Orlicz-Schatten Factorizations for Non-Commutative Sobolev Embeddings

TL;DR

This paper introduces a new factorization framework for non-commutative Sobolev space embeddings using Orlicz-Schatten ideals, with applications to quantum PDEs and quantum information theory.

Contribution

It develops a novel factorization approach for non-commutative Sobolev embeddings via Orlicz-Schatten ideals, connecting operator theory and quantum analysis.

Findings

Established existence and optimality of factorization theorems

Provided a characterization of quantum Laplacian embeddings

Linked operator ideals to non-commutative PDE regularity

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.