The minimal Mobius invariant space

In this article we study the minimal Mobius invariant space, denoted by B-1, which is compatible with the Besov spaces B-p, 1<p<. In the first part of this article, we prove that the lacunary characterization of functions in B-p(1<p<) is still true for the critical case p=1. Also, we show by counterexamples that some results for the Besov spaces B-p, 1<p<, do not remain for p=1. In the second part of this article, we give a characterization of isometric isomorphisms of B-1; that is, any isometric isomorphism of B-1 must be a composition operator.