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.
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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).
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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
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DOI: https://doi.org/10.1007/s00208-013-0935-0