Intermediate spaces between ⋂2≤τ<∞Lτ⋂2≤τ<∞Lτ and L∞L∞

In this paper we introduce an intermediate space Lψ,λ(T)Lψ,λ(T) between ⋂τ≥2Lτ(T)⋂τ≥2Lτ(T) and L∞(T)L∞(T), where Lτ(T)Lτ(T) are the LτLτ-subspaces of a nonatomic finite measure space {T,Σ,μ}{T,Σ,μ}, and study its properties. In particular, this space is Banach and rearrangement-invariant with respect to the measure μμ. Our main result states that under some conditions Lψ,λ(T)Lψ,λ(T) coincides with a special Marcinkiewicz space. In addition, we obtain Zygmund-type weak and integral estimates for functions from Lψ,λ(T)Lψ,λ(T) and apply them to estimate the norms of the conjugate, Fourier, and Littlewood–Paley operators from L∞(T)L∞(T) to Lψ,λ(T)Lψ,λ(T), where TT is the unit circle in CC. Relationships between Lψ,λ(T)Lψ,λ(T) and BMO(T)(T) are discussed as well.