. Stein, $H^{p}$ spaces of several variables, Acta Math. 129 (1972), no. 3-4, 137–193. MR 447953, DOI 10.1007/BF02392215
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, DOI 10.1090/S0002-9939-1989-0946637-8
Nobuhiko Fujii, A condition for a two-weight norm inequality for singular integral operators, Studia Math. 98 (1991), no. 3, 175–190. MR 1115188, DOI 10.4064/sm-98-3-175-190
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, DOI 10.1512/iumj.2001.50.2100
John B. Garnett and Peter W. Jones, BMO from dyadic BMO, Pacific J. Math. 99 (1982), no. 2, 351–371. MR 658065, DOI 10.2140/pjm.1982.99.351
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.
Vagif S. Guliyev, Seymur 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, DOI 10.1007/s00020-011-1904-1
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
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 312139, DOI 10.1090/S0002-9947-1973-0312139-8
Tuomas P. Hytönen, The sharp weighted bound for general Calderón-Zygmund operators, Ann. of Math. (2) 175 (2012), no. 3, 1473–1506. MR 2912709, DOI 10.4007/annals.2012.175.3.9
B. Jawerth and A. Torchinsky, Local sharp maximal functions, J. Approx. Theory 43 (1985), no. 3, 231–270. MR 779906, DOI 10.1016/0021-9045(85)90102-9
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, DOI 10.4064/sm-71-3-277-284
Douglas S. Kurtz and Richard L. Wheeden, Results on weighted norm inequalities for multipliers, Trans. Amer. Math. Soc. 255 (1979), 343–362. MR 542885, DOI 10.1090/S0002-9947-1979-0542885-8
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, DOI 10.1007/s00208-009-0473-y
A. K. Lerner, On the John-Strömberg characterization of BMO for nondoubling measures, Real Anal. Exchange 28 (2002/03), no. 2, 649–660. MR 2010347, DOI 10.14321/realanalexch.28.2.0649
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, DOI 10.1112/blms/bdq042
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.
Andrei K. Lerner, A simple proof of the $A_2$ conjecture, Int. Math. Res. Not. (2012) DOI 10.1093/imrn/rns145.
Andrei K. Lerner, On an estimate of Calderón-Zygmund operators by dyadic positive operators, preprint.
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, DOI 10.1007/s00041-005-4039-4
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
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 417671, DOI 10.4064/sm-55-3-279-294
F. Nazarov, S. Treil, and A. Volberg, The $Tb$-theorem on non-homogeneous spaces, Acta Math. 190 (2003), no. 2, 151–239. MR 1998349, DOI 10.1007/BF02392690
C. Pérez, Weighted norm inequalities for singular integral operators, J. London Math. Soc. (2) 49 (1994), no. 2, 296–308. MR 1260114, DOI 10.1112/jlms/49.2.296
Carlos Pérez, Two weighted inequalities for potential and fractional type maximal operators, Indiana Univ. Math. J. 43 (1994), no. 2, 663–683. MR 1291534, DOI 10.1512/iumj.1994.43.43028
C. 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, DOI 10.1112/plms/s3-71.1.135
Jonathan Poelhuis and Alberto Torchinsky, Medians, continuity, and vanishing oscillation, Studia Math. 213 (2012), no. 3, 227–242. MR 3024312, DOI 10.4064/sm213-3-3
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, DOI 10.1090/S0002-9939-2012-11474-1
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, DOI 10.1007/s10587-005-0032-y
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, DOI 10.1007/s10114-013-1639-9
M. Rosenthal and H. Triebel, Calderón-Zygmund operators in Morrey spaces, Rev. Mat. Complut. (2013), DOI 10.1007/s13163-013-0125-3.
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, DOI 10.1090/S0002-9947-2011-05294-3
Eric T. Sawyer, A characterization of a two-weight norm inequality for maximal operators, Studia Math. 75 (1982), no. 1, 1–11. MR 676801, DOI 10.4064/sm-75-1-1-11
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
Eric T. Sawyer, Norm inequalities relating singular integrals and the maximal function, Studia Math. 75 (1983), no. 3, 253–263. MR 722250, DOI 10.4064/sm-75-3-253-263
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
Sven Spanne, Some function spaces defined using the mean oscillation over cubes, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (3) 19 (1965), 593–608. MR 190729
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, DOI 10.1512/iumj.1979.28.28037
Jan-Olov Strömberg and Alberto Torchinsky, Weighted Hardy spaces, Lecture Notes in Mathematics, vol. 1381, Springer-Verlag, Berlin, 1989. MR 1011673, DOI 10.1007/BFb0091154
Alberto Torchinsky, Interpolation of operations and Orlicz classes, Studia Math. 59 (1976/77), no. 2, 177–207. MR 438105, DOI 10.4064/sm-59-2-177-207
Alberto Torchinsky, Real-variable methods in harmonic analysis, Pure and Applied Mathematics, vol. 123, Academic Press, Inc., Orlando, FL, 1986. MR 869816
Kôzô Yabuta, Sharp maximal function and $C_p$ condition, Arch. Math. (Basel) 55 (1990), no. 2, 151–155. MR 1064382, DOI 10.1007/BF01189135