Absolutely summing convolution operators in Lp(G)

Let G be an infinite, compact, abelian group. We investigate the Banach space ? 1 ( p ) consisting of all absolutely summing convolution operators from L p (G ) to itself. For 1?p ?2 these spaces were identified by Beauzamy (and in C (G ) by Lust-Piquard). So, we concentrate on 2 < p ? 8. More tractable than ? 1 ( p ) is the subspace ? 1 ( 1 , p ) consisting of those operators from ? 1 ( p ) which have an absolutely summing L p (G )-valued extension to L 1(G ), the reason being the availability of a factorization theorem (via L 2(G )) for operators from ? 1 ( 1 , p ) . The Banach space S p consisting of all f ? L p (G ) with an unconditionally convergent Fourier series, introduced by Bachelis, plays a crucial role. Whereas both ? 1 ( 1 , p ) and S p turn out to be reflexive Banach lattices (with an unconditional basis if G is metrizable), this is not so for ? 1 ( p ) . It is shown that S 1 1 , p ? S p ? L p . Moreover, ? 1 ( 1 , p ) has cotype 2 whereas S p has cotype p but fails to be r -concave for every 1 ? r < p . We also establish that ? 1 ( 1 , p ) ? ? 1 ( p ) ? L p and ? 1 ( p ) ? S p .