Intermediate spaces between and

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