Inclusion Mappings between Orlicz Sequence Spaces

It is shown that if ℓϕ is an Orlicz sequence space, then the space ℓw1(ℓϕ) of weakly summable sequences in ℓϕ is continuously embedded into ℓϕ(ℓ2) (resp., into ℓϕ(ℓϕ)) whenever t↦ϕ(t) is equivalent to a concave function (resp., a convex function and ϕ is a supermultiplicative function). By combining the above results with the interpolation theory we proved continuous inclusions between spaces ℓw1(ℓϕ0) and ℓφ(ℓϕ1), where ℓϕ0↪ℓϕ1 and φ is a certain Orlicz function depending on ϕ0 and ϕ1. In particular, if ϕ0 and ϕ1 are power functions we obtain the well known result on (r, 1)-summability of the inclusion mappings between ℓp-spaces proved independently by G. Bennett (1973, J. Funct. Anal.13, 20–27) and B. Carl (1974, Math. Nachr.63, 253–360).