Orlicz-Schatten Factorizations for Non-Commutative Sobolev Embeddings
Emma Sulaver
TL;DR
The paper develops a factorization framework for non-commutative Sobolev embeddings on quantum tori through Orlicz–Schatten ideals, defining $\mathcal{S}_\Phi(\mathcal{M},\tau)$ and establishing spectral-decay–driven embeddings via $L_s=(1+\Delta)^{-s/2}$. It proves a main factorization result for the embedding $W^{s,2}(\mathbb{T}^d_\theta) \hookrightarrow L^2(\mathbb{T}^d_\theta)$ through $\mathcal{S}_\Phi$, with completely $1$-summing bounds governed by $\|L_s\|_{\mathcal{S}_\Phi}$ and optimality criteria tied to the decay of the quantum spectrum. The work links operator-ideal theory to non-commutative PDE regularity, heat-kernel smoothing, and quantum information metrics via transport-type inequalities, offering a robust toolset for analyzing regularity and spectral properties in non-commutative settings. The results pave the way for broader applications in quantum geometry, PDEs on non-commutative spaces, and information-theoretic aspects of operator algebras.
Abstract
We develop a framework for factorizing embeddings of non-commutative Sobolev spaces on quantum tori through newly defined Orlicz-Schatten sequence ideals. After introducing appropriate non-commutative Sobolev norms and Orlicz spectral conditions, we establish a summing operator characterization of the quantum Laplacian embedding. Our main results provide both existence and optimality of such factorization theorems, and highlight connections to operator ideal theory. Applications to regularity of non-commutative PDEs and quantum information metrics are discussed, demonstrating the broad impact of these structures in functional analysis and mathematical physics.