Λ(p)Λ(p)-spaces

We say that a closed subspace H   of the space Lp=Lp[0,1]Lp=Lp[0,1], 1⩽p<∞1⩽p<∞, is a Λ(p)Λ(p)-space if, in H  , convergence in LpLp-norm is equivalent to convergence in measure. Mainly, we focus on the problem when the unit ball BH:={f∈H:‖f‖p⩽1}BH:={f∈H:‖f‖p⩽1} of a Λ(p)Λ(p)-space H   has equi-absolutely continuous norms in LpLp. Moreover, assuming that a rearrangement invariant space X   is embedded into LpLp, 1⩽p<∞1⩽p<∞, and that the inclusion operator I:X→LpI:X→Lp fails to be strictly singular, we are interested in what we can say about the properties of subspaces on which the norms of X   and LpLp are equivalent. We reveal their essential dependence on the value of p  , which resembles the difference in the classical theorems of Bourgain and Bachelis–Ebenstein on Λ(p)Λ(p)-sets.