Weighted norm inequalities for the continuous square function

Wilson, J. Michael

doi:10.1090/S0002-9947-1989-0972707-9

Weighted norm inequalities for the continuous square function

Author: J. Michael Wilson
Journal: Trans. Amer. Math. Soc. 314 (1989), 661-692
MSC: Primary 42B20
DOI: https://doi.org/10.1090/S0002-9947-1989-0972707-9
Erratum: Trans. Amer. Math. Soc. 321 (1990), null.
MathSciNet review: 972707
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We prove new weighted norm inequalities for real-variable analogues of the Lusin area function. We apply our results to obtain new: (i) weighted norm inequalities for singular integral operators; (ii) weighted Sobolev inequalities; (iii) eigenvalue estimates for degenerate Schrödinger operators.

[CWW] S.-Y. A. Chang, J. M. Wilson, and T. H. Wolff, Some weighted norm inequalities concerning the Schrödinger operators, Comment. Math. Helv. 60 (1985), no. 2, 217–246. MR 800004, https://doi.org/10.1007/BF02567411
[CW1] S. Chanillo and R. L. Weeden, ${L^p}$ estimates for fractional integrals and Sobolev inequalities, with applications to Schrödinger operators, preprint (1985).
[CW2] Sagun Chanillo and Richard L. Wheeden, Some weighted norm inequalities for the area integral, Indiana Univ. Math. J. 36 (1987), no. 2, 277–294. MR 891775, https://doi.org/10.1512/iumj.1987.36.36016
[F] Charles L. Fefferman, The uncertainty principle, Bull. Amer. Math. Soc. (N.S.) 9 (1983), no. 2, 129–206. MR 707957, https://doi.org/10.1090/S0273-0979-1983-15154-6
[FS1] C. Fefferman and E. M. Stein, Some maximal inequalities, Amer. J. Math. 93 (1971), 107–115. MR 0284802, https://doi.org/10.2307/2373450
[FS2] C. Fefferman and E. M. Stein, 𝐻^{𝑝} spaces of several variables, Acta Math. 129 (1972), no. 3-4, 137–193. MR 0447953, https://doi.org/10.1007/BF02392215
[RF] Robert Fefferman, Harmonic analysis on product spaces, Ann. of Math. (2) 126 (1987), no. 1, 109–130. MR 898053, https://doi.org/10.2307/1971346
[G] John B. Garnett, Bounded analytic functions, Pure and Applied Mathematics, vol. 96, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1981. MR 628971
[GJ] John B. Garnett and Peter W. Jones, The distance in BMO to 𝐿^{∞}, Ann. of Math. (2) 108 (1978), no. 2, 373–393. MR 506992, https://doi.org/10.2307/1971171
[KS] R. Kerman and E. T. Sawyer, Weighted norm inequalities for potentials with applications to Schrödinger operators, Fourier transforms and Carleson measures, preprint (1984).
[K] 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, https://doi.org/10.1090/S0002-9947-1980-0561835-X
[M] Benjamin Muckenhoupt, Weighted norm inequalities for classical operators, Harmonic analysis in Euclidean spaces (Proc. Sympos. Pure Math., Williams Coll., Williamstown, Mass., 1978) Proc. Sympos. Pure Math., XXXV, Part, Amer. Math. Soc., Providence, R.I., 1979, pp. 69–83. MR 545240
[Sch] M. Schechter, The spectrum of the Schrödinger operator, preprint (1987).
[St] Elias M. Stein, Singular integrals and differentiability properties of functions, Princeton Mathematical Series, No. 30, Princeton University Press, Princeton, N.J., 1970. MR 0290095
[U] Akihito Uchiyama, The Fefferman-Stein decomposition of smooth functions and its application to 𝐻^{𝑝}(𝑅ⁿ), Pacific J. Math. 115 (1984), no. 1, 217–255. MR 762212
[W1] J. Michael Wilson, Weighted inequalities for the dyadic square function without dyadic 𝐴_{∞}, Duke Math. J. 55 (1987), no. 1, 19–50. MR 883661, https://doi.org/10.1215/S0012-7094-87-05502-5
[W2] J. Michael Wilson, A sharp inequality for the square function, Duke Math. J. 55 (1987), no. 4, 879–887. MR 916125, https://doi.org/10.1215/S0012-7094-87-05542-6
[W3] -, ${L^p}$ weighted norm inequalities for the square function, , Illinois J. Math, (to appear).