On Certian Fine Properties of Sequence Spaces

We develop a theory of Orlicz and Lorentz--Zygmund sequence spaces. We study in detail elementary properties of both structures. In particular, we characterize the values of governing parameters for which the corresponding spaces satisfy the axioms of rearrangement-invariant Banach sequence spaces, we evaluate the asymptotic behavior of their fundamental functions, and we characterize their associate spaces. In our main result, we characterize all cases in which a Lorentz--Zygmund sequence space coincides with some Orlicz sequence space.