So, i'm trying to prove that a Banach quasinormed space E (with its quasinorm based on $L^p$ and Weak-$L^p$ norms) is between $\mathcal{S}$ and $\mathcal{S'}$, on respect to embeddings, that is, $$ \mathcal{S} \hookrightarrow E \hookrightarrow\mathcal{S'}. $$
The quasinorm that defines the space is as follows: $$ ||f||=\left(\sum_{k\in\mathbb{Z}}2^{k\alpha q}||f||_{L^{p,\infty}(A_k)}^q \right)^{1/q}, $$ where $0<p,q\leq\infty,$ $\alpha\in\mathbb{R},$ and $A_k=B(0,2^{k})\backslash B(0,2^{k-1}).$
Does anyone have any ideas of how to demonstrate it? Or at least indicate some reference of how it's done for $L^p$ and $L^{p,\infty}$?