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.
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 ⓘ |
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.