Singular integrals in quantum Euclidean spaces

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...

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Bibliographic Details
Main Authors: González-Pérez, Adrían Manuel, Junge, Marius, Parcet, Javier
Format: eBook Book
Language:English
Published: Providence, Rhode Island American Mathematical Society 2021
Edition:1
Series:Memoirs of the American Mathematical Society
Subjects:
Calderón-Zygmund operator
Calderón-Zygmund operator
Functional analysis > Selfadjoint operator algebras ($C$-algebras, von Neumann ($W$-) algebras, etc.) > General theory of von Neumann algebras. msc
Functional analysis > Selfadjoint operator algebras ($C$-algebras, von Neumann ($W$-) algebras, etc.) > Noncommutative measure and integration. msc
Harmonic analysis on Euclidean spaces > Harmonic analysis in several variables > Harmonic analysis and PDE. msc
Harmonic analysis on Euclidean spaces > Harmonic analysis in several variables > Singular and oscillatory integrals (Calderón-Zygmund, etc.). msc
Lp spaces
Noncommutative differential geometry
Operator theory > Integral, integro-differential, and pseudodifferential operators > Pseudodifferential operators. msc
Pseudodifferential operators
Quantum theory
Quantum theory > Groups and algebras in quantum theory > Noncommutative geometry. msc
Singular integrals
pseudodifferential operator
Singular integral
quantum Euclidean space
ISBN:9781470449377, 1470449374
ISSN:0065-9266, 1947-6221
Online Access:Get full text
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Table of Contents:
  • Introduction -- Quantum Euclidean spaces -- Calderón-Zygmund <inline-formula content-type="math/mathml"> L p L_p </inline-formula> theory -- Pseudodifferential <inline-formula content-type="math/mathml"> L p L_p </inline-formula> calculus -- <inline-formula content-type="math/mathml"> L p L_p </inline-formula> regularity for elliptic PDEs -- Noncommutative tori -- BMO space theory in <inline-formula content-type="math/mathml"> R Θ \mathcal {R}_\Theta </inline-formula>
  • Cover -- Title page -- Introduction -- Chapter 1. Quantum Euclidean spaces -- 1.1. Crossed product form -- 1.2. Metrics and derivations -- 1.3. Quantum Euclidean variables -- Chapter 2. Calderón-Zygmund _{ } theory -- 2.1. Kernels and symbols -- 2.2. CZ extrapolation: Model case -- 2.3. CZ extrapolation: General case -- Chapter 3. Pseudodifferential _{ } calculus -- 3.1. Adjoint and product formulae -- 3.2. ₂-boundedness: Sufficient conditions -- 3.3. _{ }-boundedness and Sobolev -estimates -- Chapter 4. _{ } regularity for elliptic PDEs -- Appendix A. Noncommutative tori -- Appendix B. BMO space theory in \RR_{Θ} -- B.1. Operator space structures on \BMO and \Hardy₁ -- B.2. The \Hardy₁-\BMO duality -- B.3. Complex interpolation -- B.4. An auxiliary density result -- Bibliography -- Back Cover