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 (1)
| Label | Occurrences |
|---|---|
| Bourgain spaces canonical | 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
NERFINISHED
ⓘ
X^{s,b} spaces NERFINISHED ⓘ |
| appliedTo |
Korteweg–de Vries equations
NERFINISHED
ⓘ
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 NERFINISHED ⓘ |
| introducedInContextOf |
Korteweg–de Vries equation
NERFINISHED
ⓘ
nonlinear Schrödinger equation ⓘ periodic boundary conditions ⓘ |
| namedAfter | Jean Bourgain NERFINISHED ⓘ |
| 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
NERFINISHED
ⓘ
Sobolev spaces NERFINISHED ⓘ Strichartz spaces NERFINISHED ⓘ modulation spaces ⓘ |
| typicalNotation |
X^{s,b}
ⓘ
X^{s,b}_ ho for a dispersion relation ρ ⓘ |
| usedFor |
Fourier restriction norm method
NERFINISHED
ⓘ
Strichartz-type estimates NERFINISHED ⓘ 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
The pipeline generated the facts above by prompting gpt-5.1 with this entity's name + description and the instruction below.
Instruction
You are a knowledge base construction expert. Given a subject entity and a description of it, return factual statements that you know for the subject as a JSON list of dictionaries(triples), where keys must be "subject", "predicate" and "object". The number of facts may be very high, between 25 to 50 or more, for very popular subjects. For less popular subjects, the number of facts can be very low, like 5 or 10. # Requirements - If you don't know the subject at all, return an empty list. - If the subject is not a named entity, return an empty list. - Include at least one triple where predicate is "instanceOf". - Do not get too wordy. - Separate several objects into multiple triples with one object.
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 (1)
Full triples — surface form annotated when it differs from this entity's canonical label.