Ebook: 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 $dtimes d$-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.