Abstract The spaces in the title are associated to a fixed representing measure m for a fixed character on a uniform algebra. It is proved that the set of representing measures for that character which are absolutely continuous with respect to m is weakly relatively compact if and only if each m-negligible closed set in the maximal ideal space of L ∞ is contained in an m-negligible peak set for H ∞. J. Chaumat's characterization of weakly relatively compact subsets in L 1 H ∞⊥ therefore remains true, and L 1 H ∞⊥ is complete, under the first conditions. In this paper we also give a direct proof. From this we obtain that L 1 H ∞⊥ has the Dunford-Pettis property.