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Graphical representation of the embedding conditions. The space W 3,p, represented by a blue dot at the point (1/p, 3), embeds into the spaces indicated by red dots, all lying on a line with slope n. The white circle at (0,0) indicates the impossibility of optimal embeddings into L ?.
Let W k,p(Rn) denote the Sobolev space consisting of all real-valued functions on Rn whose first kweak derivatives are functions in Lp. 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 real numbers such that
and the embedding is continuous. In the special case of k = 1 and l = 0, Sobolev embedding gives
If the line from the picture above intersects the x-axis at s = r + ?, the embedding into a Hölder space C r, ? (red) holds. White circles indicate intersection points at which optimal embeddings are not valid.
The second part of the Sobolev embedding theorem applies to embeddings in Hölder spacesC r,?(Rn). If n < p and
with ? ? (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.
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
On a compact manifold with C1 boundary, the Kondrachov embedding theorem states that if k > l and
then the Sobolev embedding
is completely continuous (compact). 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 W k,p(M).
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
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 Rn are then obtained by suitable iteration.
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 < ? < n and 1 < p < q < ?. Let I? = (-?)-?/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 one has two possible replacement estimates. The first is the more classical weak-type estimate:
where 1/q = 1 - ?/n. Alternatively one has the estimate
The Hardy-Littlewood-Sobolev lemma implies the Sobolev embedding essentially by the relationship between the Riesz transforms and the Riesz potentials.
Assume n < p . Then there exists a constant C, depending only on p and n, such that
for all u ? C1(Rn) ? Lp(Rn), where
Thus if u ? W 1,p(Rn), then u 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 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), then we consider two cases:
which, when integrated over the ball of radius ? gives
where ?n is the volume of the n-ball. Choosing ? 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 Lp case, in which case it is a generalization of the Gagliardo-Nirenberg-Sobolev inequality (Brezis 2011, Comments on Chapter 8). In fact, if I is a bounded interval, then for all 1 r < ? and all 1 q p < ? the following inequality holds