Proof of a conjecture of José L . Rubio de Francia

Given a compact connected abelian group G, its dual group Γ can be ordered (in a non-canonical way) so that it becomes an ordered group. It is known that, for any such ordering on Γ and p in the range 1 < p < ∞, the characteristic function χI of an interval I in Γ is a p−multiplier with a uniform bound (independent of I) on the corresponding operator SI on Lp(G). In this note it is shown that, for 1 < p, q < ∞, there is a constant Cp,q, independent of G and the particular ordering on Γ, such that ‖( ∑ j |SIjfj |)‖Lp(G) ≤ Cp,q‖( ∑ j |fj |)‖Lp(G) for all sequences {Ij} of intervals in Γ and all sequences {fj} in Lp(G). Such a result was conjectured by J.L. Rubio de Francia, who noted its validity when G = Tn. The present proof uses a transference argument, an approach which shows that any constant Cp,q for which the inequality holds when G = T will serve for every G and every ordering on Γ. An added advantage of this approach is that it adapts to give an extension of the result for functions taking values in a UMD space.