We shall establish the core of singular integral theory and pseudodifferential calculus over the archetypal algebras of noncommutative geometry: quantum forms of Euclidean spaces and tori. Our results go beyond Connes' pseudodifferential calculus for rotation algebras, thanks to a new form of Calderon-Zygmund theory over these spaces which crucially incorporates nonconvolution kernels. We deduce Lp-boundedness and Sobolev p-estimates for regular, exotic and forbidden symbols in the expected ranks. In the L2 level both Calderon-Vaillancourt and Bourdaud theorems for exotic and forbidden symbols are also generalized to the quantum setting. As a basic application of our methods, we prove Lp-regularity of solutions for elliptic PDEs.

About the Author
Adrian M. Gonzalez-Perez, Universidad Autonoma de Madrid, Spain.

Marius Junge, University of Illinois at Urbana-Champaign, IL.

Javier Parcet, Instituto de Ciencias Matematicas, Madrid, Spain.


Book Information
ISBN 9781470449377
Author Adrian M. Gonzalez-Perez
Format Paperback
Imprint American Mathematical Society
Publisher American Mathematical Society
Weight(grams) 204g