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  denote the Sobolev space consisting of all real-valued functions on  whose first  weak derivatives are functions in .  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 result should be interpreted as saying that if a function  in  has one derivative in , then  itself has improved local behavior, meaning that it belongs to the space  where . (Note that , so that .) Thus, any local singularities in  must be more mild than for a typical function in .

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. If  then  for every .

In particular, as long as , the embedding criterion will hold with  and some positive value of . That is, for a function  on , if  has  derivatives in  and , then  will be continuous (and actually Hölder continuous with some positive exponent ).

Generalizations

The Sobolev embedding theorem holds for Sobolev spaces  on other suitable domains . In particular (; ), both parts of the Sobolev embedding hold when
  is a bounded open set in  with Lipschitz boundary (or whose boundary satisfies the cone condition; )
  is a compact Riemannian manifold
  is a compact Riemannian manifold with boundary and the boundary is Lipschitz (meaning that the boundary can be locally represented as a graph of a Lipschitz continuous function).
  is a complete Riemannian manifold with injectivity radius  and bounded sectional curvature.
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 (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 .

Gagliardo–Nirenberg–Sobolev inequality
Assume that  is a continuously differentiable real-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 Lipschitz 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 . The inequality is named after Charles B. Morrey Jr.

General Sobolev inequalities
Let  be a bounded open subset of , with a  boundary. ( may also be unbounded, but in this case its boundary, if it exists, must be sufficiently well-behaved.)

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 . In particular, the condition  guarantees that  is continuous (and actually Hölder continuous with some positive exponent).

Case 
If , then  is a function of bounded mean oscillation and

for some constant  depending only on . This estimate is a corollary of the Poincaré inequality.

Nash inequality
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 (, Comments on Chapter 8). In fact, if  is a bounded interval, then for all  and all  the following inequality holds

where:

Logarithmic Sobolev inequality

The simplest of the Sobolev embedding theorems, described above, states that if a function  in  has one derivative in , then  itself is in , where

We can see that as  tends to infinity,  approaches . Thus, if the dimension  of the space on which  is defined is large, the improvement in the local behavior of  from having a derivative in  is small ( is only slightly larger than ).  In particular, for functions on an infinite-dimensional space, we cannot expect any direct analog of the classical Sobolev embedding theorems.

There is, however, a type of Sobolev inequality, established by Leonard Gross () and known as a logarithmic Sobolev inequality, that has dimension-independent constants and therefore continues to hold in the infinite-dimensional setting. The logarithmic Sobolev inequality says, roughly, that if a function is in  with respect to a Gaussian measure and has one derivative that is also in , then  is in "-log", meaning that the integral of  is finite. The inequality expressing this fact has constants that do not involve the dimension of the space and, thus, the inequality holds in the setting of a Gaussian measure on an infinite-dimensional space. It is now known that logarithmic Sobolev inequalities hold for many different types of measures, not just Gaussian measures.

Although it might seem as if the -log condition is a very small improvement over being in , this improvement is sufficient to derive an important result, namely hypercontractivity for the associated Dirichlet form operator. This result means that if a function is in the range of the exponential of the Dirichlet form operator—which means that the function has, in some sense, infinitely many derivatives in —then the function does belong to  for some  ( Theorem 6).

References

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 , , MAA review
, Translated from the Russian by T. O. Shaposhnikova.
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Inequalities
Sobolev spaces
Compactness theorems