Embeddings of Besov spaces on fractal $h$-sets

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 our earlier papers with growth envelopes of such spaces mainly and investigated the existence of traces in detail, 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 $\mathbb{R}^n$.