# $s_{*,\alpha}\hookrightarrow s_{*,\beta}$ of $p$-Schatten class?

When studying $$p$$-Schatten embeddings I considered spaces of the following kind: $$s_{*,\alpha}=\{a\in\ell^2(\mathbb{N}):\{j^{\alpha}a_j\}_{j=1}^{\infty}\in\ell^2(\mathbb{N})\}$$ for some $$\alpha>0$$.

In this paper Hanche-Olsen and Holden provide a lemma that can be used to show that $$s_{*,\alpha}\hookrightarrow \ell^2(\mathbb{N})$$ is compact for all $$\alpha>0$$. By slightly modifying the results of Hanche-Olsen and Holden I could show that a subset $$M\subset s_{*,\beta}$$ for $$0<\beta<\alpha$$ is totally bounded if and only if it is pointwise bounded and if for every $$\varepsilon>0$$ there exists an $$n(\varepsilon)\in\mathbb{N}$$ so that for all $$a\in M$$: $$\sum_{j=n(\varepsilon)+1}^{\infty}|j^{\beta}a_j|^2<\varepsilon^2.$$ From this I could conclude that $$s_{*,\alpha}\hookrightarrow s_{*,\beta}$$ are compact embeddings for $$\alpha-\beta>0$$. I could show that if $$0 and $$(\alpha-\beta)p>1$$ these embeddings are even of $$p$$-Schatten class. My question is if someone has an idea how to generalize this result to $$0 and if someone encountered these spaces in some other context. Many thanks in advance!