Sobolev, Besov and Triebel-Lizorkin spaces on quantum tori

This paper gives a systematic study of Sobolev, Besov and Triebel-Lizorkin spaces on a noncommutative d-torus \mathbb{T}^d_\theta (with \theta a skew symmetric real d\times 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 Poincar� type inequality for Sobolev spaces.