Sobolev, Besov and Triebel-Lizorkin Spaces on Quantum Tori

Gives a systematic study of Sobolev, Besov and Triebel-Lizorkin spaces on a noncommutative $d$-torus $mathbb{T}^d_ heta$ (with $ heta$ a skew symmetric real $d imes d$-matrix). These spaces share many properties with their classical counterparts. The authors prove, among other basic properties, the lifting theorem for all these spaces and a Poincare type inequality for Sobolev spaces.