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 |
How this entity was disambiguated
This entity first appeared as the object of triple T4551990 — 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: Hardy inequality Context triple: [G. H. Hardy, knownFor, Hardy inequality]
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A.
Poincaré inequality
The Poincaré inequality is a fundamental result in functional analysis and partial differential equations that bounds the average oscillation of a function by the size of its gradient, playing a key role in Sobolev space theory and the study of elliptic problems.
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B.
Young's inequality
Young's inequality is a fundamental result in mathematical analysis that provides an upper bound for the product of two nonnegative numbers in terms of their powers, playing a key role in convex analysis and functional inequalities.
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C.
Lyapunov inequality
The Lyapunov inequality is a fundamental result in stability theory and analysis that provides bounds relating norms or moments of functions or solutions to differential equations, widely used in studying the stability of dynamical systems.
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D.
Hölder inequality
Hölder inequality is a fundamental result in mathematical analysis that generalizes the Cauchy–Schwarz inequality and provides bounds for integrals or sums of products in Lᵖ spaces.
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E.
Minkowski inequality
The Minkowski inequality is a fundamental result in functional analysis and measure theory that generalizes the triangle inequality to L^p spaces, providing a key tool for studying norms and integrable functions.
- 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: Hardy inequality Target entity description: 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.
-
A.
Poincaré inequality
The Poincaré inequality is a fundamental result in functional analysis and partial differential equations that bounds the average oscillation of a function by the size of its gradient, playing a key role in Sobolev space theory and the study of elliptic problems.
-
B.
Young's inequality
Young's inequality is a fundamental result in mathematical analysis that provides an upper bound for the product of two nonnegative numbers in terms of their powers, playing a key role in convex analysis and functional inequalities.
-
C.
Lyapunov inequality
The Lyapunov inequality is a fundamental result in stability theory and analysis that provides bounds relating norms or moments of functions or solutions to differential equations, widely used in studying the stability of dynamical systems.
-
D.
Hölder inequality
Hölder inequality is a fundamental result in mathematical analysis that generalizes the Cauchy–Schwarz inequality and provides bounds for integrals or sums of products in Lᵖ spaces.
-
E.
Minkowski inequality
The Minkowski inequality is a fundamental result in functional analysis and measure theory that generalizes the triangle inequality to L^p spaces, providing a key tool for studying norms and integrable functions.
- F. None of above. chosen
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 ⓘ |
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: Hardy inequality Description of subject: 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.
Referenced by (2)
Full triples — surface form annotated when it differs from this entity's canonical label.