Sobolev inequality#Sobolev embedding theorem

File:Sobolev embedding theorem.svg

Let {{math|W^k,p(Rⁿ)}} denote the Sobolev space consisting of all real-valued functions on {{math|Rⁿ}} whose weak derivatives up to order {{mvar|k}} are functions in {{math|L^p}}. Here {{mvar|k}} is a non-negative integer and {{math|1 ≤ p < ∞}}. The first part of the Sobolev embedding theorem states that if {{math|k > ℓ}}, {{math|p < n}} and {{math|1 ≤ p < q < ∞}} are two real numbers such that

:$\backslash frac\{1\}\{p\}-\backslash frac\{k\}\{n\}\; =\; \backslash frac\{1\}\{q\}\; -\backslash frac\{\backslash ell\}\{n\},$

(given $n$, $p$, $k$ and $\backslash ell$ this is satisfied for some $q\; \backslash in\; [1,\; \backslash infty)$ provided ),

then

:$W^\{k,p\}(\backslash mathbf\{R\}^n)\backslash subseteq\; W^\{\backslash ell,q\}(\backslash mathbf\{R\}^n)$

and the embedding is continuous: for every $f\; \backslash in\; W^\{k,p\}(\backslash mathbf\{R\}^n)$, one has

$f\; \backslash in\; W^\{l,q\}(\backslash mathbf\{R\}^n)$, and

:$$

\biggl(\int_{\mathbf{R}^n} \vert \nabla^\ell f\vert^q \biggr)^\frac{1}{q}

\le C \biggl(\int_{\mathbf{R}^n} \vert \nabla^k f\vert^p \biggr)^\frac{1}{p}.

In the special case of {{math|k {{=}} 1}} and {{math|ℓ {{=}} 0}}, Sobolev embedding gives

:$W^\{1,p\}(\backslash mathbf\{R\}^n)\; \backslash subseteq\; L^\{p^*\}(\backslash mathbf\{R\}^n)$

where {{math|p^∗}} is the Sobolev conjugate of {{mvar|p}}, given by

:$\backslash frac\{1\}\{p^*\}\; =\; \backslash frac\{1\}\{p\}\; -\; \backslash frac\{1\}\{n\}$

and for every $f\; \backslash in\; W^\{1,p\}(\backslash mathbf\{R\}^n)$, one has

$f\; \backslash in\; L^\{p^*\}(\backslash mathbf\{R\}^n)$ and

:$$

\biggl(\int_{\mathbf{R}^n} \vert f\vert^{p^*} \biggr)^{\frac{1}{p^*}}

\le C \biggl(\int_{\mathbf{R}^n} \vert \nabla f\vert^p \biggr)^{\frac{1}{p}}.

This special case of the Sobolev embedding is a direct consequence of the Gagliardo–Nirenberg–Sobolev inequality. The result should be interpreted as saying that if a function $f$ in $L^p(\backslash mathbf\{R\}^n)$ has one derivative in $L^p$, then $f$ itself has improved local behavior, meaning that it belongs to the space $L^\{p^*\}$ where $p^*>p$. (Note that , so that $p^*>p$.) Thus, any local singularities in $f$ must be more mild than for a typical function in $L^p$.

File:Sobolev embedding theorem (Morrey case).svg

The second part of the Sobolev embedding theorem applies to embeddings in Hölder spaces {{math|C^r,α(Rⁿ)}}. If {{math|n < pk}} and

:$\backslash frac\{1\}\{p\}-\backslash frac\{k\}\{n\}\; =\; -\backslash frac\{r\; +\; \backslash alpha\}\{n\},\; \backslash mbox\{\; or,\; equivalently,\; \}\; r\; +\; \backslash alpha\; =\; k\; -\; \backslash frac\{n\}\{p\}$

with {{math|α ∈ (0, 1)}} then one has the embedding

:$W^\{k,p\}(\backslash mathbf\{R\}^n)\backslash subset\; C^\{r,\backslash alpha\}(\backslash mathbf\{R\}^n).$

In other words, for every $f\; \backslash in\; W^\{k,p\}(\backslash mathbf\{R\}^n)$ and $x,\; y\; \backslash in\; \backslash mathbf\{R\}^n$, one has $f\; \backslash in\; C^r\; (\backslash mathbf\{R\}^n)$, in addition,

:$$

\vert \nabla^r f (x) - \nabla^r f (y) \vert

\le C \biggl(\int_{\mathbf{R}^n} \vert \nabla^k f \vert^p\biggr)^\frac{1}{p} \vert x - y \vert^\alpha.

This part of the Sobolev embedding is a direct consequence of Morrey's inequality. Intuitively, this inclusion expresses the fact that the existence of sufficiently many weak derivatives implies some continuity of the classical derivatives. If $\backslash alpha\; =\; 1$ then $W^\{k,p\}(\backslash mathbf\{R\}^n)\backslash subset\; C^\{r,\backslash gamma\}(\backslash mathbf\{R\}^n)$ for every $\backslash gamma\; \backslash in\; (0,1)$.

In particular, as long as $pk>n$, the embedding criterion will hold with $r=0$ and some positive value of $\backslash alpha$. That is, for a function $f$ on $\backslash mathbb\; R^n$, if $f$ has $k$ derivatives in $L^p$ and $pk>n$, then $f$ will be continuous (and actually Hölder continuous with some positive exponent $\backslash alpha$).

{{Further|Souček space}}

The Sobolev embedding theorem holds for Sobolev spaces {{math|W^k,p(M)}} on other suitable domains {{mvar|M}}. In particular ({{harvnb|Aubin|1982|loc=Chapter 2}}; {{harvnb|Aubin|1976}}), both parts of the Sobolev embedding hold when

• {{mvar|M}} is a bounded open set in {{math|Rⁿ}} with Lipschitz boundary (or whose boundary satisfies the cone condition; {{harvnb|Adams|1975|loc=Theorem 5.4}})
• {{mvar|M}} is a compact Riemannian manifold
• {{mvar|M}} is a compact Riemannian manifold with boundary and the boundary is Lipschitz (meaning that the boundary can be locally represented as a graph of a Lipschitz continuous function).
• {{mvar|M}} is a complete Riemannian manifold with injectivity radius {{math|δ > 0}} and bounded sectional curvature.

If {{mvar|M}} is a bounded open set in {{math|Rⁿ}} with continuous boundary, then {{math|W^1,2(M)}} is compactly embedded in {{math|L²(M)}} ({{harvnb|Nečas|2012|loc=Section 1.1.5, Theorem 1.4}}).

{{main article|Rellich–Kondrachov theorem}}

On a compact manifold {{math|M}} with {{math|C¹}} boundary, the Kondrachov embedding theorem states that if {{math|k > ℓ}} andthen the Sobolev embedding

:$W^\{k,p\}(M)\backslash subset\; W^\{\backslash ell,q\}(M)$

is completely continuous (compact).{{cite book | author=Taylor, Michael E. | title=Partial Differential Equations I - Basic Theory | year=1997 | edition=2nd | isbn=0-387-94653-5 | page=286}} 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 {{math|W^k,p(M)}}.

Assume that {{mvar|u}} is a continuously differentiable real-valued function on {{math|Rⁿ}} with compact support. Then for {{math|1 ≤ p < n}} there is a constant {{mvar|C}} depending only on {{mvar|n}} and {{mvar|p}} such that

: $\backslash |u\backslash |\_\{L^\{p^*\}(\backslash mathbf\{R\}^n)\}\backslash leq\; C\; \backslash |Du\backslash |\_\{L^\{p\}(\backslash mathbf\{R\}^n)\}.$

with $1/p^*\; =\; 1/p\; -\; 1/n$.

The case is due to Sobolev{{cite journal |last1=Sobolev |first1=Sergeĭ L’vovich |title=Sur un théorème de l'analyse fonctionnelle |journal=Comptes Rendus (Doklady) de l'Académie des Sciences de l'URSS |series=Nouvelle Série |date=1938 |volume=20 |pages=5–9}} and the case $p\; =1$ to Gagliardo and Nirenberg independently.{{cite journal |last1=Gagliardo |first1=Emilio |title=Proprietà di alcune classi di funzioni in più variabili |journal=Ricerche di Matematica |date=1958 |volume=7 |pages=102–137}}{{cite journal |last1=Nirenberg |first1=Louis |title=On elliptic partial differential equations |journal=Annali della Scuola Normale Superiore di Pisa. Classe di Scienze. Serie III |date=1959 |volume=13 |pages=115–162}} The Gagliardo–Nirenberg–Sobolev inequality implies directly the Sobolev embedding

:$W^\{1,p\}(\backslash mathbf\{R\}^n)\; \backslash sub\; L^\{p^*\}(\backslash mathbf\{R\}^n).$

The embeddings in other orders on {{math|Rⁿ}} are then obtained by suitable iteration.

Sobolev's original proof of the Sobolev embedding theorem relied on the following, sometimes known as the Hardy–Littlewood–Sobolev fractional integration theorem. An equivalent statement is known as the Sobolev lemma in {{harv|Aubin|1982|loc=Chapter 2}}. A proof is in {{harv|Stein|1970|loc=Chapter V, §1.3}}.

Let {{math|0 < α < n}} and {{math|1 < p < q < ∞}}. Let {{math|I_α {{=}} (−Δ)^−α/2}} be the Riesz potential on {{math|Rⁿ}}. Then, for {{mvar|q}} defined by

:$\backslash frac\; 1\; q\; =\; \backslash frac\; 1\; p\; -\; \backslash frac\; \backslash alpha\; n$

there exists a constant {{mvar|C}} depending only on {{mvar|p}} such that

:$\backslash left\; \backslash |I\_\backslash alpha\; f\; \backslash right\; \backslash |\_q\; \backslash le\; C\; \backslash |f\backslash |\_p.$

If {{math|p {{=}} 1}}, then one has two possible replacement estimates. The first is the more classical weak-type estimate:

:$m\; \backslash left\; \backslash \{x\; :\; \backslash left\; |I\_\backslash alpha\; f(x)\; \backslash right\; |\; >\; \backslash lambda\; \backslash right\; \backslash \}\; \backslash le\; C\; \backslash left(\; \backslash frac\{\backslash |f\backslash |\_1\}\{\backslash lambda\}\; \backslash right\; )^q,$

where {{math|1/q {{=}} 1 − α/n}}. Alternatively one has the estimate$$\backslash left\; \backslash |I\_\backslash alpha\; f\; \backslash right\; \backslash |\_q\; \backslash le\; C\; \backslash |Rf\backslash |\_1,$$where $Rf$ is the vector-valued Riesz transform, cf. {{harv|Schikorra | Spector | Van Schaftingen|2017}}. 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.

The Hardy–Littlewood–Sobolev lemma implies the Sobolev embedding essentially by the relationship between the Riesz transforms and the Riesz potentials.

Assume {{math|n < p ≤ ∞}}. Then there exists a constant {{mvar|C}}, depending only on {{mvar|p}} and {{mvar|n}}, such that

:$\backslash |u\backslash |\_\{C^\{0,\backslash gamma\}(\backslash mathbf\{R\}^n)\}\backslash leq\; C\; \backslash |u\backslash |\_\{W^\{1,p\}(\backslash mathbf\{R\}^n)\}$

for all {{math|u ∈ C¹(Rⁿ) ∩ L^p(Rⁿ)}}, where

:$\backslash gamma=1-\backslash frac\{n\}\{p\}.$

Thus if {{math|u ∈ W^1,p(Rⁿ)}}, then {{mvar|u}} is in fact Hölder continuous of exponent {{mvar|γ}}, after possibly being redefined on a set of measure 0.

A similar result holds in a bounded domain {{mvar|U}} with Lipschitz boundary. In this case,

:$\backslash |u\backslash |\_\{C^\{0,\backslash gamma\}(U)\}\backslash leq\; C\; \backslash |u\backslash |\_\{W^\{1,p\}(U)\}$

where the constant {{mvar|C}} depends now on {{math|n, p}} and {{mvar|U}}. This version of the inequality follows from the previous one by applying the norm-preserving extension of {{math|W^1,p(U)}} to {{math|W^1,p(Rⁿ)}}. The inequality is named after Charles B. Morrey Jr.

Let {{mvar|U}} be a bounded open subset of {{math|Rⁿ}}, with a {{math|C¹}} boundary. ({{mvar|U}} may also be unbounded, but in this case its boundary, if it exists, must be sufficiently well-behaved.)

Assume {{math|u ∈ W^k,p(U)}}. Then we consider two cases:

In this case we conclude that {{math|u ∈ L^q(U)}}, where

:$\backslash frac\{1\}\{q\}=\backslash frac\{1\}\{p\}-\backslash frac\{k\}\{n\}.$

We have in addition the estimate

:$\backslash |u\backslash |\_\{L^q(U)\}\backslash leq\; C\; \backslash |u\backslash |\_\{W^\{k,p\}(U)\}$,

the constant {{mvar|C}} depending only on {{math|k, p, n}}, and {{mvar|U}}.

Here, we conclude that {{mvar|u}} belongs to a Hölder space, more precisely:

:$u\; \backslash in\; C^\{k-\backslash left[\backslash frac\{n\}\{p\}\backslash right]-1,\backslash gamma\}(U),$

where

:$\backslash gamma\; =\; \backslash begin\{cases\}$

\left[\frac{n}{p}\right]+1-\frac{n}{p} & \frac{n}{p} \notin \mathbf{Z} \\

\text{any element in } (0, 1) & \frac{n}{p} \in \mathbf{Z}

\end{cases}

We have in addition the estimate

:$\backslash |u\backslash |\_\{C^\{k-\backslash left[\backslash frac\{n\}\{p\}\backslash right]-1,\backslash gamma\}(U)\}\backslash leq\; C\; \backslash |u\backslash |\_\{W^\{k,p\}(U)\},$

the constant {{mvar|C}} depending only on {{math|k, p, n, γ}}, and {{mvar|U}}. In particular, the condition $k>n/p$ guarantees that $u$ is continuous (and actually Hölder continuous with some positive exponent).

If $u\backslash in\; W^\{1,n\}(\backslash mathbf\{R\}^n)$, then {{mvar|u}} is a function of bounded mean oscillation and

:$\backslash |u\backslash |\_\{BMO\}\; \backslash leq\; C\; \backslash |Du\backslash |\_\{L^n(\backslash mathbf\{R\}^n)\},$

for some constant {{mvar|C}} depending only on {{mvar|n}}.{{cite journal |last1=Brezis |first1=H. |last2=Nirenberg |first2=L. |title=Degree theory and BMO; part I: Compact manifolds without boundaries |journal=Selecta Mathematica |date=September 1995 |volume=1 |issue=2 |pages=197–263 |doi=10.1007/BF01671566|s2cid=195270732 }}{{rp|§I.2}} This estimate is a corollary of the Poincaré inequality.

The Nash inequality, introduced by {{harvs|first=John|last=Nash|authorlink=John Forbes Nash, Jr.|year=1958|txt}}, states that there exists a constant {{math|C > 0}}, such that for all {{math|u ∈ L¹(Rⁿ) ∩ W^1,2(Rⁿ)}},

:$\backslash |u\backslash |\_\{L^2(\backslash mathbf\{R\}^n)\}^\{1+2/n\}\; \backslash leq\; C\backslash |u\backslash |\_\{L^1(\backslash mathbf\{R\}^n)\}^\{2/n\}\; \backslash |\; Du\backslash |\_\{L^2(\backslash mathbf\{R\}^n)\}.$

The inequality follows from basic properties of the Fourier transform. Indeed, integrating over the complement of the ball of radius {{mvar|ρ}},

{{NumBlk|:|$\backslash int\_\{|x|\backslash ge\backslash rho\}\; \backslash left\; |\backslash hat\{u\}(x)\; \backslash right\; |^2\backslash ,dx\; \backslash le\; \backslash int\_\{|x|\backslash ge\backslash rho\}\; \backslash frac\{|x|^2\}\{\backslash rho^2\}\; \backslash left\; |\backslash hat\{u\}(x)\; \backslash right\; |^2\backslash ,dx\backslash le\; \backslash rho^\{-2\}\backslash int\_\{\backslash mathbf\{R\}^n\}|D\; u|^2\backslash ,dx$|{{EquationRef|1}}}}

because $1\backslash le|x|^2/\backslash rho^2$. On the other hand, one has

:$|\backslash hat\{u\}|\; \backslash le\; \backslash |u\backslash |\_\{L^1\}$

which, when integrated over the ball of radius {{mvar|ρ}} gives

{{NumBlk|:|$\backslash int\_\{|x|\backslash le\backslash rho\}\; |\backslash hat\{u\}(x)|^2\backslash ,dx\; \backslash le\; \backslash rho^n\backslash omega\_n\; \backslash |u\backslash |\_\{L^1\}^2$|{{EquationRef|2}}}}

where {{math|ω_n}} is the volume of the n sphere. Choosing {{mvar|ρ}} to minimize the sum of ({{EquationNote|1}}) and ({{EquationNote|2}}) and applying Parseval's theorem:

:$\backslash |\backslash hat\{u\}\backslash |\_\{L^2\}\; =\; \backslash |u\backslash |\_\{L^2\}$

gives the inequality.

In the special case of {{math|n {{=}} 1}}, the Nash inequality can be extended to the {{math|L^p}} case, in which case it is a generalization of the Gagliardo-Nirenberg-Sobolev inequality ({{harvnb|Brezis|2011}}, Comments on Chapter 8). In fact, if {{mvar|I}} is a bounded interval, then for all {{math|1 ≤ r < ∞}} and all {{math|1 ≤ q ≤ p < ∞}} the following inequality holds

:$\backslash |\; u\backslash |\_\{L^p(I)\}\backslash le\; C\backslash |\; u\backslash |^\{1-a\}\_\{L^q(I)\}\; \backslash |u\backslash |^a\_\{W^\{1,r\}(I)\},$

where:

:$a\backslash left(\backslash frac\{1\}\{q\}-\backslash frac\{1\}\{r\}+1\backslash right)=\backslash frac\{1\}\{q\}-\backslash frac\{1\}\{p\}.$

{{main|Logarithmic Sobolev inequalities}}

The simplest of the Sobolev embedding theorems, described above, states that if a function $f$ in $L^p(\backslash mathbb\; R^n)$ has one derivative in $L^p$, then $f$ itself is in $L^\{p^*\}$, where

:$1/p^*=1/p-1/n.$

We 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 $p$). In 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 ({{harvnb|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 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 $|f|^p\backslash log|f|$ is finite. The 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. It is now known that logarithmic Sobolev inequalities hold for many different types of measures, not just Gaussian measures.

Although 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 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 $L^p$—then the function does belong to $L^\{p^*\}$ for some $p^*>p$ ({{harvnb|Gross|1975}} Theorem 6).

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{{Functional analysis}}

Category:Inequalities (mathematics)

Category:Sobolev spaces

Category:Compactness theorems