On preduals and Kothe duals of some subspaces of Morrey spaces

For 1 <= q <= alpha < infinity, {(L-q, l(p))(alpha)(R-d): alpha <= p <= infinity} is an increasing familyof Banach spaces such that (L-q, l(alpha))(alpha)(R-d) is the Lebesgue space L-alpha(R-d) and (L-q, l(infinity))(alpha)(R-d) is the Morrey space M-q(alpha)(R-d). A predual space H(q', p', alpha')(R-d) of ( L-q, l(p))(alpha)(R-d) is known. We study the normed Kothe space structure of H(q', p', alpha')(R-d) and show that it is the Kothe dual space of (L-q, l(p))(alpha)(R-d) and the dual space of the closure of L-alpha(R-d) in this space when 1 < q <= alpha < p <= infinity. These results are applied to the study of extensions of classical operators. (C) 2020 Elsevier Inc. All rights reserved.