Singular integrals in quantum Euclidean spaces

Adrían Manuel González-Pérez, Marius Junge, Javier Parcet

Research output: Contribution to journalArticlepeer-review


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 Calderón-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 Calderón-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.

Original languageEnglish (US)
Pages (from-to)1-87
Number of pages87
JournalMemoirs of the American Mathematical Society
Issue number1334
StatePublished - 2021


  • Pseudodifferential operator
  • Quantum Euclidean space
  • Singular integral

ASJC Scopus subject areas

  • General Mathematics
  • Applied Mathematics


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