Hausdorff-Young-type inequalities for vector-valued Dirichlet series

[EN] We study Hausdorff-Young-type inequalities for vector-valued Dirichlet series which allow us to compare the norm of a Dirichlet series in the Hardy space H-p(X) with the q-norm of its coefficients. In order to obtain inequalities completely analogous to the scalar case, a Banach space must satisfy the restrictive notion of Fourier type/cotype. We show that variants of these inequalities hold for the much broader range of spaces enjoying type/cotype. We also consider Hausdorff-Young-type inequalities for functions defined on the infinite torus T-infinity or the boolean cube {- 1, 1}(infinity). As a fundamental tool we show that type and cotype are equivalent to a hypercontractive homogeneous polynomial type and cotype, a result of independent interest.The third author was supported by MICINN and FEDER Project MTM2017-83262-C2-1-P and MECD grant PRX17/00040.Carando, D.; Marceca, F.; Sevilla Peris, P. (2020). Hausdorff-Young-type inequalities for vector-valued Dirichlet series. Transactions of the American Mathematical Society. 373(8):5627-5652. https://doi.org/10.1090/tran/8147562756523738BLASCO, O., & PAVLOVIC, M. (2003). COMPLEX CONVEXITY AND VECTOR-VALUED LITTLEWOOD–PALEY INEQUALITIES. Bulletin of the London Mathematical Society, 35(06), 749-758. doi:10.1112/s0024609303002479Carando, D., Defant, A., & Sevilla-Peris, P. (2014). Bohr’s absolute convergence problem for ℋp-Dirichlet series in Banach spaces. Analysis & PDE, 7(2), 513-527. doi:10.2140/apde.2014.7.513Carando, D., Defant, A., & Sevilla-Peris, P. (2016). Some polynomial versions of cotype and applications. Journal of Functional Analysis, 270(1), 68-87. doi:10.1016/j.jfa.2015.09.017Davis, W. J., Garling, D. J. ., & Tomczak-Jaegermann, N. (1984). The complex convexity of quasi-normed linear spaces. Journal of Functional Analysis, 55(1), 110-150. doi:10.1016/0022-1236(84)90021-1De la Peña, V. H., & Giné, E. (1999). Decoupling. Probability and its Applications. doi:10.1007/978-1-4612-0537-1Defant, A., García, D., Maestre, M., & Pérez-García, D. (2008). Bohr’s strip for vector valued Dirichlet series. Mathematische Annalen, 342(3), 533-555. doi:10.1007/s00208-008-0246-zDefant, A., García, D., Maestre, M., & Sevilla-Peris, P. (2019). Dirichlet Series and Holomorphic Functions in High Dimensions. doi:10.1017/9781108691611Defant, A., Maestre, M., & Schwarting, U. (2012). Bohr radii of vector valued holomorphic functions. Advances in Mathematics, 231(5), 2837-2857. doi:10.1016/j.aim.2012.07.016Defant, A., Mastyło, M., & Pérez, A. (2018). On the Fourier spectrum of functions on Boolean cubes. Mathematische Annalen, 374(1-2), 653-680. doi:10.1007/s00208-018-1756-yDefant, A., & Pérez, A. (2018). Hardy spaces of vector-valued Dirichlet series. Studia Mathematica, 243(1), 53-78. doi:10.4064/sm170303-26-7Diestel, J., Jarchow, H., & Tonge, A. (1995). Absolutely Summing Operators. doi:10.1017/cbo9780511526138Garcia-Cuerva, J., Kazarian, K. S., Kolyada, V. I., & Torrea, J. L. (1998). Vector-valued Hausdorff-Young inequality and applications. Russian Mathematical Surveys, 53(3), 435-513. doi:10.1070/rm1998v053n03abeh000018Hedenmalm, H., Lindqvist, P., & Seip, K. (1997). A Hilbert space of Dirichlet series and systems of dilated functions in L2(0,1). Duke Mathematical Journal, 86(1). doi:10.1215/s0012-7094-97-08601-4Kwapień, S., & Woyczyński, W. A. (1992). Random Series and Stochastic Integrals: Single and Multiple. doi:10.1007/978-1-4612-0425-1Pelczynski, A. (1988). Commensurate Sequences of Characters. Proceedings of the American Mathematical Society, 104(2), 525. doi:10.2307/2047005Pisier, G. (1982). Holomorphic Semi-Groups and the Geometry of Banach Spaces. The Annals of Mathematics, 115(2), 375. doi:10.2307/1971396Queffélec, H., & Queffélec, M. (2013). Diophantine Approximation and Dirichlet Series. doi:10.1007/978-93-86279-61-3Maciej Rzeszut and Michał Wojciechowski, Hoeffding Decomposition in ¹ spaces. arXiv:1906.01405.Wilf, H. S. (1970). Finite Sections of Some Classical Inequalities. doi:10.1007/978-3-642-86712-