Algebra structure for Lp of a vector measure

In this paper, we present new results about the space Lp(ν) for ν being a vector measure defined in the Borel σ-algebra of a compact abelian group G and satisfying certain property concerning translation of simple functions. Namely, we show that Lp(ν) is a translation invariant space which can be endowed with an algebra structure via usual convolution product. We apply these results to the optimal domain of the Fourier transform and convolution operators.