On the two-weight problem for singular integral operators

Cruz-Uribe, David; Pérez, Carlos

Cruz-Uribe, David; Pérez, Carlos

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Serie 5, Volume 1 (2002) no. 4, p. 821-849

Abstract

We give $A_{p}$ type conditions which are sufficient for two-weight, strong $(p, p)$ inequalities for Calderón-Zygmund operators, commutators, and the Littlewood-Paley square function $g_{λ}^{*}$ . Our results extend earlier work on weak $(p, p)$ inequalities in [13].

MR 1991004 | Zbl 1072.42010

Classification: 42B20, 42B25

@article{ASNSP_2002_5_1_4_821_0,
     author = {Cruz-Uribe, David and P\'erez, Carlos},
     title = {On the two-weight problem for singular integral operators},
     journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze},
     publisher = {Scuola normale superiore},
     volume = {Ser. 5, 1},
     number = {4},
     year = {2002},
     pages = {821-849},
     zbl = {1072.42010},
     mrnumber = {1991004},
     language = {en},
     url = {http://www.numdam.org/item/ASNSP_2002_5_1_4_821_0}
}

Cruz-Uribe, David; Pérez, Carlos. On the two-weight problem for singular integral operators. Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Serie 5, Volume 1 (2002) no. 4, pp. 821-849. http://www.numdam.org/item/ASNSP_2002_5_1_4_821_0/

References

[1] D. Cruz-Uribe, SFO - A. Fiorenza | MR 1876947 | Zbl 1041.42009

[2] D. Cruz-Uribe, SFO - C. Pérez, Sharp two-weight, weak-type norm inequalities for singular integral operators, Math. Res. Let. 6 (1999), 417-428. | MR 1713140 | Zbl 0961.42013

[3] D. Cruz-Uribe, SFO - C. Pérez, Two-weight, weak-type norm inequalities for fractional integrals, Calderón-Zygmund operators and commutators, Indiana Math. J. 49 (2000), 697-721. | MR 1793688 | Zbl 1033.42009

[4] Duoandikoetxea, J., “Fourier Analysis”, Grad. Studies Math. 29, Amer. Math. Soc., Providence, 2000. | MR 1800316 | Zbl 0969.42001

[5] C. Fefferman, Inequalities for strongly singular convolution operators, Acta Math. 124 (1970), 9-36. | MR 257819 | Zbl 0188.42601

[6] N. Fujii, A condition for a two-weight norm inequality for singular integral operators, Studia Math. 98 (1991), 175-190. | MR 1115188 | Zbl 0732.42012

[7] J. García-Cuerva - E. Harboure - C. Segovia - J. L. Torrea, Weighted norm inequalities for commutators of strongly singular integrals, Indiana Univ. Math. J. 40 (1991), 1398-1420. | MR 1142721 | Zbl 0765.42012

[8] J. García-Cuerva - J. L. Rubio De Francia, “Weighted Norm Inequalities and Related Topics”, North Holland Math. Studies 116, North Holland, Amsterdam, 1985. | MR 848136 | Zbl 0578.46046

[9] L. Hörmander, Pseudo-differential operators and hypoelliptic equations, Proc. Symp. Pure Math. 10 (1967), 138-183. | MR 383152 | Zbl 0167.09603

[10] S. Janson, Mean oscillation and commutators of singular integrals, Ark. Mat. 16 (1978), 263-270. | MR 524754 | Zbl 0404.42013

[11] J.-L. Journé, “Zygmund Operators, Pseudo-Differential Operators and the Cauchy Integral of Calderón”, Lecture Notes in Mathematics, 994, Springer Verlag, Berlin, 1983. | Zbl 0508.42021

[12] N. H. Katz - C. Pereyra, On the two weights problem for the Hilbert transform, Rev. Mat. Iberoamericana 13 (1997), 189-210. | MR 1462332 | Zbl 0908.49029

[13] M. A. Leckband, Structure results on the maximal Hilbert transform and two-weight norm inequalities, Indiana Math. J. 34 (1985), 259-275. | MR 783915 | Zbl 0586.42010

[14] B. Muckenhoupt, Weighted norm inequalities for the Hardy-Littlewood maximal function, Trans. Amer. Math. Soc. 165 (1972), 207-226. | MR 293384 | Zbl 0236.26016

[15] B. Muckenhoupt - R. L. Wheeden, Norm inequalities for Littlewood-Paley function $g_{λ}^{*}$ , Trans. Amer. Math. Soc. 191 (1974), 95-111. | MR 387973 | Zbl 0289.44005

[16] B. Muckenhoupt - R. Wheeden, Two weight function norm inequalities for the Hardy-Littlewood maximal function and the Hilbert transform, Studia Math. 60 (1976), 279-294. | MR 417671 | Zbl 0336.44006

[17] C. J. Neugebauer, Inserting $A_{p}$ -weights, Proc. Amer. Math. Soc. 87 (1983), 644-648. | MR 687633 | Zbl 0521.42019

[18] R. O'Neil, Fractional integration in Orlicz spaces, Trans. Amer. Math. Soc. 115 (1965), 300-328. | MR 194881 | Zbl 0132.09201

[19] C. Pérez, Two weighted inequalities for potential and fractional type maximal operators, Indiana Math. J. 43 (1994), 663-683. | MR 1291534 | Zbl 0809.42007

[20] 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. 71 (1995), 135-57. | MR 1327936 | Zbl 0829.42019

[21] C. Pérez, Endpoint estimates for commutators of singular integral operators, J. Func. Anal. 128 (1995), 163-185. | MR 1317714 | Zbl 0831.42010

[22] C. Pérez, Sharp estimates for commutators of singular integrals via iterations of the Hardy-Littlewood maximal function, J. Fourier Anal. Appl. 3 (6) (1997), 743-756. | MR 1481632 | Zbl 0894.42006

[23] C. Pérez - G. Pradolini, Sharp weighted endpoint estimates for commutators of singular integral operators, Michigan Math. J. 49 (2001), 23-37. | MR 1827073 | Zbl 1010.42007

[24] C. Pérez - R. Wheeden, Uncertainty principle estimates for vector fields, J. Func. Anal. 181 (2001), 146-188. | MR 1818113 | Zbl 0982.42010

[25] D. H. Phong - E. M. Stein, Hilbert integrals, singular integrals and Radon transforms, Acta Math. 157 (1985), 99-157. | MR 857680 | Zbl 0622.42011

[26] M. M. Rao - Z. D. Ren, “Theory of Orlicz Spaces”, Marcel Dekker, New York, 1991. | MR 1113700 | Zbl 0724.46032

[27] Y. Rakotondratsimba, Two weight norm inequality for Calderón-Zygmund operators, Acta Math. Hungar. 80 (1998), 39-54. | MR 1624522 | Zbl 0914.47007

[28] Y. Rakotondratsimba, Two-weight inequality for commutators of singular integral operators, Kobe J. Math. 16 (1999), 1-20. | MR 1723543 | Zbl 0940.42011

[29] J. L. Rubio De Francia - F. J. Ruiz - J. L. Torrea, Calderón-Zygmund theory for operator-valued kernels, Adv. in Math. 62, (1986), 7-48. | MR 859252 | Zbl 0627.42008

[30] E. T. Sawyer, A characterization of a two weight norm weight inequality for maximal operators, Studia Math. 75 (1982), 1-11. | MR 676801 | Zbl 0508.42023

[31] C. Segovia - J. L. Torrea, Higher order commutators for vector-valued Calderón-Zygmund operators, Trans. Amer. Math. Soc. 336 (1993), 537-556. | MR 1074151 | Zbl 0799.42009

[32] E. M. Stein, Note on the class L logL, Studia Math. 32 (1969), 305-310. | MR 247534 | Zbl 0182.47803

[33] E. M. Stein, “Singular Integrals and Differentiability Properties of Functions”, Princeton University Press, Princeton, 1970. | MR 290095 | Zbl 0207.13501

[34] J.-O. Strömberg - A. Torchinsky, “Weighted Hardy Spaces”, Lecture Notes in Mathematics, 1381, Springer Verlag, Berlin, 1989. | MR 1011673 | Zbl 0676.42021

[35] A. Torchinsky, “Real-Variable Methods in Harmonic Analysis”, Academic Press, New York, 1986. | MR 869816 | Zbl 0621.42001

[36] S. Treil - A. Volberg - D. Zheng, Hilbert Transform 13 (1997), 319-360. | MR 1617653 | Zbl 0896.42009

[37] G. Weiss, A note on Orlicz spaces, Portugal. Math. 15 (1950), 35-47. | MR 82645 | Zbl 0071.33001

[38] J. M. Wilson, Weighted norm inequalities for the continuous square functions, Trans. Amer. Math. Soc. 314 (1989), 661-692. | MR 972707 | Zbl 0689.42016