Open AccessSingular integrals in quantum Euclidean spaces
Author(s) -
Adrián M. González-Pérez,
Marius Junge,
Javier Parcet
Publication year - 2021
Publication title -
memoirs of the american mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 3.034
H-Index - 70
eISSN - 1947-6221
pISSN - 0065-9266
DOI - 10.1090/memo/1334
Subject(s) - algorithm , type (biology) , euclidean geometry , mathematics , annotation , sobolev space , semantics (computer science) , computer science , artificial intelligence , algebra over a field , pure mathematics , geometry , ecology , biology , programming language
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 L p L_p -boundedness and Sobolev p p -estimates for regular, exotic and forbidden symbols in the expected ranks. In the L 2 L_2 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 L p L_p -regularity of solutions for elliptic PDEs.