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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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
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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
AbstractList 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
Author Junge, Marius
González-Pérez, Adrían Manuel
Parcet, Javier
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Keywords pseudodifferential operator
Singular integral
quantum Euclidean space
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Notes Includes bibliographical references (p. 87-90)
July 2021, volume 272, number 1334 (fourth of 7 numbers)
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Snippet We shall establish the core of singular integral theory and pseudodifferential calculus over the archetypal algebras of noncommutative geometry: quantum forms...
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ams
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SubjectTerms 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
TableOfContents 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
Title Singular integrals in quantum Euclidean spaces
URI https://www.ams.org/memo/1334/
https://cir.nii.ac.jp/crid/1130008896098931077
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Volume 272
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