Embeddings of Besov Spaces on fractal h-sets

TL;DR

This paper investigates the conditions under which Besov-type trace spaces on fractal h-sets embed into each other or into L_r spaces, providing necessary and sufficient criteria and complete characterizations in some cases.

Contribution

It offers new necessary and sufficient conditions for embeddings between Besov trace spaces on fractal h-sets, including exact criteria for the existence of such spaces.

Findings

Established necessary and sufficient conditions for embeddings.

Provided complete characterizations in specific cases.

Derived exact conditions for the existence of trace spaces.

Abstract

Let $\Gamma$ be a fractal $h$-set and ${\mathbb{B}}^{{\sigma}}_{p,q}(\Gamma)$ be a trace space of Besov type defined on $\Gamma$. While we dealt in [9] with growth envelopes of such spaces mainly and investigated the existence of traces in detail in [12], we now study continuous embeddings between different spaces of that type on $\Gamma$. We obtain necessary and sufficient conditions for such an embedding to hold, and can prove in some cases complete characterisations. It also includes the situation when the target space is of type $L_r(\Gamma)$ and, as a by-product, under mild assumptions on the $h$-set $\Gamma$ we obtain the exact conditions on $\sigma$, $p$ and $q$ for which the trace space ${\mathbb{B}}^{{\sigma}}_{p,q}(\Gamma)$ exists. We can also refine some embedding results for spaces of generalised smoothness on $\mathbb R^n$.