Abstract.
We present a framework that yields a variety of weighted and vector-valued estimates for maximally modulated Calderón-Zygmund singular (and maximal singular) integrals from a single a priori weak type unweighted estimate for the maximal modulations of such operators. We discuss two approaches, one based on the good-λ method of Coifman and Fefferman [CF] and an alternative method employing the sharp maximal operator. As an application we obtain new weighted and vector-valued inequalities for the Carleson operator proving that it is controlled by a natural maximal function associated with the Orlicz space L(log L)(log log log L). This control is in the sense of a good-λ inequality and yields strong and weak type estimates as well as vector-valued and weighted estimates for the operator in question.
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References
Antonov, N.Y.: Convergence of Fourier series. East J. Approx. 2, 187–196 (1996)
Arias de Reyna, J.: Pointwise convergence of Fourier series. J. London Math. Soc. (2) 65, 139–153 (2002)
Bennett, C., Sharpley, R.C.: Interpolation of Operators. Pure and Applied Mathematics 129. Academic Press, Inc. 1988
Carleson, L.: On convergence and growth of partial sums of Fourier series. Acta Math. 116, 135–157 (1966)
Coifman, R.R., Fefferman, C.: Weighted norm inequalities for maximal functions and singular integrals. Studia Math. 51, 241–250 (1974)
Cruz-Uribe, D., Martell, J.M., Pérez, C.: Extrapolation from A∞ weights and applications. J. Funct. Anal. 213, 412–439 (2004)
Curbera, G., García-Cuerva, J., Martell, J.M., Pérez, C.: Extrapolation with Weights, Rearrangement Invariant Function Spaces, Modular inequalities and applications to Singular Integrals. Preprint 2004
Fefferman, C., Stein, E.M.: Some maximal inequalities. Amer. J. Math. 93, 107–115 (1971)
García-Cuerva, J., Rubio de Francia, J.L.: Weighted Norm Inequalities and Related Topics. North-Holland Math. Stud. 116, North-Holland, 1985
Hunt, R.: On the convergence of Fourier series. Orthogonal Expansions and Their Continuous Analogues (Edwardsville, Ill., 1967), pp. 235–255, D. T. Haimo (ed.), Southern Illinois Univ. Press, Carbondale IL, 1968
Hunt, R., Young, W.-S.: A weighted norm inequality for Fourier series. Bull. Amer. Math. Soc. 80, 274–277 (1974)
Kalton, N.J.: Convexity, type, and the three space problem. Studia Math. 69, 247–287 (1981)
Lacey, M.: Carleson’s theorem with quadratic phase functions. Studia Math. 153(3), 249–267 (2002)
Muckenhoupt, B.: Weighted norm inequalities for the Hardy maximal function. Trans. Amer. Math. Soc. 165, 207–226 (1972)
Rao, M., Ren, Z.: Theory of Orlicz Spaces. Monographs and Textbooks in Pure and Applied Mathematics 146, Marcel Dekker, Inc. New York, 1991
Rubio de Francia, J.L., Ruiz, F.J., Torrea, J.L.: Calderón-Zygmund theory for operator-valued kernels. Adv. Math. 62, 7–48 (1986)
Sjölin, P.: On the convergence almost everywhere of certain singular integrals and multiple Fourier series. Arkiv f. Math. 9, 65–90 (1971)
Sjölin, P., Soria, F.: Remarks on a theorem by N.Yu. Antonov. Studia Math. 158(1), 79–97 (2003)
Soria, F.: Note on differentiation of integrals and the halo conjecture. Studia Math. 81, 29–36 (1985)
Soria, F.: On an extrapolation theorem of Carleson-Sjölin with applications to a.e. convergence of Fourier sereis. Studia Math. 94, 235–244 (1989)
Stein, E.M., Wainger, S.: Oscillatory integrals related to Carleson’s theorem. Math. Res. Lett. 8, 789–800 (2001)
Stein, E.M., Weiss, N.J.: On the convergence of Poisson integrals. Trans. Amer. Math. Soc. 140, 35–54 (1969)
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Mathematics Subject Classification (2000): 42B20, 42B25
The first author is supported by the National Science Foundation under grant DMS 0099881.
Part of this work was carried out while the second author was a Postdoctoral Fellow at University of Missouri-Columbia. The second author would like to thank this department for its support and hospitality.
The second and the third authors are partially supported by MCYT Grant BFM2001-0189.
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Grafakos, L., Martell, J. & Soria, F. Weighted norm inequalities for maximally modulated singular integral operators. Math. Ann. 331, 359–394 (2005). https://doi.org/10.1007/s00208-004-0586-2
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DOI: https://doi.org/10.1007/s00208-004-0586-2