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This paper gives a systematic study of Sobolev, Besov and Triebel-Lizorkin spaces on a noncommutative $d$-torus $mathbb^d_ heta$ (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.

This volume contains the notes of lectures given at the School Quantum Potential Theory: Structure and Applications to Physics . This school was held at the Alfried Krupp Wissenschaftskolleg in Greifswald from February 26 to March 9, 2007. We thank the lecturers for the hard work they - complished in preparing and giving these lectures and in writing these notes. Theirlecturesgiveanintroductiontocurrentresearchintheirdomains, which is essentially self-contained and should be accessible to Ph.D. students. We hope that this volume will help to bring together researchers from the areas of...