# 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}*(

**R**

^{n}) denote the Sobolev space consisting of all real-valued functions on

**R**

^{n}whose first Template:Mvar weak derivatives are functions in

*L*. Here Template:Mvar is a non-negative integer and 1 ≤

^{p}*p*≤ ∞. The first part of the Sobolev embedding theorem states that if

*k*>

*ℓ*and 1 ≤

*p*<

*q*≤ ∞ are two extended real numbers such that (

*k*−

*ℓ*)

*p*<

*n*and:

then

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

where *p*^{∗} is the **Sobolev conjugate** of Template:Mvar, given by

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}*(

**R**

^{n}). If (

*k*−

*r*−

*α*)/

*n*= 1/

*p*with

*α*∈ (0, 1), then one has the embedding

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

- Template:Mvar is a bounded open set in
**R**^{n}with Lipschitz boundary (or whose boundary satisfies the cone condition; Template:Harvnb) - Template:Mvar is a compact Riemannian manifold
- Template:Mvar is a compact Riemannian manifold with boundary with Lipschitz boundary
- Template:Mvar is a complete Riemannian manifold with injectivity radius
*δ*> 0 and bounded sectional curvature.

### Kondrachov embedding theorem

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On a compact manifold with *C*^{1} boundary, the **Kondrachov embedding theorem** states that if *k* > *ℓ* and *k* − *n*/*p* > *ℓ* − *n*/*q* then the Sobolev embedding

is completely continuous (compact).

## Gagliardo–Nirenberg–Sobolev inequality

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

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

The embeddings in other orders on **R**^{n} 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 *I _{α}* = (−Δ)

^{−α/2}be the Riesz potential on

**R**

^{n}. Then, for Template:Mvar defined by

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

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

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

for all *u* ∈ *C*^{1}(**R**^{n}) ∩ *L ^{p}*(

**R**

^{n}), where

Thus if *u* ∈ *W*^{ 1,p}(**R**^{n}), 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 *C*^{1} boundary. In this case,

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}(**R**^{n}).

## General Sobolev inequalities

Let Template:Mvar be a bounded open subset of **R**^{n}, with a *C*^{1} boundary. (Template:Mvar may also be unbounded, but in this case its boundary, if it exists, must be sufficiently well-behaved.) Assume *u* ∈ *W ^{ k,p}*(

*U*), then we consider two cases:

*k* < *n*/*p*

In this case *u* ∈ *L ^{q}*(

*U*), where

We have in addition the estimate

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:

where

We have in addition the estimate

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

## Case

If , then Template:Mvar is a function of bounded mean oscillation and

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 *u* ∈ *L*^{1}(**R**^{n}) ∩ *W*^{ 1,2}(**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

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

where *ω _{n}* 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:

gives the inequality.

In the special case of *n* = 1, the Nash inequality can be extended{{ safesubst:#invoke:Unsubst||date=__DATE__ |$B=
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}} to the *L ^{p}* 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 ≤

*q*≤

*p*< ∞ the following inequality holds

where:

## References

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