Duality for α-Mobius invariant Besov spaces

For 1 <= p <= infinity and alpha > 0, Besov spaces B-alpha(p) play a key role in the theory of alpha-Mobius invariant function spaces. In some sense, B-alpha(1) is the minimal alpha-Mobius invariant function space, B-alpha(2) is the unique alpha-Mobius invariant Hilbert space, and B-alpha(infinity) is the maximal alpha-Mobius invariant function space. In this paper, under the alpha-Mobius invariant pairing and by the space B-alpha(infinity), we identify the predual and dual spaces of B-alpha(1). In particular, the corresponding identifications are isometric isomorphisms. The duality theorem via the alpha-Mobius invariant pairing for B-alpha(p) with p > 1 is also given.