A(p)-spaces

作者：Astashkin S V^*

来源：Journal of Functional Analysis, 2014, 266(8): 5174-5198.

DOI：10.1016/j.jfa.2014.02.011

• L-p-space
• Orlicz space
• Rearrangement invariant space
• Disjoint ly strictly singular operator

We say that a closed subspace H of the space L-p = L-p[0,1], 1 <= p < infinity, is a A(p)-space if, in H, convergence in L-p-norm is equivalent to convergence in measure. Mainly, we focus on the problem when the unit ball B-H := {f is an element of H: parallel to f parallel to(p) <= 1} of a A(p)-space H has equi-absolutely continuous norms in L. Moreover, assuming that a rearrangement invariant space X is embedded into L-p, 1 <= p < infinity, and that the inclusion operator I : X -> L-p 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 L-p 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 A (p)-sets.

• 出版日期2014-4-15