Abstract
We prove a vector-valued version of Carleson’s theorem: let \(Y=[X,H]_\theta \) be a complex interpolation space between an unconditionality of martingale differences (UMD) space \(X\) and a Hilbert space \(H\). For \(p\in (1,\infty )\) and \(f\in L^p(\mathbb T ;Y)\), the partial sums of the Fourier series of \(f\) converge to \(f\) pointwise almost everywhere. Apparently, all known examples of UMD spaces are of this intermediate form \(Y=[X,H]_\theta \). In particular, we answer affirmatively a question of Rubio de Francia on the pointwise convergence of Fourier series of Schatten class valued functions.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Barrionuevo, J., Lacey, M.T.: A weak-type orthogonality principle. Proc. Am. Math. Soc. 131(6), 1763–1769 (2003) (electronic)
Carleson, L.: On convergence and growth of partial sums of Fourier series. Acta Math. 116, 135–157 (1966)
Figiel, T.: Singular integral operators: a martingale approach. In: Geometry of Banach Spaces (Strobl, 1989). London Mathematical Society Lecture Note Series, vol. 158, pp. 95–110. Cambridge University Press, Cambridge (1990)
García-Cuerva, J., Rubio de Francia, J.L.: Weighted Norm Inequalities and Related Topics. North-Holland Mathematics Studies, vol. 116. Notas de Matemática (Mathematical Notes), vol. 104. North-Holland Publishing Co., Amsterdam (1985)
Haase, M.: Identification of some real interpolation spaces. Proc. Am. Math. Soc. 134(8), 2349–2358 (2006)
Hytönen, T.P., Lacey, M.T.: Pointwise convergence of Walsh–Fourier series of vector-valued functions. Preprint (2012). arXiv:1202.0209
Lacey, M., Thiele, C.: A proof of boundedness of the Carleson operator. Math. Res. Lett. 7(4), 361–370 (2000)
Martínez, T., Torrea, J.L., Xu, Q.: Vector-valued Littlewood–Paley–Stein theory for semigroups. Adv. Math. 203(2), 430–475 (2006)
Parcet, J., Soria, F., Xu, Q.: On the growth of vector-valued Fourier series. Preprint (2011). arXiv:1109.4313
Rubio de Francia, J.L.: Martingale and integral transforms of Banach space valued functions. In: Probability and Banach Spaces (Zaragoza, 1985). Lecture Notes in Mathematics, vol. 1221, pp. 195–222. Springer, Berlin (1986)
Thiele, C.: Wave Packet Analysis. CBMS Regional Conference Series in Mathematics, vol. 105. Conference Board of the Mathematical Sciences, Washington, DC (2006)
Torrea, J.L., Zhang, C.: Fractional vector-valued Littlewood–Paley–Stein theory for semigroups. Preprint (2011). arXiv:1105.6022
Acknowledgments
T.H. is supported by the European Union through the ERC Starting Grant “Analytic–probabilistic methods for borderline singular integrals”, and by the Academy of Finland through projects 130166 and 133264. M.L. is supported in part by the NSF Grant 0968499, and a Grant from the Simons Foundation (#229596 to Michael Lacey).
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Hytönen, T.P., Lacey, M.T. Pointwise convergence of vector-valued Fourier series. Math. Ann. 357, 1329–1361 (2013). https://doi.org/10.1007/s00208-013-0935-0
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00208-013-0935-0