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 realvalued functions on R^{n} whose first k weak derivatives are functions in L^{p}. Here k is a nonnegative integer and 1 ≤ p < ∞. The first part of the Sobolev embedding theorem states that if k > ℓ and 1 ≤ p < q < ∞ are two 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 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}(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 M. In particular (Aubin 1982, Chapter 2; Aubin 1976), both parts of the Sobolev embedding hold when
 M is a bounded open set in R^{n} with Lipschitz boundary (or whose boundary satisfies the cone condition; Adams 1975, Theorem 5.4)
 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.
Kondrachov embedding theorem
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 u is a continuously differentiable realvalued function on R^{n} with compact support. Then for 1 ≤ p < n there is a constant C depending only on n and p such that
with 1/p* = 1/p  1/n. The case is due to Sobolev, to Gagliardo and Nirenberg independently. 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 (Aubin 1982, Chapter 2). A proof is in (Stein, Chapter V, §1.3).
Let 0 < α < n and 1 < p < q < ∞. Let I_{α} = (−Δ)^{−α/2} be the Riesz potential on R^{n}. Then, for q defined by
there exists a constant C depending only on p such that
If p = 1, then one has two possible replacement estimates. The first is the more classical weaktype estimate:
where 1/q = 1 − α/n. Alternatively one has the estimate
where is the vectorvalued Riesz transform, c.f. (Schikorra & Spector Van Schaftingen). Interestingly the boundedness of the Riesz transforms implies that the latter inequality gives a unified way to write the family.
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 ∈ C^{1}(R^{n}) ∩ L^{p}(R^{n}), where
Thus if u ∈ W^{ 1,p}(R^{n}), 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 C^{1} 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 normpreserving extension of W^{ 1,p}(U) to W^{ 1,p}(R^{n}).
General Sobolev inequalities
Let U be a bounded open subset of R^{n}, with a C^{1} boundary. (U may also be unbounded, but in this case its boundary, if it exists, must be sufficiently wellbehaved.) 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 C depending only on k, p, n, and U.
k > n/p
Here, u belongs to a Hölder space, more precisely:
where
We have in addition the estimate
the constant C depending only on k, p, n, γ, and U.
Case
If , then u is a function of bounded mean oscillation and
for some constant C depending only on n. This estimate is a corollary of the Poincaré inequality.
Nash inequality
The Nash inequality, introduced by John Nash (1958), 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 ρ,

(1)
by Parseval's theorem. On the other hand, one has
which, when integrated over the ball of radius ρ gives

(2)
where ω_{n} is the volume of the nball. Choosing ρ to minimize the sum of (1) and (2) and again applying Parseval's theorem:
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 GagliardoNirenbergSobolev inequality (Brezis 1999). In fact, if I is a bounded interval, then for all 1 ≤ r < ∞ and all 1 ≤ q ≤ p < ∞ the following inequality holds
where:
References
 Adams, Robert A. (1975), Sobolev spaces, Pure and Applied Mathematics, 65., New YorkLondon: Academic Press, pp. xviii+268, ISBN 9780120441501, MR 0450957.
 Aubin, Thierry (1976), "Espaces de Sobolev sur les variétés riemanniennes", Bulletin des Sciences Mathématiques. 2e Série, 100 (2): 149–173, ISSN 00074497, MR 0488125
 Aubin, Thierry (1982), Nonlinear analysis on manifolds. MongeAmpère equations, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 252, Berlin, New York: SpringerVerlag, ISBN 9780387907048, MR 681859.
 Brezis, Haïm (1983), Analyse fonctionnelle : théorie et applications, Paris: Masson, ISBN 0821807722
 Evans, Lawrence (1998), Partial Differential Equations, American Mathematical Society, Providence, ISBN 0821807722
 Leoni, Giovanni (2009), A First Course in Sobolev Spaces, Graduate Studies in Mathematics, American Mathematical Society, pp. xvi+607 ISBN 9780821847688, MR2527916, Zbl 1180.46001, MAA
 Vladimir G., Maz'ja (1985), Sobolev spaces, Springer Series in Soviet Mathematics, Berlin: SpringerVerlag, Translated from the Russian by T. O. Shaposhnikova.
 Nash, J. (1958), "Continuity of solutions of parabolic and elliptic equations", Amer. J. Math., American Journal of Mathematics, Vol. 80, No. 4, 80 (4): 931–954, doi:10.2307/2372841, JSTOR 2372841.
 Nikol'skii, S.M. (2001), "Imbedding theorems", in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer, ISBN 9781556080104
 Stein, Elias (1970), Singular integrals and differentiability properties of functions, Princeton, NJ: Princeton University Press, ISBN 0691080798
 Schikorra, Armin; Spector, Daniel; Van Schaftingen, Jean, An type estimate for Riesz potentials, arXiv:1411.2318