Fourier Transform and Convolutions on L p of a Vector Measure on a Compact Hausdorff Abelian Group

Let nu 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 nu. 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.