Duality for $\alpha$-M\"obius invariant Besov spaces

TL;DR

This paper characterizes the dual and predual spaces of $eta$-M"obius invariant Besov spaces $B^p_eta$, establishing isometric isomorphisms and duality relations, thereby deepening the understanding of their functional-analytic structure.

Contribution

It provides a complete duality and preduality characterization for $eta$-M"obius invariant Besov spaces, including explicit isometric isomorphisms, which was previously unknown.

Findings

Identified the predual and dual spaces of $B^1_eta$ using the $eta$-M"obius invariant pairing.

Established duality relations for $B^p_eta$ with $p>1$ via the $eta$-M"obius invariant pairing.

Proved that the duality and preduality mappings are isometric isomorphisms.

Abstract

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