Riesz–Thorin interpolation theorem
E746578
The Riesz–Thorin interpolation theorem is a fundamental result in functional analysis that provides bounds for linear operators between Lᵖ spaces by interpolating their behavior between two known endpoint estimates.
All labels observed (1)
| Label | Occurrences |
|---|---|
| Riesz–Thorin interpolation theorem canonical | 1 |
Statements (48)
| Predicate | Object |
|---|---|
| instanceOf |
interpolation theorem
ⓘ
mathematical theorem ⓘ |
| appearsIn |
graduate textbooks on functional analysis
ⓘ
graduate textbooks on harmonic analysis ⓘ |
| applicableWhen | operator is linear ⓘ |
| appliesTo | linear operators between Lp and Lq spaces ⓘ |
| assumes | operator is bounded on two endpoint Lp spaces ⓘ |
| characteristic | interpolates exponents linearly in 1/p and 1/q ⓘ |
| concerns |
Lp spaces
ⓘ
bounded linear operators ⓘ complex interpolation ⓘ norm estimates ⓘ |
| consequence | convexity of log of operator norm in interpolation parameter ⓘ |
| contrastWith | real interpolation methods ⓘ |
| field |
functional analysis
ⓘ
harmonic analysis ⓘ operator theory ⓘ |
| generalizes | Riesz convexity theorem NERFINISHED ⓘ |
| gives | bounds for operator norms between Lp spaces ⓘ |
| hasVersion |
finite measure space version
ⓘ
sigma-finite measure space version ⓘ |
| historicalNote | proved independently by Marcel Riesz and Gunnar Thorin ⓘ |
| implies |
Lp boundedness from Lp0 and Lp1 bounds
ⓘ
intermediate operator norm estimate is log-convex in 1/p and 1/q ⓘ operator is bounded on intermediate Lp spaces ⓘ |
| influenced | development of modern interpolation theory ⓘ |
| involves |
holomorphic families of operators
ⓘ
strip in the complex plane ⓘ |
| isPartOf | interpolation theory of operators ⓘ |
| namedAfter |
Gunnar Thorin
NERFINISHED
ⓘ
Marcel Riesz NERFINISHED ⓘ |
| notApplicableTo | nonlinear operators in its standard form ⓘ |
| relatedConcept |
Banach space interpolation
NERFINISHED
ⓘ
Lp interpolation scale ⓘ |
| relatedTo |
Marcinkiewicz interpolation theorem
NERFINISHED
ⓘ
Stein interpolation theorem NERFINISHED ⓘ |
| requires | measure spaces to be sigma-finite in standard formulations ⓘ |
| typeOf | complex method interpolation result ⓘ |
| typicalAssumption | operator acts on simple functions and extends by density ⓘ |
| usedIn |
Fourier analysis
NERFINISHED
ⓘ
Lp-boundedness of Fourier transform related operators ⓘ ergodic theory ⓘ estimates for convolution operators ⓘ partial differential equations ⓘ probability theory ⓘ study of singular integral operators ⓘ |
| uses |
Hadamard three-lines theorem
NERFINISHED
ⓘ
complex analytic methods ⓘ |
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.