Abstract
We prove weak (2, 2) bounds for maximally modulated anisotropically homogeneous smooth multipliers on \(\mathbb {R}^n\). These can be understood as generalizing the classical one-dimensional Carleson operator. For the proof we extend the time-frequency method by Lacey and Thiele to the anisotropic setting. We also discuss a related open problem concerning Carleson operators along monomial curves.
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Notes
We can think of all the \(\mathcal {P}_\ell \) as being initialized by the empty set.
References
Carleson, L.: On convergence and growth of partial sums of Fourier series. Acta Math. 116, 135–157 (1966)
Duoandikoetxea, J.: Fourier Analysis, vol. 29. Graduate Studies in Mathematics. American Mathematical Society, Providence (2001). Translated and revised from the 1995 Spanish original by D. Cruz-Uribe
Fabes, E., Rivière, N.: Singular integrals with mixed homogeneity. Stud. Math. 27(1), 19–38 (1966)
Fefferman, C.: Pointwise convergence of Fourier series. Ann. Math. 2(98), 551–571 (1973)
Grafakos, L., Tao, T., Terwilleger, E.: \(L^p\) bounds for a maximal dyadic sum operator. Math. Z. 246(1–2), 321–337 (2004)
Guo, S., Pierce, L.B., Roos, J., Yung, P.-L.: Polynomial Carleson operators along monomial curves in the plane. J. Geom. Anal. 27(4), 2977–3012 (2017)
Kenig, C., Tomas, P.A.: Maximal operators defined by Fourier multipliers. Stud. Math. 68, 79–83 (1980)
Lacey, M.T.: Carleson’s theorem: proof, complements, variations. Publ. Mat. 48(2), 251–307 (2004)
Lacey, M.T., Thiele, C.: A proof of boundedness of the Carleson operator. Math. Res. Lett. 7(4), 361–370 (2000)
Lie, V.: The (weak-\(L^2\)) boundedness of the quadratic Carleson operator. Geom. Funct. Anal. 19, 457–497 (2009)
Pierce, L.B., Yung, P.-L.: A polynomial Carleson operator along the paraboloid (2015). arXiv:1505.03882. Rev. Mat. Iberoam. (To appear)
Pramanik, M., Terwilleger, E.: A weak \(L^2\) estimate for a maximal dyadic sum operator on \({\mathbb{R}}^n\). Ill. J. Math. 47(3), 775–813 (2003)
Sjölin, P.: Convergence almost everywhere of certain singular integrals and multiple Fourier series. Ark. Mat. 9, 65–90 (1971)
Stein, E.M.: Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals, vol. 43. Princeton Mathematical Series. Princeton University Press (1993)
Thiele, C.: Wave Packet Analysis, vol. 105. CBMS Regional Conference Series in Mathematics. AMS, Providence (2006)
Acknowledgements
The author thanks his doctoral advisor Christoph Thiele for encouragement and countless valuable comments and discussions on this project. He is also deeply indebted to Po-Lam Yung for many important and fruitful conversations. This work was carried out while the author was supported by the German Academic Scholarship Foundation.
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Communicated by Loukas Grafakos.
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Roos, J. Bounds for Anisotropic Carleson Operators. J Fourier Anal Appl 25, 2324–2355 (2019). https://doi.org/10.1007/s00041-018-09657-7
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DOI: https://doi.org/10.1007/s00041-018-09657-7