Singular integrals in quantum Euclidean spaces

In this paper, we 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 PDEsA. Gonz ́alez-P ́erez was partially supported by European Research Council Consolidator Grant 614195. M. Junge was partially supported by the NSF DMS-1501103. J. Parcet was partially supported by European Research Council Starting Grant 256997 and CSIC Grant PIE 201650E030. All authors are also supported in part by ICMAT Severo Ochoa Grant SEV-2015-0554 (Spain)