 Sobolev Inequality
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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 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 k weak 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

${\frac {1}{p}}-{\frac {k}{n}}={\frac {1}{q}}-{\frac {\ell }{n}},$ then

$W^{k,p}(\mathbf {R} ^{n})\subseteq W^{\ell ,q}(\mathbf {R} ^{n})$ and the embedding is continuous. In the special case of k = 1 and l = 0, Sobolev embedding gives

$W^{1,p}(\mathbf {R} ^{n})\subseteq L^{p^{*}}(\mathbf {R} ^{n})$ where p* is the Sobolev conjugate of p, given by

${\frac {1}{p^{*}}}={\frac {1}{p}}-{\frac {1}{n}}.$ This special case of the Sobolev embedding is a direct consequence of the Gagliardo-Nirenberg-Sobolev inequality. 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 spaces C r,?(Rn). If n < p and

${\frac {1}{p}}-{\frac {k}{n}}=-{\frac {r+\alpha }{n}},$ with ? ? (0, 1] then one has the embedding

$W^{k,p}(\mathbf {R} ^{n})\subset C^{r,\alpha }(\mathbf {R} ^{n}).$ 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

### Kondrachov embedding theorem

On a compact manifold with C1 boundary, the Kondrachov embedding theorem states that if k > l and

${\frac {1}{p}}-{\frac {k}{n}}<{\frac {1}{q}}-{\frac {\ell }{n}}$ then the Sobolev embedding
$W^{k,p}(M)\subset W^{\ell ,q}(M)$ 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).

## 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

$\|u\|_{L^{p^{*}}(\mathbf {R} ^{n})}\leq C\|Du\|_{L^{p}(\mathbf {R} ^{n})}.$ with 1/p* = 1/p - 1/n. The case $1 is due to Sobolev, $p=1$ to Gagliardo and Nirenberg independently. The Gagliardo-Nirenberg-Sobolev inequality implies directly the Sobolev embedding

$W^{1,p}(\mathbf {R} ^{n})\subset L^{p^{*}}(\mathbf {R} ^{n}).$ 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 (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

${\frac {1}{q}}={\frac {1}{p}}-{\frac {\alpha }{n}}$ there exists a constant C depending only on p such that

$\left\|I_{\alpha }f\right\|_{q}\leq C\|f\|_{p}.$ If p = 1, then one has two possible replacement estimates. The first is the more classical weak-type estimate:

$m\left\{x:\left|I_{\alpha }f(x)\right|>\lambda \right\}\leq C\left({\frac {\|f\|_{1}}{\lambda }}\right)^{q},$ where 1/q = 1 - ?/n. Alternatively one has the estimate

$\left\|I_{\alpha }f\right\|_{q}\leq C\|Rf\|_{1},$ where $Rf$ is the vector-valued Riesz transform, c.f. (Schikorra, Spector & Van Schaftingen). 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 n < p . Then there exists a constant C, depending only on p and n, such that

$\|u\|_{C^{0,\gamma }(\mathbf {R} ^{n})}\leq C\|u\|_{W^{1,p}(\mathbf {R} ^{n})}$ for all u ? C1(Rn) ? Lp(Rn), where

$\gamma =1-{\frac {n}{p}}.$ 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,

$\|u\|_{C^{0,\gamma }(U)}\leq C\|u\|_{W^{1,p}(U)}$ 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:

### k < n/p

In this case u ? Lq(U), where

${\frac {1}{q}}={\frac {1}{p}}-{\frac {k}{n}}.$ We have in addition the estimate

$\|u\|_{L^{q}(U)}\leq C\|u\|_{W^{k,p}(U)}$ ,

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

### k > n/p

Here, u belongs to a Hölder space, more precisely:

$u\in C^{k-\left[{\frac {n}{p}}\right]-1,\gamma }(U),$ where

$\gamma ={\begin{cases}\left[{\frac {n}{p}}\right]+1-{\frac {n}{p}}&{\frac {n}{p}}\notin \mathbf {Z} \\{\text{any element in }}(0,1)&{\frac {n}{p}}\in \mathbf {Z} \end{cases}}$ We have in addition the estimate

$\|u\|_{C^{k-\left[{\frac {n}{p}}\right]-1,\gamma }(U)}\leq C\|u\|_{W^{k,p}(U)},$ the constant C depending only on k, p, n, ?, and U.

## Case $p=n,k=1$ If $u\in W^{1,n}(\mathbf {R} ^{n})$ , then u is a function of bounded mean oscillation and

$\|u\|_{BMO}\leq C\|Du\|_{L^{n}(\mathbf {R} ^{n})},$ 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 ? L1(Rn) ? W 1,2(Rn),

$\|u\|_{L^{2}(\mathbf {R} ^{n})}^{1+2/n}\leq C\|u\|_{L^{1}(\mathbf {R} ^{n})}^{2/n}\|Du\|_{L^{2}(\mathbf {R} ^{n})}.$ The inequality follows from basic properties of the Fourier transform. Indeed, integrating over the complement of the ball of radius ?,

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

$|{\hat {u}}|\leq \|u\|_{L^{1}}$ 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:

$\|{\hat {u}}\|_{L^{2}}=\|u\|_{L^{2}}$ 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

$\|u\|_{L^{p}(I)}\leq C\|u\|_{L^{q}(I)}^{1-a}\|u\|_{W^{1,r}(I)}^{a},$ where:

$a\left({\frac {1}{q}}-{\frac {1}{r}}+1\right)={\frac {1}{q}}-{\frac {1}{p}}.$ 