Bourgain spaces
E547402
Bourgain spaces are function spaces introduced by Jean Bourgain that are tailored to study the well-posedness and regularity of nonlinear dispersive partial differential equations.
All labels observed (3)
| Label | Occurrences |
|---|---|
| Bourgain spaces canonical | 1 |
| Fourier restriction norm spaces | 1 |
| X^{s,b} spaces | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T5790536 — resolving that mention is where its identity was fixed. The disambiguator weighed these candidate entities and picked the highlighted one (or “None”, minting a new entity). This is how homonymy is resolved: the same surface form can point to different entities.
NED1
Entity disambiguation (via context triple)
gpt-5-mini-2025-08-07
Target entity: Bourgain spaces Context triple: [Jean Bourgain, knownFor, Bourgain spaces]
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A.
Three regularity results in harmonic analysis
"Three regularity results in harmonic analysis" is the doctoral thesis of mathematician Terence Tao, focusing on advanced problems in harmonic analysis and the study of regularity properties of functions and operators.
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B.
Littlewood–Paley theory
Littlewood–Paley theory is a collection of techniques in harmonic analysis that decompose functions into frequency-localized pieces to study their behavior in L^p spaces and related function spaces.
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C.
Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals
"Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals" is a foundational graduate-level textbook by Elias Stein that systematically develops modern harmonic analysis using real-variable techniques, emphasizing singular integrals, Littlewood–Paley theory, and oscillatory integral methods.
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D.
Sobolev spaces
Sobolev spaces are function spaces that incorporate both functions and their weak derivatives, providing a fundamental framework for studying partial differential equations and variational problems.
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E.
Orlicz spaces
Orlicz spaces are a class of function spaces that generalize Lebesgue spaces by measuring integrability via convex Orlicz functions rather than fixed power exponents.
- F. None of above. chosen
- G. Unsure - the case is ambiguous/there is not enough information to decide.
NED2
Entity disambiguation (via description)
gpt-5-mini-2025-08-07
Target entity: Bourgain spaces Target entity description: Bourgain spaces are function spaces introduced by Jean Bourgain that are tailored to study the well-posedness and regularity of nonlinear dispersive partial differential equations.
-
A.
Three regularity results in harmonic analysis
"Three regularity results in harmonic analysis" is the doctoral thesis of mathematician Terence Tao, focusing on advanced problems in harmonic analysis and the study of regularity properties of functions and operators.
-
B.
Littlewood–Paley theory
Littlewood–Paley theory is a collection of techniques in harmonic analysis that decompose functions into frequency-localized pieces to study their behavior in L^p spaces and related function spaces.
-
C.
Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals
"Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals" is a foundational graduate-level textbook by Elias Stein that systematically develops modern harmonic analysis using real-variable techniques, emphasizing singular integrals, Littlewood–Paley theory, and oscillatory integral methods.
-
D.
Sobolev spaces
Sobolev spaces are function spaces that incorporate both functions and their weak derivatives, providing a fundamental framework for studying partial differential equations and variational problems.
-
E.
Orlicz spaces
Orlicz spaces are a class of function spaces that generalize Lebesgue spaces by measuring integrability via convex Orlicz functions rather than fixed power exponents.
- F. None of above. chosen
Statements (50)
| Predicate | Object |
|---|---|
| instanceOf |
Banach space
ⓘ
function space ⓘ mathematical concept ⓘ |
| advantage |
allows sharp low-regularity results
ⓘ
captures dispersive smoothing effects ⓘ well-suited to Picard iteration ⓘ |
| alsoKnownAs |
Fourier restriction norm spaces
ⓘ
linked to:
Bourgain spaces
X^{s,b} spaces ⓘ
linked to:
Bourgain spaces
|
| appliedTo |
Korteweg–de Vries equations
ⓘ
linked to:
Korteweg–De Vries equation
general dispersive equations ⓘ non-periodic dispersive equations ⓘ nonlinear Schrödinger equations ⓘ periodic dispersive equations ⓘ |
| builtFrom |
dispersion relation ω(ξ)
ⓘ
solution operator of the linear dispersive equation ⓘ |
| characterizedBy |
adaptation to linear dispersive flows
ⓘ
dependence on spatial regularity index s ⓘ dependence on temporal regularity index b ⓘ frequency-time localization ⓘ space-time Fourier transform norms ⓘ use of modulation variable τ−ω(ξ) ⓘ weighted L2 norms in (ξ,τ)-space ⓘ |
| field |
dispersive PDE theory
ⓘ
functional analysis ⓘ harmonic analysis ⓘ partial differential equations ⓘ |
| introducedBy | Jean Bourgain ⓘ |
| introducedInContextOf |
Korteweg–de Vries equation
ⓘ
linked to:
Korteweg–De Vries equation
nonlinear Schrödinger equation ⓘ periodic boundary conditions ⓘ |
| namedAfter | Jean Bourgain ⓘ |
| normDefinedBy | L2 norm of weighted space-time Fourier transform ⓘ |
| property |
Banach space structure for fixed s and b
ⓘ
invariance under linear flow of the associated dispersive equation ⓘ |
| relatedTo |
Besov spaces
ⓘ
Sobolev spaces ⓘ Strichartz spaces ⓘ
linked to:
Strichartz-type estimates
modulation spaces ⓘ |
| typicalNotation |
X^{s,b}
ⓘ
X^{s,b}_ ho for a dispersion relation ρ ⓘ |
| usedFor |
Fourier restriction norm method
ⓘ
linked to:
Fourier restriction theory
Strichartz-type estimates ⓘ bilinear and multilinear estimates ⓘ contraction mapping principle in PDE ⓘ fixed-point arguments in PDE ⓘ global well-posedness results ⓘ local well-posedness results ⓘ regularity theory for nonlinear PDEs ⓘ study of nonlinear dispersive partial differential equations ⓘ well-posedness of nonlinear PDEs ⓘ |
How these facts were elicited
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Instruction
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Input
Subject: Bourgain spaces Description of subject: Bourgain spaces are function spaces introduced by Jean Bourgain that are tailored to study the well-posedness and regularity of nonlinear dispersive partial differential equations.
Referenced by (3)
Full triples — surface form annotated when it differs from this entity's canonical label.
linked to: Bourgain spaces
linked to: Bourgain spaces