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 Kondrashov theorem (or Rellich-Kondrashov 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 k weak derivatives are functions in L^p. Here k is a non-negative integer and 1 ≤ 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:

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

• M is a bounded open set in Rⁿ with Lipschitz boundary (or whose boundary satisfies the cone condition; )

• M is a compact Riemannian manifold

• M is a compact Riemannian manifold with boundary with Lipschitz boundary

• M is a complete Riemannian manifold with injectivity radius δ > 0 and bounded sectional curvature.

Kondrashov embedding theorem
On a compact manifold with C¹ boundary, the Kondrashov 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 u is a continuously differentiable real-valued function on Rⁿ with compact support. Then for 1 ≤ p < n there is a constant C depending only on n and p such that

where

is the Sobolev conjugate of p.

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

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 . A proof is in .

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

there exists a constant C depending only on p 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 C, depending only on p and n, such that

for all u ∈ C¹(Rⁿ) ∩ L^p(Rⁿ), where

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

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

where the constant C depends now on n, p and U. 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 U be a bounded open subset of Rⁿ, with a C¹ boundary. (U may also be unbounded, but in this case its boundary, if it exists, must be sufficiently well-behaved.) Assume u ∈ W^k,p(U).

(i) If

then $u\backslash in\; L^q(U)$, where

We have in addition the estimate

the constant C depending only on k, p, n, and U.

(ii) If

then u belongs to the Hölder space $C^\{k-[n/p]-1,\backslash gamma\}(U)\backslash ,$, where

We have in addition the estimate

the constant C depending only on k, p, n, γ, and U.

Case $p=n$

If $u\backslash in\; W^\{1,n\}(R^n)\backslash cap\; L^1\_\{loc\}(R^n)$, then $u$ is a function of bounded mean oscillation and

This estimate is a corollary of the Poincaré inequality.

Nash inequality

The Nash inequality, introduced by , states that there exists a constant C > 0, such that for all u ∈ L¹(Rⁿ) ∩ W^1,2(Rⁿ),

The inequality follows from basic properties of the Fourier transform. On the one hand, integrating over the complement of the ball of radius ρ,

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

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

which, when integrated over the ball of radius ρ gives
{{NumBlk|:|$\backslash int\_\{|x|\backslash le\backslash rho\}\; |\backslash hat\{u\}(x)|^2\backslash ,dx\; \backslash le\; \backslash rho^n\backslash omega\_n\; \backslash |u\backslash |\_\{L^1\}^2$|}}
where ω_n is the volume of the n-ball. Choosing ρ to minimize the sum of () and () and again applying Parseval's theorem $\backslash scriptstyle\{\backslash |\backslash hat\{u\}\backslash |\_\{L^2\}\; =\; \backslash |u\backslash |\_\{L^2\}\}$ gives the inequality.

In the special case of n =1, the Nash inequality can be extended to the L^p case, in which case it is a generalization of the Gagliardo-Nirenberg-Sobolev inequality . In fact, if I is a bounded interval, then for all 1 ≤ r < ∞ and all 1 ≤ q ≤ p < ∞ the following inequality holds

where a is defined by