Main embedding theorems for symmetric spaces of measurable functions

摘要

Let m be the usual Lebesgue measure on R+= [0, +∞). Dealing with symmetric(rearrangement invariant)spaces E on the standard measure space(R+, m), we treat the following embeddings: {formula}, where E~0= cl_E(L_1 ∩ L∞) is the closure of L_1 ∩ L∞ in E, E~(11)= (E~1)~1 is the second associate space of E, V(x)=||1_([0,x]||E) is the fundamental function of the symmetric space {formula} and {~ under letter V} is the least concave majorant of V, Λ{~under letter V} and M_(V*) are the Lorentz and Marcinkiewicz spaces with the weights V and V, respectively and {formula}. The embeddings and natural inequalities for corresponding norms are studied in detail.