# Sobolev inequality

In mathematics, there is in mathematical analysis a class of Sobolev inequalities, relating norms including those of Sobolev spaces. These are used to prove the Sobolev embedding theorem, giving inclusions between certain Sobolev spaces, 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

Let W k,p(Rn) denote the Sobolev space consisting of all real-valued functions on Rn whose first Template:Mvar weak derivatives are functions in Lp. Here Template:Mvar is a non-negative integer and 1 ≤ p ≤ ∞. The first part of the Sobolev embedding theorem states that if k > and 1 ≤ p < q ≤ ∞ are two extended real numbers such that (k)p < n and:

${\displaystyle {\frac {1}{q}}={\frac {1}{p}}-{\frac {k-\ell }{n}},}$

then

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

and the embedding is continuous. In the special case of k = 1 and = 0, Sobolev embedding gives

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

where p is the Sobolev conjugate of Template:Mvar, given by

${\displaystyle {\frac {1}{p^{*}}}={\frac {1}{p}}-{\frac {1}{n}}.}$

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

The second part of the Sobolev embedding theorem applies to embeddings in Hölder spaces C r,α(Rn). If (krα)/n = 1/p with α ∈ (0, 1), then one has the embedding

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

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.

### Generalizations

The Sobolev embedding theorem holds for Sobolev spaces W k,p(M) on other suitable domains Template:Mvar. In particular (Template:Harvnb; Template:Harvnb), both parts of the Sobolev embedding hold when

### Kondrachov embedding theorem

{{#invoke:main|main}} On a compact manifold with C1 boundary, the Kondrachov embedding theorem states that if k > and kn/p > n/q then the Sobolev embedding

${\displaystyle W^{k,p}(M)\subset W^{\ell ,q}(M)}$

is completely continuous (compact).

## Gagliardo–Nirenberg–Sobolev inequality

Assume that Template:Mvar is a continuously differentiable real-valued function on Rn with compact support. Then for 1 ≤ p < n there is a constant Template:Mvar depending only on Template:Mvar and Template:Mvar such that

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

The Gagliardo–Nirenberg–Sobolev inequality implies directly the Sobolev embedding

${\displaystyle W^{1,p}(\mathbf {R} ^{n})\subset L^{p^{*}}(\mathbf {R} ^{n}).}$

The embeddings in other orders on Rn 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 Template:Harv. A proof is in Template:Harv.

Let 0 < α < n and 1 < p < q < ∞. Let Iα = (−Δ)α/2 be the Riesz potential on Rn. Then, for Template:Mvar defined by

${\displaystyle q={\frac {pn}{n-\alpha p}}}$

there exists a constant Template:Mvar depending only on Template:Mvar such that

${\displaystyle \left\|I_{\alpha }f\right\|_{q}\leq C\|f\|_{p}.}$

If p = 1, then the weak-type estimate holds:

${\displaystyle m\left\{x:\left|I_{\alpha }f(x)\right|>\lambda \right\}\leq C\left({\frac {\|f\|_{1}}{\lambda }}\right)^{q}}$

where 1/q = 1 − α/n.

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

## Morrey's inequality

Assume n < p ≤ ∞. Then there exists a constant Template:Mvar, depending only on Template:Mvar and Template:Mvar, such that

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

for all uC1(Rn) ∩ Lp(Rn), where

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

Thus if uW 1,p(Rn), then Template:Mvar is in fact Hölder continuous of exponent Template:Mvar, after possibly being redefined on a set of measure 0.

A similar result holds in a bounded domain Template:Mvar with C1 boundary. In this case,

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

where the constant Template:Mvar depends now on n, p and Template:Mvar. 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).

## General Sobolev inequalities

Let Template:Mvar be a bounded open subset of Rn, with a C1 boundary. (Template:Mvar may also be unbounded, but in this case its boundary, if it exists, must be sufficiently well-behaved.) Assume uW k,p(U), then we consider two cases:

### k < n/p

In this case uLq(U), where

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

We have in addition the estimate

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

the constant Template:Mvar depending only on k, p, n, and Template:Mvar.

### k > n/p

Here, Template:Mvar belongs to a Hölder space, more precisely:

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

where

${\displaystyle \gamma ={\begin{cases}\left[{\frac {n}{p}}\right]+1-{\frac {n}{p}}&{\frac {n}{p}}\notin \mathbf {Z} \\{\text{any element in }}(0,1)&{\frac {n}{p}}\in \mathbf {Z} \end{cases}}}$

We have in addition the estimate

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

the constant Template:Mvar depending only on k, p, n, γ, and Template:Mvar.

## Case ${\displaystyle p=n,k=1}$

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

for some constant Template:Mvar depending only on Template:Mvar. This estimate is a corollary of the Poincaré inequality.

## Nash inequality

The Nash inequality, introduced by Template:Harvs, states that there exists a constant C > 0, such that for all uL1(Rn) ∩ W 1,2(Rn),

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

The inequality follows from basic properties of the Fourier transform. Indeed, integrating over the complement of the ball of radius Template:Mvar,

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

${\displaystyle |{\hat {u}}|\leq \|u\|_{L^{1}}}$

which, when integrated over the ball of radius Template:Mvar gives

where ωn is the volume of the [[n sphere|Template:Mvar-ball]]. Choosing Template:Mvar to minimize the sum of (Template:EquationNote) and (Template:EquationNote) and again applying Parseval's theorem:

${\displaystyle \|{\hat {u}}\|_{L^{2}}=\|u\|_{L^{2}}}$

gives the inequality.

In the special case of n = 1, the Nash inequality can be extended{{ safesubst:#invoke:Unsubst||date=__DATE__ |\$B= {{#invoke:Category handler|main}}{{#invoke:Category handler|main}}[citation needed] }} to the Lp case, in which case it is a generalization of the Gagliardo-Nirenberg-Sobolev inequality Template:Harv. In fact, if Template:Mvar is a bounded interval, then for all 1 ≤ r < ∞ and all 1 ≤ qp < ∞ the following inequality holds

${\displaystyle \|u\|_{L^{p}(I)}\leq C\|u\|_{L^{q}(I)}^{1-a}\|u\|_{W^{1,r}(I)}^{a},}$

where:

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

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