Duality for $\alpha$-Möbius invariant Besov spaces

For $1\le p\le \mathrm{\infty }$ and $\alpha >0$, Besov spaces $B_{\alpha }^p$ play a key role in the theory of $\alpha$-Möbius invariant function spaces. In some sense, $B_{\alpha }^1$ is the minimal $\alpha$-Möbius invariant function space, $B_{\alpha }^2$ is the unique $\alpha$-Möbius invariant Hilbert space, and $B_{\alpha }^{\mathrm{\infty }}$ is the maximal $\alpha$-Möbius invariant function space. In this paper, under the $\alpha$-Möbius invariant pairing and by the space $B_{\alpha }^{\mathrm{\infty }}$, we identify the predual and dual spaces of $B_{\alpha }^1$. In particular, the corresponding identifications are isometric isomorphisms. The duality theorem via the $\alpha$-Möbius invariant pairing for $B_{\alpha }^p$ with $p>1$ is also given.