{"history":[],"article":{"id":"bb8fbf9a-69fc-4585-800a-969210c57f94","title":"Sobolev inequality","slug":"sobolev-inequality","content":":::summary\n:::\n\n# \nIn [mathematics](https://bliptext.com/articles/mathematics), there is in [mathematical analysis](https://bliptext.com/articles/mathematical-analysis) a class of **Sobolev inequalities**, relating norms including those of [Sobolev spaces](https://bliptext.com/articles/sobolev-space).\nThese are used to prove the **Sobolev embedding theorem**, giving inclusions between certain [Sobolev spaces](https://bliptext.com/articles/sobolev-space), and the [Rellich–Kondrachov theorem](https://bliptext.com/articles/rellich-kondrachov-theorem) showing that under slightly stronger conditions some [Sobolev spaces](https://bliptext.com/articles/sobolev-space) are [compactly embedded](https://bliptext.com/articles/compactly-embedded) in others.\nThey are named after [Sergei Lvovich Sobolev](https://bliptext.com/articles/sergei-lvovich-sobolev).\n\n# Sobolev embedding theorem\n[[File:Sobolev embedding theorem.svg|thumb|250px|Graphical representation of the embedding conditions. The space\nW 3,p\n, represented by a blue dot at the point\n(1/p, 3)\n, embeds into the spaces indicated by red dots, all lying on a line with slope\nn\n. The white circle at\n(0,0)\nindicates the impossibility of optimal embeddings into\nL ∞\n.]] Let\nW k,p(Rn)\ndenote the [Sobolev space](https://bliptext.com/articles/sobolev-space) consisting of all real-valued functions on\nRn\nwhose [weak derivatives](https://bliptext.com/articles/weak-derivative) up to order k are functions in\nLp\n. Here k is a non-negative integer and\n1 ≤ p < ∞\n. The first part of the Sobolev embedding theorem states that if\nk > ℓ\n,\np < n\nand\n1 ≤ p < q < ∞\nare two real numbers such that\n(given n, p, k and \\ell this is satisfied for some provided ),\nthen\nand the embedding is continuous: for every, one has\n, and\nIn the special case of\nand\n, Sobolev embedding gives\nwhere\np∗\nis the **[Sobolev conjugate](https://bliptext.com/articles/sobolev-conjugate)** of p, given by\nand for every, one has\nand\nThis special case of the Sobolev embedding is a direct consequence of the [Gagliardo–Nirenberg–Sobolev inequality](https://bliptext.com/articles/).\nThe result should be interpreted as saying that if a function f in has one derivative in L^p, then f itself has improved local behavior, meaning that it belongs to the space L^{p^*} where p^*>p.\n(Note that 1/p^*<1/p, so that p^*>p.) Thus, any local singularities in f must be more mild than for a typical function in L^p.\n[[File:Sobolev embedding theorem (Morrey case).svg|thumb|250px|If the line from the picture above intersects the y-axis at s = r + α, the embedding into a Hölder space\nC r, α\n(red) holds.\nWhite circles indicate intersection points at which optimal embeddings are not valid.]]\nThe second part of the Sobolev embedding theorem applies to embeddings in [Hölder spaces](https://bliptext.com/articles/h-lder-space)\nC r,α(Rn)\n. If\nn < pk\nand\nwith\nα ∈ (0, 1)\nthen one has the embedding\nIn other words, for every and, one has\nThis part of the Sobolev embedding is a direct consequence of [Morrey's inequality](https://bliptext.com/articles/).\nIntuitively, this inclusion expresses the fact that the existence of sufficiently many weak derivatives implies some continuity of the classical derivatives.\nIf \\alpha = 1 then for every.\nIn particular, as long as pk>n, the embedding criterion will hold with r=0 and some positive value of \\alpha.\nThat is, for a function f on \\mathbb R^n, if f has k derivatives in L^p and pk>n, then f will be continuous (and actually Hölder continuous with some positive exponent \\alpha).\n\n## Generalizations\nThe Sobolev embedding theorem holds for Sobolev spaces\nW k,p(M)\non other suitable domains M. In particular, both parts of the Sobolev embedding hold when\nRn\nwith [Lipschitz](https://bliptext.com/articles/lipschitz-continuity) boundary (or whose boundary satisfies the [cone condition](https://bliptext.com/articles/cone-condition); )\nδ > 0\nand bounded [sectional curvature](https://bliptext.com/articles/sectional-curvature).\nIf M is a bounded open set in\nRn\nwith continuous boundary, then\nW 1,2(M)\nis compactly embedded in\nL2(M)\n.\n\n## Kondrachov embedding theorem\nOn a compact manifold\nM\nwith\nC1\nboundary, the **Kondrachov embedding theorem** states that if\nk > ℓ\nandthen the Sobolev embedding\nis [completely continuous](https://bliptext.com/articles/completely-continuous) (compact).\nNote 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\nW k,p(M)\n.\n\n# Gagliardo–Nirenberg–Sobolev inequality\nAssume that u is a continuously differentiable real-valued function on\nRn\nwith [compact support](https://bliptext.com/articles/compact-support).\nThen for\n1 ≤ p < n\nthere is a constant C depending only on n and p such that\nwith.\nThe case 1< p < n is due to Sobolev and the case p =1 to Gagliardo and Nirenberg independently.\nThe Gagliardo–Nirenberg–Sobolev inequality implies directly the Sobolev embedding\nThe embeddings in other orders on\nRn\nare then obtained by suitable iteration.\n\n# Hardy–Littlewood–Sobolev lemma\nSobolev's original proof of the Sobolev embedding theorem relied on the following, sometimes known as the Hardy–Littlewood–Sobolev [fractional integration](https://bliptext.com/articles/fractional-integration) theorem.\nAn equivalent statement is known as the **Sobolev lemma** in.\nA proof is in.\nLet\n0 < α < n\nand\n1 < p < q < ∞\n. Let\nbe the [Riesz potential](https://bliptext.com/articles/riesz-potential) on\nRn\n. Then, for q defined by\nthere exists a constant C depending only on p such that\nIf\n, then one has two possible replacement estimates.\nThe first is the more classical weak-type estimate:\nwhere\n. Alternatively one has the estimatewhere Rf is the vector-valued [Riesz transform](https://bliptext.com/articles/riesz-transform), c.f. . The boundedness of the [Riesz transform](https://bliptext.com/articles/riesz-transform)s implies that the latter inequality gives a unified way to write the family of inequalities for the Riesz potential.\nThe Hardy–Littlewood–Sobolev lemma implies the Sobolev embedding essentially by the relationship between the [Riesz transforms](https://bliptext.com/articles/riesz-transform) and the Riesz potentials.\n\n# Morrey's inequality\nAssume\nn < p ≤ ∞\n. Then there exists a constant C, depending only on p and n, such that\nfor all\nu ∈ C1(Rn) ∩ Lp(Rn)\n, where\nThus if\nu ∈ W 1,p(Rn)\n, then u is in fact [Hölder continuous](https://bliptext.com/articles/h-lder-continuous) of exponent γ, after possibly being redefined on a set of measure 0.\nA similar result holds in a bounded domain U with Lipschitz boundary.\nIn this case,\nwhere the constant C depends now on\nn, p\nand U. This version of the inequality follows from the previous one by applying the norm-preserving extension of\nW 1,p(U)\nto\nW 1,p(Rn)\n. The inequality is named after [Charles B. Morrey Jr.](https://bliptext.com/articles/charles-b-morrey-jr)\n\n# General Sobolev inequalities\nLet U be a bounded open subset of\nRn\n, with a\nC1\nboundary.\n(U may also be unbounded, but in this case its boundary, if it exists, must be sufficiently well-behaved.)\nAssume\nu ∈ W k,p(U)\n. Then we consider two cases:\n\n## k < n/p\n\n or \n\nk {{=}} n\n, \n\np {{=}} 1\nIn this case we conclude that\nu ∈ Lq(U)\n, where\nWe have in addition the estimate\nthe constant C depending only on\nk, p, n\n, and U.\n\n## k > n/p\nHere, we conclude that u belongs to a [Hölder space](https://bliptext.com/articles/h-lder-space), more precisely:\nwhere\nWe have in addition the estimate\nthe constant C depending only on\nk, p, n, γ\n, and U. In particular, the condition k>n/p guarantees that u is continuous (and actually Hölder continuous with some positive exponent).\n\n# Case \n\np=n, k=1\nIf, then u is a function of [bounded mean oscillation](https://bliptext.com/articles/bounded-mean-oscillation) and\nfor some constant C depending only on n. This estimate is a corollary of the [Poincaré inequality](https://bliptext.com/articles/poincar-inequality).\n\n# Nash inequality\nThe Nash inequality, introduced by, states that there exists a constant\nC > 0\n, such that for all\nu ∈ L1(Rn) ∩ W 1,2(Rn)\n,\nThe inequality follows from basic properties of the [Fourier transform](https://bliptext.com/articles/fourier-transform).\nIndeed, integrating over the complement of the ball of radius ρ,\nbecause.\nOn the other hand, one has\nwhich, when integrated over the ball of radius ρ gives\nwhere\nωn\nis the volume of the [n-ball](https://bliptext.com/articles/n-sphere).\nChoosing ρ to minimize the sum of and and applying Parseval's theorem:\ngives the inequality.\nIn the special case of\n, the Nash inequality can be extended to the\nLp\ncase, in which case it is a generalization of the Gagliardo-Nirenberg-Sobolev inequality (, Comments on Chapter 8).\nIn fact, if I is a bounded interval, then for all\n1 ≤ r < ∞\nand all\n1 ≤ q ≤ p < ∞\nthe following inequality holds\nwhere:\n\n# Logarithmic Sobolev inequality\nThe simplest of the Sobolev embedding theorems, described above, states that if a function f in has one derivative in L^p, then f itself is in L^{p^*}, where\nWe can see that as n tends to infinity, p^* approaches p. Thus, if the dimension n of the space on which f is defined is large, the improvement in the local behavior of f from having a derivative in L^p is small (p^* is only slightly larger than p).\nIn particular, for functions on an infinite-dimensional space, we cannot expect any direct analog of the classical Sobolev embedding theorems.\nThere is, however, a type of Sobolev inequality, established by [Leonard Gross](https://bliptext.com/articles/leonard-gross) and known as a [logarithmic Sobolev inequality](https://bliptext.com/articles/logarithmic-sobolev-inequalities), that has dimension-independent constants and therefore continues to hold in the infinite-dimensional setting.\nThe logarithmic Sobolev inequality says, roughly, that if a function is in L^p with respect to a Gaussian measure and has one derivative that is also in L^p, then f is in \"L^p-log\", meaning that the integral of is finite.\nThe 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.\nIt is now known that logarithmic Sobolev inequalities hold for many different types of measures, not just Gaussian measures.\nAlthough it might seem as if the L^p-log condition is a very small improvement over being in L^p, this improvement is sufficient to derive an important result, namely hypercontractivity for the associated [Dirichlet form](https://bliptext.com/articles/dirichlet-form) operator.\nThis 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 L^p—then the function does belong to L^{p^*} for some p^*>p ( Theorem 6).","current_revision":null,"created_by":"kACnuQmH5k9Nr5VNVEpVeg7XpvUmMctD","created_at":"2025-01-26T07:31:40.748Z","updated_at":"2025-01-26T07:31:40.748Z","search_vector":"'inequ':2 'sobolev':1","search_content":" \n \n\n# \nIn [mathematics](https://bliptext.com/articles/mathematics), there is in [mathematical analysis](https://bliptext.com/articles/mathematical-analysis) a class of **Sobolev inequalities**, relating norms including those of [Sobolev spaces](https://bliptext.com/articles/sobolev-space).\nThese are used to prove the **Sobolev embedding theorem**, giving inclusions between certain [Sobolev spaces](https://bliptext.com/articles/sobolev-space), and the [Rellich–Kondrachov theorem](https://bliptext.com/articles/rellich-kondrachov-theorem) showing that under slightly stronger conditions some [Sobolev spaces](https://bliptext.com/articles/sobolev-space) are [compactly embedded](https://bliptext.com/articles/compactly-embedded) in others.\nThey are named after [Sergei Lvovich Sobolev](https://bliptext.com/articles/sergei-lvovich-sobolev).\n\n# Sobolev embedding theorem\n[[File:Sobolev embedding theorem.svg|thumb|250px|Graphical representation of the embedding conditions. The space\nW 3,p\n, represented by a blue dot at the point\n(1/p, 3)\n, embeds into the spaces indicated by red dots, all lying on a line with slope\nn\n. The white circle at\n(0,0)\nindicates the impossibility of optimal embeddings into\nL ∞\n.]] Let\nW k,p(Rn)\ndenote the [Sobolev space](https://bliptext.com/articles/sobolev-space) consisting of all real-valued functions on\nRn\nwhose [weak derivatives](https://bliptext.com/articles/weak-derivative) up to order k are functions in\nLp\n. Here k is a non-negative integer and\n1 ≤ p ℓ\n,\np p.\n(Note that 1/p^* p.) Thus, any local singularities in f must be more mild than for a typical function in L^p.\n[[File:Sobolev embedding theorem (Morrey case).svg|thumb|250px|If the line from the picture above intersects the y-axis at s = r + α, the embedding into a Hölder space\nC r, α\n(red) holds.\nWhite circles indicate intersection points at which optimal embeddings are not valid.]]\nThe second part of the Sobolev embedding theorem applies to embeddings in [Hölder spaces](https://bliptext.com/articles/h-lder-space)\nC r,α(Rn)\n. If\nn n, the embedding criterion will hold with r=0 and some positive value of \\alpha.\nThat is, for a function f on \\mathbb R^n, if f has k derivatives in L^p and pk>n, then f will be continuous (and actually Hölder continuous with some positive exponent \\alpha).\n\n## Generalizations\nThe Sobolev embedding theorem holds for Sobolev spaces\nW k,p(M)\non other suitable domains M. In particular, both parts of the Sobolev embedding hold when\nRn\nwith [Lipschitz](https://bliptext.com/articles/lipschitz-continuity) boundary (or whose boundary satisfies the [cone condition](https://bliptext.com/articles/cone-condition); )\nδ > 0\nand bounded [sectional curvature](https://bliptext.com/articles/sectional-curvature).\nIf M is a bounded open set in\nRn\nwith continuous boundary, then\nW 1,2(M)\nis compactly embedded in\nL2(M)\n.\n\n## Kondrachov embedding theorem\nOn a compact manifold\nM\nwith\nC1\nboundary, the **Kondrachov embedding theorem** states that if\nk > ℓ\nandthen the Sobolev embedding\nis [completely continuous](https://bliptext.com/articles/completely-continuous) (compact).\nNote 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\nW k,p(M)\n.\n\n# Gagliardo–Nirenberg–Sobolev inequality\nAssume that u is a continuously differentiable real-valued function on\nRn\nwith [compact support](https://bliptext.com/articles/compact-support).\nThen for\n1 ≤ p n/p\nHere, we conclude that u belongs to a [Hölder space](https://bliptext.com/articles/h-lder-space), more precisely:\nwhere\nWe have in addition the estimate\nthe constant C depending only on\nk, p, n, γ\n, and U. In particular, the condition k>n/p guarantees that u is continuous (and actually Hölder continuous with some positive exponent).\n\n# Case \n\np=n, k=1\nIf, then u is a function of [bounded mean oscillation](https://bliptext.com/articles/bounded-mean-oscillation) and\nfor some constant C depending only on n. This estimate is a corollary of the [Poincaré inequality](https://bliptext.com/articles/poincar-inequality).\n\n# Nash inequality\nThe Nash inequality, introduced by, states that there exists a constant\nC > 0\n, such that for all\nu ∈ L1(Rn) ∩ W 1,2(Rn)\n,\nThe inequality follows from basic properties of the [Fourier transform](https://bliptext.com/articles/fourier-transform).\nIndeed, integrating over the complement of the ball of radius ρ,\nbecause.\nOn the other hand, one has\nwhich, when integrated over the ball of radius ρ gives\nwhere\nωn\nis the volume of the [n-ball](https://bliptext.com/articles/n-sphere).\nChoosing ρ to minimize the sum of and and applying Parseval's theorem:\ngives the inequality.\nIn the special case of\n, the Nash inequality can be extended to the\nLp\ncase, in which case it is a generalization of the Gagliardo-Nirenberg-Sobolev inequality (, Comments on Chapter 8).\nIn fact, if I is a bounded interval, then for all\n1 ≤ r p ( Theorem 6).","revisionCount":0},"hasMore":false,"nextCursor":null}

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