On the Predual of a Morrey-Lorentz Space and Its Applications to the Linear Calderon-Zygmund Operators

Abstract

Our main purpose in this paper is to construct a predual of Morrey–Lorentz space via the block spaces, defined in [Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 1999, 28(1): 31–40]. As a direct application of duality, we obtain the Morrey–Lorentz boundedness of linear Calderón–Zygmund operators. Moreover, we study a weak Hardy factorization in terms of linear Calderón–Zygmund operators in Morrey–Lorentz spaces. As a consequence of this result, we obtain a characterization of functions in BMO(ℝn) through the boundedness of commutator [b, T], where T is a homogeneous Calderón–Zygmund operator. Finally, we prove a Morrey–Lorentz compactness characterization of [b, T]. Precisely, [b, T] is a compact operator on Morrey–Lorentz spaces if and only if b ∈ CMO(ℝn).