On continuous embeddings of quantum Sobolev spaces into Schatten classes $\mathfrak{H}_γ^{s,p}(G,H) \hookrightarrow S_p(H)$
Alexander Plakhotnikov
TL;DR
The paper addresses continuous embeddings of quantum Sobolev spaces into Schatten classes for non-Hilbert Schatten indices by extending the operator-valued Fourier approach to p≠2. It develops the Banach-space structure of \\mathfrak{H}_\\gamma^{s,p}(G,H) for 1<p<2 and establishes a topological isomorphism with a closed subspace of \\mathcal{L}^q(\\hat{G}) via the quantum Fourier transform \\mathcal{F}_U, leveraging integrable projective representations and the Hausdorff-Young inequality. For p>2, it constructs dual spaces \\mathfrak{H}^{-s,p'}_\\gamma and demonstrates that \\mathrm{S}_p(H) embeds continuously into the dual, allowing the Sobolev spaces to be realized as closures of Schatten operators under this dual pairing. The results also provide weighted Hölder/Hausdorff-Young embeddings into \\mathrm{S}_\\beta(H) with explicit thresholds, and show that such embeddings generally fail for \\beta<p without additional assumptions, though improved embeddings are possible under stronger integrability or representation-bounds. These findings extend Lakmon and Mensah’s p=2 theory to a broader Schatten-class regime, clarifying when operator-valued Sobolev regularity propagates to Schatten-class regularity.
Abstract
This work investigates continuous embeddings for quantum Sobolev spaces $\mathfrak{H}_γ^{s,p}(G,H)$ 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 $p>2$.