On Preduals and Köthe Duals of some subspaces of Morrey spaces

Abstract For 1 ≤ q ≤ α ∞ , { ( L q , l p ) α ( R d ) : α ≤ p ≤ ∞ } is an increasing familly of Banach spaces such that ( L q , l α ) α ( R d ) is the Lebesgue space L α ( R d ) and ( L q , l ∞ ) α ( R d ) is the Morrey space M q α ( R d ) . A predual space H ( q ′ , p ′ , α ′ ) ( R d ) of ( L q , l p ) α ( R d ) is known. We study the normed Kothe space structure of H ( q ′ , p ′ , α ′ ) ( R d ) and show that it is the Kothe dual space of ( L q , l p ) α ( R d ) and the dual space of the closure of L α ( R d ) in this space when 1 q ≤ α p ≤ ∞ . These results are applied to the study of extensions of classical operators.