Fourier transform and convolutions on L(p) of a vector measure on a compact Hausdorff abelian group

The final publication is available at Springer via http://dx.doi.org/10.1007/s00041-012-9252-3"Let ν be a countably additive vector measure defined on the Borel subsets of a compact Hausdorff abelian group G. In this paper we define and study a vector valued Fourier transform and a vector valued convolution for functions which are (weakly) integrable with respect to ν. A form of the Riemann Lebesgue Lemma and a Uniqueness Theorem are established in this context. In order to study the vector valued convolution we discuss the invariance under reflection in G of these spaces of integrable functions. Finally we present a Young’s type inequality in this setting and several relevant examples, namely related with the vector measure associated to different important classical operators coming from Harmonic Analysis.J.M. Calabuig was supported by Ministerio de Economia y Competitividad and FEDER (project MTM2008-04594) and MEC "Jose Castillejo 2008". E. A. Sanchez Perez was supported by Ministerio de Economia y Competitividad and FEDER (project MTM2012-36740-c02-02). J.M. Calabuig and E. A. Sanchez Perez were also supported by "Ayuda para Estancias de PDI de la UPV en centros de investigacion de prestigio" (PAID-00-11).Calabuig Rodriguez, JM.; Galaz-Fontes, F.; Navarrete, EM.; Sánchez Pérez, EA. (2013). Fourier transform and convolutions on L(p) of a vector measure on a compact Hausdorff abelian group. Journal of Fourier Analysis and Applications. 19(2):312-332. https://doi.org/10.1007/s00041-012-9252-3312332192Blasco, O., Calabuig, J.M.: Fourier analysis with respect to bilinear maps. Acta Math. Sin. Engl. Ser. 25(4), 519–530 (2009)del Campo, R., Fernández, A., Ferrando, I., Mayoral, F., Naranjo, F.: Multiplication operators on spaces on integrable functions with respect to a vector measure. J. Math. Anal. Appl. 343(1), 514–524 (2008)Delgado, O., Miana, P.J.: Algebra structure for L p of a vector measure. J. Math. Anal. Appl. 358(2), 355–363 (2009)Diestel, J., Uhl, J.J.: Vector Measures. Math. Surveys, vol. 15. Amer. Math. Soc., Providence (1977)Ferrando, I., Galaz-Fontes, F.: Multiplication operators on vector measure Orlicz spaces. Indag. Math. 20(1), 57–71 (2009)Fernández, A., Mayoral, F., Naranjo, F., Sánchez Pérez, E.A.: Complex interpolation of spaces of integrable functions with respect to a vector measure. Collect. Math. 61(3), 241–252 (2010)Katznelson, Y.: An Introduction to Harmonic Analysis. Cambridge University Press, Cambridge (2004)Mockenhaupt, G., Ricker, W.J.: Optimal extension of the Hausdorff-Young inequality. J. Reine Angew. Math. 620, 195–211 (2008)Pap, E.: Handbook of Measure Theory. North-Holland/Elsevier, Amsterdam (2002)Talagrand, M.: Pettis Integral and Measure Theory. Memoirs of the AMS, vol. 307 (1984)Okada, S., Ricker, W., Sánchez Pérez, E.A.: Optimal Domain and Integral Extension of Operators acting in Function Spaces. Oper. Theory Adv. Appl., vol. 180. Birkhäuser, Basel (2008)Rudin, W.: Fourier Analysis on Groups. Interscience, New York (1967)Luxemburg, W.A.J., Zaanen, A.C.: Riesz Spaces I. North Holland, Amsterdam (1971