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Singular integrals in quantum Euclidean spaces
About this Title
Adrían Manuel González-Pérez, Marius Junge and Javier Parcet
Publication: Memoirs of the American Mathematical Society
Publication Year:
2021; Volume 272, Number 1334
ISBNs: 978-1-4704-4937-7 (print); 978-1-4704-6750-0 (online)
DOI: https://doi.org/10.1090/memo/1334
Published electronically: September 27, 2021
Keywords: Singular integral,
pseudodifferential operator,
quantum Euclidean space
Table of Contents
Chapters
- Introduction
- 1. Quantum Euclidean spaces
- 2. Calderón-Zygmund $L_p$ theory
- 3. Pseudodifferential $L_p$ calculus
- 4. $L_p$ regularity for elliptic PDEs
- A. Noncommutative tori
- B. BMO space theory in $\mathcal {R}_\Theta$
Abstract
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 $L_p$-boundedness and Sobolev $p$-estimates for regular, exotic and forbidden symbols in the expected ranks. In the $L_2$ 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 $L_p$-regularity of solutions for elliptic PDEs.- Saad Baaj, Calcul pseudo-différentiel et produits croisés de $C^*$-algèbres. I, C. R. Acad. Sci. Paris Sér. I Math. 307 (1988), no. 11, 581–586 (French, with English summary). MR 967366
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