We discuss the latticial structure of projection sets and antiprojection sets in Orlicz-Musielak spaces LΦ, with the distance given by modular ρΦ and classical F-norms generated by ρΦ. In particular, we show that the case of Amemiya F-norm is essentially different from others. This extends results given
for sets of best approximants by Landers and Rogge, Kilmer and Kozlowski, and Mazzone.
Motivated by the well known Kadec-Pełczynski disjointification theorem, we undertake an analysis of the supports of non-zero functions in strongly embedded subspaces of Banach functions spaces. The main aim is to isolate those properties that bring additional information on strongly embedded subspaces. This is the case of the support localization property, which is a necessary condition fulfilled by all strongly embedded subspaces. Several examples that involve Rademacher functions, the Volterra operator, Lorentz spaces or Orlicz spaces are provided.
Moduli of p-continuity provide a measure of fractional smoothness of functions via p-variation. We prove a sharp estimate of the modulus of p-continuity in terms of the modulus of q-continuity (1<p<q<∞).