Norm estimates and representations for Calderón-Zygmund operators using averages over starlike sets

Watson, David; Wheeden, Richard

doi:10.1090/S0002-9947-99-02313-2

Norm estimates and representations for Calderón-Zygmund operators using averages over starlike sets
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by David K. Watson and Richard L. Wheeden

Trans. Amer. Math. Soc. 351 (1999), 4127-4171

DOI: https://doi.org/10.1090/S0002-9947-99-02313-2

Published electronically: July 1, 1999

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Abstract:

We show that homogeneous singular integrals may be represented in terms of averages over starlike sets. This permits us to use the geometry of starlike sets to derive operator-specific weighted norm inequalities.

References

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, DOI 10.4064/sm-69-1-19-31
Cahit Arf, Untersuchungen über reinverzweigte Erweiterungen diskret bewerteter perfekter Körper, J. Reine Angew. Math. 181 (1939), 1–44 (German). MR 18, DOI 10.1515/crll.1940.181.1
Sagun Chanillo, David K. Watson, and Richard L. Wheeden, Some integral and maximal operators related to starlike sets, Studia Math. 107 (1993), no. 3, 223–255. MR 1247201, DOI 10.4064/sm-107-3-223-255
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
Javier Duoandikoetxea, Weighted norm inequalities for homogeneous singular integrals, Trans. Amer. Math. Soc. 336 (1993), no. 2, 869–880. MR 1089418, DOI 10.1090/S0002-9947-1993-1089418-5
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
C. Fefferman and E. M. Stein, Some maximal inequalities, Amer. J. Math. 93 (1971), 107–115. MR 284802, DOI 10.2307/2373450
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, Weighted norm inequalities and vector valued inequalities for certain rough operators, Indiana Univ. Math. J. 42 (1993), no. 1, 1–14. MR 1218703, DOI 10.1512/iumj.1993.42.42001
S. Hofmann, D. K. Watson, Erratum to Steve Hofmann’s paper, “Weighted norm inequalities and vector-valued inequalities for certain rough operators”, to appear.
Peter W. Jones, Factorization of $A_{p}$ weights, Ann. of Math. (2) 111 (1980), no. 3, 511–530. MR 577135, DOI 10.2307/1971107
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
Elias M. Stein, Singular integrals and differentiability properties of functions, Princeton Mathematical Series, No. 30, Princeton University Press, Princeton, N.J., 1970. MR 0290095
Saunders MacLane and O. F. G. Schilling, Infinite number fields with Noether ideal theories, Amer. J. Math. 61 (1939), 771–782. MR 19, DOI 10.2307/2371335
Elias M. Stein and Guido Weiss, Introduction to Fourier analysis on Euclidean spaces, Princeton Mathematical Series, No. 32, Princeton University Press, Princeton, N.J., 1971. MR 0304972
H. Triebel, Interpolation theory, function spaces, differential operators, VEB Deutscher Verlag der Wissenschaften, Berlin, 1978. MR 500580
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
—, ‘Dual’ weighted bounds for maximal operators from smoothness or Fourier transform estimates, preprint.
—, $A_{1}$ weights and weak type (1, 1) estimates for rough operators, preprint.
David K. Watson, Vector-valued inequalities, factorization, and extrapolation for a family of rough operators, J. Funct. Anal. 121 (1994), no. 2, 389–415. MR 1272132, DOI 10.1006/jfan.1994.1053
A. Zygmund, Trigonometric series. 2nd ed. Vols. I, II, Cambridge University Press, New York, 1959. MR 0107776