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In [[mathematics]], there is in [[mathematical analysis]] a class of '''Sobolev inequalities''', relating norms including those of [[Sobolev space]]s. These are used to prove the '''Sobolev embedding theorem''', giving inclusions between certain [[Sobolev space]]s, and the [[Rellich–Kondrachov theorem]] showing that under slightly stronger conditions some Sobolev spaces are [[compactly embedded]] in others. They are named after [[Sergei Lvovich Sobolev]].

==Sobolev embedding theorem==

[[File:Sobolev embedding theorem.svg|thumb|250px|Graphical representation of the embedding conditions. The space {{math|''W<sup>&thinsp;3,p</sup>''}}, represented by a blue dot at the point {{math|''(1/p, 3)''}}, embeds into the spaces indicated by red dots, all lying on a line with slope {{math|''n''}}. The white circle at {{math|''(0,0)''}} indicates the impossibility of optimal embeddings into {{math|''L<sup>&thinsp;∞</sup>''}}.]]
Let {{math|''W<sup>&thinsp;k,p</sup>''('''R'''<sup>''n''</sup>)}} denote the Sobolev space consisting of all real-valued functions on {{math|'''R'''<sup>''n''</sup>}} whose first {{mvar|k}} [[weak derivative]]s are functions in [[Lp space|{{math|''L<sup>p</sup>''}}]].  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''}} and {{math|1 ≤ ''p'' < ''q'' < ∞}} are two real numbers such that

:<math>\frac{1}{p}-\frac{k}{n} = \frac{1}{q} -\frac{\ell}{n},</math>

then

:<math>W^{k,p}(\mathbf{R}^n)\subseteq W^{\ell,q}(\mathbf{R}^n)</math>

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

:<math>W^{1,p}(\mathbf{R}^n) \subseteq L^{p^*}(\mathbf{R}^n)</math>

where {{math|''p''<sup>∗</sup>}} is the '''[[Sobolev conjugate]]''' of {{mvar|p}}, given by<!-- only if p<n! -->

:<math>\frac{1}{p^*} = \frac{1}{p} - \frac{1}{n}.</math>

This special case of the Sobolev embedding is a direct consequence of the [[#Gagliardo–Nirenberg–Sobolev inequality|Gagliardo–Nirenberg–Sobolev inequality]].

[[File:Sobolev embedding theorem (Morrey case).svg|thumb|250px|If the line from the picture above intersects the x-axis at ''s = r + α'', the embedding into a Hölder space {{math|''C<sup>&thinsp;r, α</sup>''}} (red) holds. White circles indicate intersection points at which ''optimal'' embeddings are not valid.]]

The second part of the Sobolev embedding theorem applies to embeddings in [[Hölder space]]s {{math|''C<sup>&thinsp;r,α</sup>''('''R'''<sup>''n''</sup>)}}. If {{math|''n'' < ''p''}} and

:<math>\frac{1}{p}-\frac{k}{n} = -\frac{r + \alpha}{n},</math>

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

:<math>W^{k,p}(\mathbf{R}^n)\subset C^{r,\alpha}(\mathbf{R}^n).</math>

This part of the Sobolev embedding is a direct consequence of [[#Morrey's inequality|Morrey's inequality]]. Intuitively, this inclusion expresses the fact that the existence of sufficiently many weak derivatives implies some continuity of the classical derivatives.

===Generalizations===
The Sobolev embedding theorem holds for Sobolev spaces {{math|''W<sup>&thinsp;k,p</sup>''(''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 set|bounded]] [[open set]] in {{math|'''R'''<sup>''n''</sup>}} with [[Lipschitz continuity|Lipschitz]] boundary (or whose boundary satisfies the [[cone condition]]; {{harvnb|Adams|1975|loc=Theorem 5.4}})
* {{mvar|M}} is a [[compact space|compact]] [[Riemannian manifold]]
* {{mvar|M}} is a compact Riemannian [[manifold with boundary]] with Lipschitz boundary
* {{mvar|M}} is a [[complete manifold|complete]] Riemannian manifold with [[injectivity radius]] {{math|''δ'' > 0}} and bounded [[sectional curvature]].

===Kondrachov embedding theorem===
{{main article|Rellich–Kondrachov theorem}}
On a compact manifold with {{math|''C''<sup>1</sup>}} boundary, the '''Kondrachov embedding theorem''' states that if {{math|''k'' > ''ℓ''}} and<math display="block">\frac{1}{p}-\frac{k}{n} < \frac{1}{q} -\frac{\ell}{n}</math>then the Sobolev embedding

:<math>W^{k,p}(M)\subset W^{\ell,q}(M)</math>

is [[completely continuous]] (compact). 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<sup>&thinsp;k,p</sup>''('''M''')}}.

==Gagliardo–Nirenberg–Sobolev inequality==
Assume that {{mvar|u}} is a continuously differentiable real-valued function on {{math|'''R'''<sup>''n''</sup>}} with [[compact support]]. Then for {{math|1 &le; ''p'' < ''n''}} there is a constant {{mvar|C}} depending only on {{mvar|n}} and {{mvar|p}} such that

: <math>\|u\|_{L^{p^*}(\mathbf{R}^n)}\leq C \|Du\|_{L^{p}(\mathbf{R}^n)}.</math>

with 1/p* = 1/p - 1/n.
The case <math> 1< p < n </math> is due to Sobolev, <math>  p =1 </math> to Gagliardo and Nirenberg independently.  The Gagliardo–Nirenberg–Sobolev inequality implies directly the Sobolev embedding

:<math>W^{1,p}(\mathbf{R}^n) \sub L^{p^*}(\mathbf{R}^n).</math>

The embeddings in other orders on {{math|'''R'''<sup>''n''</sup>}} are then obtained by suitable iteration.

==Hardy–Littlewood–Sobolev lemma==
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|loc=Chapter V, §1.3}}.

Let {{math|0 < ''α'' < ''n''}} and {{math|1 < ''p'' < ''q'' < ∞}}.  Let {{math|''I<sub>α</sub>'' {{=}} (−Δ)<sup>−''α''/2</sup>}} be the [[Riesz potential]] on {{math|'''R'''<sup>''n''</sup>}}.  Then, for {{mvar|q}} defined by

:<math>\frac 1 q = \frac 1 p - \frac \alpha n</math>

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

:<math>\left \|I_\alpha f \right \|_q \le C \|f\|_p.</math>

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

:<math>m \left \{x : \left |I_\alpha f(x) \right | > \lambda \right \} \le C \left( \frac{\|f\|_1}{\lambda} \right )^q,</math>

where {{math|1/''q'' {{=}} 1 − ''α''/''n''}}.  Alternatively one has the estimate<math display="block">\left \|I_\alpha f \right \|_q \le C \|Rf\|_1,</math>where <math> Rf </math> is the vector-valued [[Riesz transform]], c.f.   {{harv|Schikorra | Spector | Van Schaftingen}}.  The boundedness of the [[Riesz transform]]s 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 transform]]s and the Riesz potentials.

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

:<math>\|u\|_{C^{0,\gamma}(\mathbf{R}^n)}\leq C \|u\|_{W^{1,p}(\mathbf{R}^n)}</math>

for all {{math|''u'' ∈ ''C''<sup>1</sup>('''R'''<sup>''n''</sup>) ∩ ''L<sup>p</sup>''('''R'''<sup>''n''</sup>)}}, where

:<math>\gamma=1-\frac{n}{p}.</math>

Thus if {{math|''u'' ∈ ''W''<sup>&thinsp;1,''p''</sup>('''R'''<sup>''n''</sup>)}}, 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 {{math|''C''<sup>1</sup>}} boundary.  In this case,

:<math>\|u\|_{C^{0,\gamma}(U)}\leq C \|u\|_{W^{1,p}(U)}</math>

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''<sup>&thinsp;1,''p''</sup>(''U'')}} to {{math|''W''<sup>&thinsp;1,''p''</sup>('''R'''<sup>''n''</sup>)}}.

==General Sobolev inequalities==
Let {{mvar|U}} be a bounded open subset of {{math|'''R'''<sup>''n''</sup>}}, with a {{math|''C''<sup>1</sup>}} 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<sup>&thinsp;k,p</sup>''(''U'')}}, then we consider two cases:

==={{math|''k'' < ''n''/''p''}}===
In this case {{math|''u'' ∈ ''L<sup>q</sup>''(''U'')}}, where

:<math>\frac{1}{q}=\frac{1}{p}-\frac{k}{n}.</math>

We have in addition the estimate

:<math>\|u\|_{L^q(U)}\leq C \|u\|_{W^{k,p}(U)}</math>,

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

==={{math|''k'' > ''n''/''p''}}===
Here, {{mvar|u}} belongs to a [[Hölder space]], more precisely:

:<math> u \in C^{k-\left[\frac{n}{p}\right]-1,\gamma}(U),</math>

where

:<math>\gamma = \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}</math>

We have in addition the estimate

:<math>\|u\|_{C^{k-\left[\frac{n}{p}\right]-1,\gamma}(U)}\leq C \|u\|_{W^{k,p}(U)},</math>

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

==Case <math>p=n, k=1</math>==
If <math>u\in W^{1,n}(\mathbf{R}^n)</math>, then {{mvar|u}} is a function of [[bounded mean oscillation]] and

:<math>\|u\|_{BMO} \leq C \|Du\|_{L^n(\mathbf{R}^n)},</math>

for some constant {{mvar|C}} depending only on {{mvar|n}}. This estimate is a corollary of the [[Poincaré inequality]].

==Nash 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'' ∈ ''L''<sup>1</sup>('''R'''<sup>''n''</sup>) ∩ ''W''<sup>&thinsp;1,2</sup>('''R'''<sup>''n''</sup>)}},

:<math>\|u\|_{L^2(\mathbf{R}^n)}^{1+2/n} \leq C\|u\|_{L^1(\mathbf{R}^n)}^{2/n} \| Du\|_{L^2(\mathbf{R}^n)}.</math>

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

{{NumBlk|:|<math>\int_{|x|\ge\rho} \left |\hat{u}(x) \right |^2\,dx \le \int_{|x|\ge\rho} \frac{|x|^2}{\rho^2} \left |\hat{u}(x) \right |^2\,dx\le \rho^{-2}\int_{\mathbf{R}^n}|D u|^2\,dx</math>|{{EquationRef|1}}}}

by [[Parseval's theorem]]. On the other hand, one has

:<math>|\hat{u}| \le \|u\|_{L^1}</math>

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

{{NumBlk|:|<math>\int_{|x|\le\rho} |\hat{u}(x)|^2\,dx \le \rho^n\omega_n \|u\|_{L^1}^2</math>|{{EquationRef|2}}}}

where {{math|''ω<sub>n</sub>''}} is the volume of the [[n sphere|{{mvar|n}}-ball]].  Choosing {{mvar|ρ}} to minimize the sum of ({{EquationNote|1}}) and ({{EquationNote|2}}) and again applying Parseval's theorem:

:<math>\|\hat{u}\|_{L^2} = \|u\|_{L^2}</math>

gives the inequality.

In the special case of {{math|''n'' {{=}} 1}}, the Nash inequality can be extended to the {{math|''L<sup>p</sup>''}} 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

:<math>\| u\|_{L^p(I)}\le C\| u\|^{1-a}_{L^q(I)} \|u\|^a_{W^{1,r}(I)},</math>

where:

:<math>a\left(\frac{1}{q}-\frac{1}{r}+1\right)=\frac{1}{q}-\frac{1}{p}.</math>

==References==
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*{{citation|first=Lawrence|last=Evans|authorlink=Lawrence C. Evans| title=Partial Differential Equations | publisher= [[American Mathematical Society]] |location=Providence RI | year=1998 | isbn=0-8218-0772-2}}
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*{{citation|last=Maz'ja|first=Vladimir G.|authorlink= Vladimir Maz'ja|title=Sobolev Spaces|series=Springer Series in Soviet Mathematics|publisher=Springer-Verlag|year=1985}}, Translated from the Russian by T. O. Shaposhnikova.
*{{citation|last=Nash|first=J.|authorlink=John Forbes Nash, Jr.|title=Continuity of solutions of parabolic and elliptic equations|journal= [[American Journal of Mathematics]] |volume=80|year=1958|pages=931&ndash;954| doi=10.2307/2372841 |jstor=2372841| issue=4|hdl=10338.dmlcz/101876}}.
*{{springer|id=i/i050230|title=Imbedding theorems|first=S.M.|last= Nikol'skii}}
*{{Citation| title=An <math> L^1 </math>-type estimate for Riesz potentials|last1=Schikorra|last2=Spector|last3= Van Schaftingen| first1=Armin | first2=Daniel | first3=Jean |journal= Revista Matemática Iberoamericana |volume=33 |year=2017 |issue=1 |pages=291–304|arxiv=1411.2318 | doi=10.4171/rmi/937}}
*{{citation|first=Elias|last=Stein|authorlink=Elias Stein|title=Singular Integrals and Differentiability Properties of Functions|publisher=[[Princeton University Press]]|location=Princeton, NJ|year=1970|isbn=0-691-08079-8}}

[[Category:Inequalities]]
[[Category:Sobolev spaces]]
[[Category:Compactness theorems]]