Let denote the Sobolev spaceconsisting of all real-valued functions on whose first weak derivatives are functions in Lp space|. Here is a non-negative integer and. The first part of the Sobolev embedding theorem states that if and are two real numbers such that then and the embedding is continuous. In the special case of and, Sobolev embedding gives where is the Sobolev conjugate of, 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. If and with 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 on other suitable domains. In particular, both parts of the Sobolev embedding hold when
If is a bounded open set in with continuous boundary, then is compactly embedded in .
Kondrachov embedding theorem
On a compact manifold with boundary, the Kondrachov embedding theorem states that if andthen the Sobolev embedding is completely continuous. 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.
Gagliardo–Nirenberg–Sobolev inequality
Assume that is a continuously differentiablereal-valued function on with compact support. Then for there is a constant depending only on and 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 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 and. Let be the Riesz potential on. Then, for defined by there exists a constant depending only on such that If, then one has two possible replacement estimates. The first is the more classical weak-type estimate: where. Alternatively one has the estimatewhere is the vector-valued Riesz transform, c.f. . The boundedness of the Riesz transforms 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 transforms and the Riesz potentials.
Morrey's inequality
Assume. Then there exists a constant, depending only on and, such that for all, where Thus if, then is in fact Hölder continuous of exponent, after possibly being redefined on a set of measure 0. A similar result holds in a bounded domain with boundary. In this case, where the constant depends now on and. This version of the inequality follows from the previous one by applying the norm-preserving extension of to.
General Sobolev inequalities
Let be a bounded open subset of, with a boundary. Assume. Then we consider two cases:In this case we conclude that, where We have in addition the estimate the constant depending only on, and.Here, we conclude that belongs to a Hölder space, more precisely: where We have in addition the estimate the constant depending only on, and.
The Nash inequality, introduced by, states that there exists a constant, such that for all, The inequality follows from basic properties of the Fourier transform. Indeed, integrating over the complement of the ball of radius, because. On the other hand, one has which, when integrated over the ball of radius gives where is the volume of the -ball. Choosing to minimize the sum of and and applying Parseval's theorem: gives the inequality. In the special case of, the Nash inequality can be extended to the case, in which case it is a generalization of the Gagliardo-Nirenberg-Sobolev inequality. In fact, if is a bounded interval, then for all and all the following inequality holds where:
