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9781470449377 Academic Inspection Copy

Singular Integrals in Quantum Euclidean Spaces

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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 Calder?on-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 Calder?on-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.
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.
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