Fefferman–Phong inequality
E537775
The Fefferman–Phong inequality is a fundamental result in harmonic analysis and partial differential equations that provides weighted \(L^2\) estimates controlling functions by their gradients and associated potentials.
Statements (48)
| Predicate | Object |
|---|---|
| instanceOf |
mathematical inequality
ⓘ
result in harmonic analysis ⓘ result in partial differential equations ⓘ |
| appliesTo |
Schrödinger-type operators
ⓘ
second-order elliptic operators ⓘ |
| assumes |
measurable potentials
ⓘ
nonnegative weight functions ⓘ |
| concerns | control of integrals of |u|^2 by integrals of |∇u|^2 and V|u|^2 ⓘ |
| context |
analysis of singular or rough potentials
ⓘ
local and global estimates for PDE solutions ⓘ |
| ensures |
control of potential energy by kinetic energy terms
ⓘ
lower bounds for quadratic forms involving potentials ⓘ |
| field |
harmonic analysis
ⓘ
partial differential equations ⓘ |
| hasGeneralizations |
inequalities for magnetic Schrödinger operators
ⓘ
inequalities with matrix-valued potentials ⓘ non-Euclidean settings such as manifolds and Lie groups ⓘ |
| involves |
nonnegative potentials
ⓘ
quadratic forms associated with differential operators ⓘ weighted integral inequalities ⓘ |
| mainConcept |
Schrödinger operators
NERFINISHED
ⓘ
potential terms in PDEs ⓘ weighted L^2 estimates ⓘ |
| mathematicalArea |
functional analysis
ⓘ
spectral theory ⓘ |
| namedAfter |
Charles Fefferman
NERFINISHED
ⓘ
Dinh H. Phong NERFINISHED ⓘ |
| provides | control of L^2 norms of functions by L^2 norms of gradients and potentials ⓘ |
| relatedTo |
Caccioppoli inequality
ⓘ
Fefferman–Phong class of weights NERFINISHED ⓘ Hardy inequality NERFINISHED ⓘ Kato class potentials ⓘ |
| relates |
functions
ⓘ
gradients of functions ⓘ potentials ⓘ |
| type |
coercivity estimate
ⓘ
weighted energy inequality ⓘ |
| typicalDomain | Euclidean space R^n NERFINISHED ⓘ |
| typicalFunctionSpace |
L^2 spaces
ⓘ
Sobolev spaces NERFINISHED ⓘ |
| usedFor |
Carleman-type estimates
ⓘ
a priori estimates for PDE solutions ⓘ regularity theory in PDE ⓘ spectral theory of Schrödinger operators ⓘ unique continuation problems ⓘ |
| usedIn |
boundedness of Riesz transforms
ⓘ
estimates for eigenvalues of Schrödinger operators ⓘ study of self-adjointness of Schrödinger operators ⓘ |
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.