Characterization of non-linear Besov spaces

TL;DR

This paper investigates non-linear Besov spaces, demonstrating the equivalence of different norms and exploring their embeddings into non-linear p-variation spaces without assuming UMD or separability.

Contribution

It establishes norm equivalences and embedding results for non-linear Besov spaces, extending classical theory to more general metric space-valued functions.

Findings

Canonical norms are equivalent in non-linear Besov spaces.

Non-linear Besov spaces embed into non-linear p-variation spaces and vice versa.

Results hold without assuming UMD property or separability.

Abstract

The canonical generalizations of two classical norms on Besov spaces are shown to be equivalent even in the case of non-linear Besov spaces, that is, function spaces consisting of functions taking values in a metric space and equipped with some Besov-type topology. The proofs are based on atomic decomposition techniques and metric embeddings. Additionally, we provide embedding results showing how non-linear Besov spaces embed into non-linear $p$-variation spaces and vice versa. We emphasize that we neither assume the UMD property of the involved spaces nor their separability.