On preduals and Köthe duals of some subspaces of Morrey spaces

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For 1≤q≤α<∞, {(Lq,lp)α(Rd):α≤p≤∞} is an increasing family of Banach spaces such that (Lq,lα)α(Rd) is the Lebesgue space Lα(Rd) and (Lq,l∞)α(Rd) is the Morrey space Mqα(Rd). A predual space H(q′,p′,α′)(Rd) of (Lq,lp)α(Rd) is known. We study the normed Köthe space structure of H(q′,p′,α′)(Rd) and show that it is the Köthe dual space of (Lq,lp)α(Rd) and the dual space of the closure of Lα(Rd) in this space when 1<q≤α<p≤∞. These results are applied to the study of extensions of classical operators.