Riesz transforms
E746579
Riesz transforms are fundamental singular integral operators in harmonic analysis that generalize the Hilbert transform to higher dimensions and play a key role in studying function spaces and partial differential equations.
All labels observed (1)
| Label | Occurrences |
|---|---|
| Riesz transforms canonical | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T8640756 — 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: Riesz transforms Context triple: [Frigyes Riesz, knownFor, Riesz transforms]
-
A.
Bochner–Riesz means
Bochner–Riesz means are a family of summability methods in harmonic analysis used to improve the convergence of Fourier series and Fourier integrals by smoothing their partial sums.
-
B.
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.
-
C.
Calderón–Zygmund theory
Calderón–Zygmund theory is a branch of harmonic analysis that studies singular integral operators and their boundedness properties on function spaces such as L^p.
-
D.
Singular Integrals and Differentiability Properties of Functions
"Singular Integrals and Differentiability Properties of Functions" is a landmark mathematical monograph by Elias M. Stein that developed the modern theory of singular integral operators and their role in harmonic analysis and differentiability.
-
E.
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.
- 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: Riesz transforms Target entity description: Riesz transforms are fundamental singular integral operators in harmonic analysis that generalize the Hilbert transform to higher dimensions and play a key role in studying function spaces and partial differential equations.
-
A.
Bochner–Riesz means
Bochner–Riesz means are a family of summability methods in harmonic analysis used to improve the convergence of Fourier series and Fourier integrals by smoothing their partial sums.
-
B.
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.
-
C.
Calderón–Zygmund theory
Calderón–Zygmund theory is a branch of harmonic analysis that studies singular integral operators and their boundedness properties on function spaces such as L^p.
-
D.
Singular Integrals and Differentiability Properties of Functions
"Singular Integrals and Differentiability Properties of Functions" is a landmark mathematical monograph by Elias M. Stein that developed the modern theory of singular integral operators and their role in harmonic analysis and differentiability.
-
E.
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.
- F. None of above. chosen
Statements (50)
| Predicate | Object |
|---|---|
| instanceOf |
Fourier multiplier operator
ⓘ
linear operator ⓘ singular integral operator ⓘ |
| actsOn |
functions on R^n
ⓘ
tempered distributions ⓘ |
| characterizes |
BMO via L^∞-control of transforms
ⓘ
Hardy space H^1 via L^1-boundedness ⓘ |
| commutesWith |
dilations on R^n
ⓘ
translations on R^n ⓘ |
| definedFor | dimension n ≥ 1 ⓘ |
| extendsTo | weighted L^p spaces with Muckenhoupt A_p weights ⓘ |
| field |
functional analysis
ⓘ
harmonic analysis ⓘ partial differential equations ⓘ |
| forms | Riesz transform vector (R_1,…,R_n) ⓘ |
| FourierMultiplierSymbol | -i ξ_i / |ξ| ⓘ |
| generalizes | Hilbert transform ⓘ |
| hasApplicationIn |
fluid mechanics
ⓘ
geometric measure theory ⓘ quantitative rectifiability of sets ⓘ |
| hasComponent | i-th Riesz transform ⓘ |
| hasIndex | i = 1,…,n ⓘ |
| isBoundedFrom | H^1(R^n) to L^1(R^n) ⓘ |
| isBoundedOn |
BMO(R^n) modulo constants
ⓘ
Hardy space H^1(R^n) ⓘ L^p(R^n) for 1 < p < ∞ ⓘ |
| isConvolutionOperator | true (in distribution sense) ⓘ |
| isOfStrongType | (p,p) for 1 < p < ∞ ⓘ |
| isPrototypeOf | Calderón–Zygmund theory NERFINISHED ⓘ |
| isSelfAdjoint | false ⓘ |
| isSkewAdjoint | true on L^2(R^n) ⓘ |
| isTranslationInvariant | true ⓘ |
| isUnboundedOn |
L^1(R^n)
ⓘ
L^∞(R^n) ⓘ |
| isVectorValued | true ⓘ |
| isWeakType | (1,1) ⓘ |
| kernel | c_n x_i / |x|^{n+1} ⓘ |
| kernelType |
homogeneous kernel of degree -n
ⓘ
odd kernel ⓘ |
| namedAfter | Frigyes Riesz NERFINISHED ⓘ |
| operatorType | Calderón–Zygmund operator NERFINISHED ⓘ |
| relatedTo |
Cauchy–Riemann system in higher dimensions
NERFINISHED
ⓘ
Laplace operator NERFINISHED ⓘ Poisson kernel ⓘ harmonic functions ⓘ |
| satisfies | R_1^2 + … + R_n^2 = -I on suitable spaces ⓘ |
| usedToStudy |
Sobolev spaces
ⓘ
potential theory ⓘ regularity of solutions to elliptic PDEs ⓘ singular integrals ⓘ |
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: Riesz transforms Description of subject: Riesz transforms are fundamental singular integral operators in harmonic analysis that generalize the Hilbert transform to higher dimensions and play a key role in studying function spaces and partial differential equations.
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.