Sobolev inequality

Let W k,p(Rn) denote the Sobolev space consisting of all real-valued functions on Rn whose weak derivatives up to order k are functions in Lp.The first part of the Sobolev embedding theorem states that if k > ℓ, p < n and 1 ≤ p < q < ∞ are two real numbers such that (givenand This special case of the Sobolev embedding is a direct consequence of the Gagliardo–Nirenberg–Sobolev inequality.itself has improved local behavior, meaning that it belongs to the space, in addition, This part of the Sobolev embedding is a direct consequence of Morrey's inequality.The Sobolev embedding theorem holds for Sobolev spaces W k,p(M) on other suitable domains M. In particular (Aubin 1982, Chapter 2; Aubin 1976), both parts of the Sobolev embedding hold when If M is a bounded open set in Rn with continuous boundary, then W 1,2(M) is compactly embedded in L2(M) (Nečas 2012, Section 1.1.5, Theorem 1.4).On a compact manifold M with C1 boundary, the Kondrachov embedding theorem states that if k > ℓ and[1] Note that the condition is just as in the first part of the Sobolev embedding theorem, with the equality replaced by an inequality, thus requiring a more regular space W k,p(M).Assume that u is a continuously differentiable real-valued function on Rn with compact support.An equivalent statement is known as the Sobolev lemma in (Aubin 1982, Chapter 2).Then, for q defined by there exists a constant C depending only on p such that If p = 1, then one has two possible replacement estimates.The boundedness of the Riesz transforms implies that the latter inequality gives a unified way to write the family of inequalities for the Riesz potential.A similar result holds in a bounded domain U with Lipschitz boundary.This version of the inequality follows from the previous one by applying the norm-preserving extension of W 1,p(U) to W 1,p(Rn).The inequality is named after Charles B. Morrey Jr. Let U be a bounded open subset of Rn, with a C1 boundary.(U may also be unbounded, but in this case its boundary, if it exists, must be sufficiently well-behaved.)Here, we conclude that u belongs to a Hölder space, more precisely: where We have in addition the estimate the constant C depending only on k, p, n, γ, and U., then u is a function of bounded mean oscillation and for some constant C depending only on n.[5]: §I.2  This estimate is a corollary of the Poincaré inequality.The Nash inequality, introduced by John Nash (1958), states that there exists a constant C > 0, such that for all u ∈ L1(Rn) ∩ W 1,2(Rn), The inequality follows from basic properties of the Fourier transform.Indeed, integrating over the complement of the ball of radius ρ, becauseOn the other hand, one has which, when integrated over the ball of radius ρ gives where ωn is the volume of the n-ball.Choosing ρ to minimize the sum of (1) and (2) and applying Parseval's theorem: gives the inequality.In fact, if I is a bounded interval, then for all 1 ≤ r < ∞ and all 1 ≤ q ≤ p < ∞ the following inequality holds where: The simplest of the Sobolev embedding theorems, described above, states that if a functionis defined is large, the improvement in the local behavior ofIn particular, for functions on an infinite-dimensional space, we cannot expect any direct analog of the classical Sobolev embedding theorems.There is, however, a type of Sobolev inequality, established by Leonard Gross (Gross 1975) and known as a logarithmic Sobolev inequality, that has dimension-independent constants and therefore continues to hold in the infinite-dimensional setting.The logarithmic Sobolev inequality says, roughly, that if a function is inThe inequality expressing this fact has constants that do not involve the dimension of the space and, thus, the inequality holds in the setting of a Gaussian measure on an infinite-dimensional space., this improvement is sufficient to derive an important result, namely hypercontractivity for the associated Dirichlet form operator.This result means that if a function is in the range of the exponential of the Dirichlet form operator—which means that the function has, in some sense, infinitely many derivatives in
Graphical representation of the embedding conditions. The space W 3,p , represented by a blue dot at the point (1/ p , 3) , embeds into the spaces indicated by red dots, all lying on a line with slope n . The white circle at (0,0) indicates the impossibility of optimal embeddings into L ∞ .
If the line from the picture above intersects the y-axis at s = r + α , the embedding into a Hölder space C r, α (red) holds. White circles indicate intersection points at which optimal embeddings are not valid.
mathematicsmathematical analysisSobolev spacesRellich–Kondrachov theoremcompactly embeddedSergei Lvovich SobolevSobolev spaceweak derivativesSobolev conjugateHölder spacesSouček spaceboundedopen setLipschitzcone conditioncompactRiemannian manifoldmanifold with boundarycompleteinjectivity radiussectional curvaturecompletely continuouscompact supportfractional integrationRiesz potentialRiesz transformRiesz transformsHölder continuousCharles B. Morrey Jr.Hölder spacebounded mean oscillationPoincaré inequalityJohn NashFourier transformn-ballLogarithmic Sobolev inequalitiesLeonard Grosslogarithmic Sobolev inequalityDirichlet formSpringer-VerlagBrezis, HaïmMassonSpringer Science & Business MediaEvans, LawrenceAmerican Mathematical SocietyGross, LeonardMaz'ja, Vladimir G.Nash, J.American Journal of MathematicsBibcodeEncyclopedia of MathematicsEMS PressStein, EliasPrinceton University PressFunctional analysistopicsglossaryBanachFréchetHilbertHölderNuclearOrliczSchwartzSobolevTopological vectorBarrelledLocally convexReflexiveSeparableHahn–BanachRiesz representationClosed graphUniform boundedness principleKrein–MilmanMin–maxGelfand–NaimarkBanach–AlaogluAdjointHilbert–SchmidtNormalTrace classTransposeUnboundedUnitaryBanach algebraC*-algebraSpectrum of a C*-algebraOperator algebraGroup algebra of a locally compact groupVon Neumann algebraInvariant subspace problemMahler's conjectureHardy spaceSpectral theory of ordinary differential equationsHeat kernelIndex theoremCalculus of variationsFunctional calculusIntegral linear operatorJones polynomialTopological quantum field theoryNoncommutative geometryRiemann hypothesisDistributionGeneralized functionsApproximation propertyBalanced setChoquet theoryWeak topologyBanach–Mazur distanceTomita–Takesaki theory