On continuous embeddings of quantum Sobolev spaces into Schatten classes $\mathfrak{H}_\gamma^{s,p}(G,H) \hookrightarrow S_p(H)$

This work investigates continuous embeddings for quantum Sobolev spaces $\mathfrak{H}_\gamma^{s,p}(G,H)$ into Schatten--von Neumann classes $S_r(H)$. We try to extend the results of Lakmon and Mensah to the case where the operators belong to Schatten classes $S_p(H)$ for $p \neq 2$. We establish that these quantum Sobolev spaces are Banach spaces and, by employing a duality argument, we define spaces for $p>2$.