Weighted norm inequalities for homogeneous singular integrals

Duoandikoetxea, Javier

doi:10.1090/S0002-9947-1993-1089418-5

Weighted norm inequalities for homogeneous singular integrals
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by Javier Duoandikoetxea

Trans. Amer. Math. Soc. 336 (1993), 869-880

DOI: https://doi.org/10.1090/S0002-9947-1993-1089418-5

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Abstract:

We prove weighted norm inequalities for homogeneous singular integrals when only a size condition is assumed on the restriction of the kernel to the unit sphere. The same results hold for the operator obtained by modifying the centered Hardy-Littlewood maximal operator over balls with a degree zero homogeneous function and also for the maximal singular integral.

References

Jöran Bergh and Jörgen Löfström, Interpolation spaces. An introduction, Grundlehren der Mathematischen Wissenschaften, No. 223, Springer-Verlag, Berlin-New York, 1976. MR 0482275
A. P. Calderón and A. Zygmund, On singular integrals, Amer. J. Math. 78 (1956), 289–309. MR 84633, DOI 10.2307/2372517
Osvaldo N. Capri and Carlos Segovia, Behaviour of $L^r$-Dini singular integrals in weighted $L^1$ spaces, Studia Math. 92 (1989), no. 1, 21–36. MR 984847, DOI 10.4064/sm-92-1-21-36

Estimates for the Kakeya maximal operator on radial functions in

Michael Christ, Weak type $(1,1)$ bounds for rough operators, Ann. of Math. (2) 128 (1988), no. 1, 19–42. MR 951506, DOI 10.2307/1971461
Michael Christ and José Luis Rubio de Francia, Weak type $(1,1)$ bounds for rough operators. II, Invent. Math. 93 (1988), no. 1, 225–237. MR 943929, DOI 10.1007/BF01393693
Javier Duoandikoetxea, Multiple singular integrals and maximal functions along hypersurfaces, Ann. Inst. Fourier (Grenoble) 36 (1986), no. 4, 185–206 (English, with French summary). MR 867920
Javier Duoandikoetxea and José L. Rubio de Francia, Maximal and singular integral operators via Fourier transform estimates, Invent. Math. 84 (1986), no. 3, 541–561. MR 837527, DOI 10.1007/BF01388746
Javier Duoandikoetxea and José L. Rubio de Francia, Estimations indépendantes de la dimension pour les transformées de Riesz, C. R. Acad. Sci. Paris Sér. I Math. 300 (1985), no. 7, 193–196 (French, with English summary). MR 780616
R. Fefferman, A note on singular integrals, Proc. Amer. Math. Soc. 74 (1979), no. 2, 266–270. MR 524298, DOI 10.1090/S0002-9939-1979-0524298-3
José García-Cuerva and José L. Rubio de Francia, Weighted norm inequalities and related topics, North-Holland Mathematics Studies, vol. 116, North-Holland Publishing Co., Amsterdam, 1985. Notas de Matemática [Mathematical Notes], 104. MR 807149
Steve Hofmann, Weak $(1,1)$ boundedness of singular integrals with nonsmooth kernel, Proc. Amer. Math. Soc. 103 (1988), no. 1, 260–264. MR 938680, DOI 10.1090/S0002-9939-1988-0938680-9
Steve Hofmann, Weighted weak-type $(1,1)$ inequalities for rough operators, Proc. Amer. Math. Soc. 107 (1989), no. 2, 423–435. MR 984795, DOI 10.1090/S0002-9939-1989-0984795-X
Douglas S. Kurtz, Littlewood-Paley and multiplier theorems on weighted $L^{p}$ spaces, Trans. Amer. Math. Soc. 259 (1980), no. 1, 235–254. MR 561835, DOI 10.1090/S0002-9947-1980-0561835-X
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
Douglas S. Kurtz and Richard L. Wheeden, A note on singular integrals with weights, Proc. Amer. Math. Soc. 81 (1981), no. 3, 391–397. MR 597648, DOI 10.1090/S0002-9939-1981-0597648-9
Benjamin Muckenhoupt and Richard L. Wheeden, Weighted norm inequalities for singular and fractional integrals, Trans. Amer. Math. Soc. 161 (1971), 249–258. MR 285938, DOI 10.1090/S0002-9947-1971-0285938-7
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
Elias M. Stein, Singular integrals and differentiability properties of functions, Princeton Mathematical Series, No. 30, Princeton University Press, Princeton, N.J., 1970. MR 0290095
David K. Watson, Weighted estimates for singular integrals via Fourier transform estimates, Duke Math. J. 60 (1990), no. 2, 389–399. MR 1047758, DOI 10.1215/S0012-7094-90-06015-6