Hardy inequality
E451926
The Hardy inequality is a fundamental result in mathematical analysis that provides bounds on integrals or sums involving a function and its distance from a point, with important applications in functional analysis and partial differential equations.
All labels observed (1)
| Label | Occurrences |
|---|---|
| Hardy inequality canonical | 2 |
Statements (49)
| Predicate | Object |
|---|---|
| instanceOf |
mathematical inequality
ⓘ
result in mathematical analysis ⓘ |
| appliesTo |
functions on Euclidean space
ⓘ
functions on domains with boundary ⓘ sequences of real or complex numbers ⓘ |
| describes |
bounds on integrals involving a function and its distance from a point
ⓘ
bounds on sums involving a sequence and its index ⓘ |
| field |
Sobolev spaces
NERFINISHED
ⓘ
functional analysis ⓘ mathematical analysis ⓘ partial differential equations ⓘ spectral theory ⓘ |
| generalizedBy |
Hardy–Littlewood inequalities
NERFINISHED
ⓘ
Hardy–Sobolev inequalities NERFINISHED ⓘ |
| hasApplication |
boundary behavior of harmonic functions
ⓘ
quantum mechanics with inverse-square potentials ⓘ stability analysis of PDE solutions ⓘ weighted norm inequalities ⓘ |
| hasProperty |
extremal functions often do not exist in critical case
ⓘ
scale invariant in critical cases ⓘ sharp constants known in many cases ⓘ |
| hasVariant |
Hardy inequality in L^p spaces
ⓘ
Hardy inequality on R^n NERFINISHED ⓘ Hardy inequality on bounded domains ⓘ Hardy inequality with remainder term NERFINISHED ⓘ Hardy–Rellich inequality NERFINISHED ⓘ continuous Hardy inequality ⓘ discrete Hardy inequality ⓘ improved Hardy inequality ⓘ |
| holdsFor |
1-dimensional domains
ⓘ
n-dimensional Euclidean space ⓘ radial functions in R^n ⓘ |
| involves |
distance to a point or boundary
ⓘ
inverse-square type weights ⓘ singular weights ⓘ weighted L^p norms ⓘ |
| namedAfter | G. H. Hardy NERFINISHED ⓘ |
| originatedIn | early 20th century ⓘ |
| relatedTo |
Caffarelli–Kohn–Nirenberg inequalities
NERFINISHED
ⓘ
Poincaré inequality NERFINISHED ⓘ Sobolev inequality NERFINISHED ⓘ uncertainty principle NERFINISHED ⓘ |
| usedIn |
analysis of singular potentials
ⓘ
control of behavior near singularities ⓘ estimates for solutions of elliptic PDEs ⓘ regularity theory for PDEs ⓘ spectral estimates for differential operators ⓘ study of Schrödinger operators ⓘ study of critical exponents in Sobolev embeddings ⓘ |
Referenced by (2)
Full triples — surface form annotated when it differs from this entity's canonical label.