In
mathematics, there is in
mathematical analysis
Analysis is the branch of mathematics dealing with continuous functions, limit (mathematics), limits, and related theories, such as Derivative, differentiation, Integral, integration, measure (mathematics), measure, infinite sequences, series (m ...
a class of Sobolev inequalities, relating norms including those of
Sobolev space
In mathematics, a Sobolev space is a vector space of functions equipped with a norm that is a combination of ''Lp''-norms of the function together with its derivatives up to a given order. The derivatives are understood in a suitable weak sense ...
s. These are used to prove the Sobolev embedding theorem, giving inclusions between certain
Sobolev space
In mathematics, a Sobolev space is a vector space of functions equipped with a norm that is a combination of ''Lp''-norms of the function together with its derivatives up to a given order. The derivatives are understood in a suitable weak sense ...
s, and the
Rellich–Kondrachov theorem In mathematics, the Rellich–Kondrachov theorem is a compact embedding theorem concerning Sobolev spaces. It is named after the Austrian-German mathematician Franz Rellich and the Russian mathematician Vladimir Iosifovich Kondrashov. Rellich pr ...
showing that under slightly stronger conditions some Sobolev spaces are
compactly embedded In mathematics, the notion of being compactly embedded expresses the idea that one set or space is "well contained" inside another. There are versions of this concept appropriate to general topology and functional analysis.
Definition (topological ...
in others. They are named after
Sergei Lvovich Sobolev.
Sobolev embedding theorem
![Sobolev embedding theorem](https://upload.wikimedia.org/wikipedia/commons/2/20/Sobolev_embedding_theorem.svg)
Let denote the Sobolev space consisting of all real-valued functions on whose first
weak derivative
In mathematics, a weak derivative is a generalization of the concept of the derivative of a function (''strong derivative'') for functions not assumed differentiable, but only integrable, i.e., to lie in the L''p'' space L^1( ,b.
The method ...
s 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
.
![Sobolev embedding theorem (Morrey case)](https://upload.wikimedia.org/wikipedia/commons/2/25/Sobolev_embedding_theorem_%28Morrey_case%29.svg)
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 mathematics, open sets are a generalization of open intervals in the real line.
In a metric space (a set along with a distance defined between any two points), open sets are the sets that, with every point , contain all points that are su ...
in with
Lipschitz boundary (or whose boundary satisfies the
cone condition In mathematics, the cone condition is a property which may be satisfied by a subset of a Euclidean space. Informally, it requires that for each point in the subset a cone with vertex in that point must be contained in the subset itself, and so the s ...
; )
* is a
compact
Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to:
* Interstate compact
* Blood compact, an ancient ritual of the Philippines
* Compact government, a type of colonial rule utilized in British ...
Riemannian manifold
* is a compact Riemannian
manifold with boundary
In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a ne ...
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
This is a glossary of some terms used in Riemannian geometry and metric geometry — it doesn't cover the terminology of differential topology.
The following articles may also be useful; they either contain specialised vocabulary or provi ...
and bounded
sectional curvature In Riemannian geometry, the sectional curvature is one of the ways to describe the curvature of Riemannian manifolds. The sectional curvature ''K''(σ''p'') depends on a two-dimensional linear subspace σ''p'' of the tangent space at a poi ...
.
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 and
then the Sobolev embedding
:
is
completely continuous
In functional analysis, a branch of mathematics, a compact operator is a linear operator T: X \to Y, where X,Y are normed vector spaces, with the property that T maps bounded subsets of X to relatively compact subsets of Y (subsets with compact c ...
(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
In fractional calculus, an area of mathematical analysis, the differintegral (sometime also called the derivigral) is a combined differentiation/ integration operator. Applied to a function ƒ, the ''q''-differintegral of ''f'', here denoted ...
theorem. An equivalent statement is known as the Sobolev lemma in . A proof is in .
Let and . Let be the
Riesz potential In mathematics, the Riesz potential is a potential named after its discoverer, the Hungarian mathematician Marcel Riesz. In a sense, the Riesz potential defines an inverse for a power of the Laplace operator on Euclidean space. They generalize to ...
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 estimate
where
is the vector-valued
Riesz transform In the mathematical theory of harmonic analysis, the Riesz transforms are a family of generalizations of the Hilbert transform to Euclidean spaces of dimension ''d'' > 1. They are a type of singular integral operator, meaning that they a ...
, c.f. . The boundedness of the
Riesz transform In the mathematical theory of harmonic analysis, the Riesz transforms are a family of generalizations of the Hilbert transform to Euclidean spaces of dimension ''d'' > 1. They are a type of singular integral operator, meaning that they a ...
s 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 transform In the mathematical theory of harmonic analysis, the Riesz transforms are a family of generalizations of the Hilbert transform to Euclidean spaces of dimension ''d'' > 1. They are a type of singular integral operator, meaning that they a ...
s 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 Hölder:
* ''Hölder, Hoelder'' as surname
* Hölder condition
* Hölder's inequality
* Hölder mean
In mathematics, generalized means (or power mean or Hölder mean from Otto Hölder) are a family of functions for aggregating sets of number ...
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 In harmonic analysis in mathematics, a function of bounded mean oscillation, also known as a BMO function, is a real-valued function whose mean oscillation is bounded (finite). The space of functions of bounded mean oscillation (BMO), is a functi ...
and
:
for some constant depending only on . This estimate is a corollary of the
Poincaré inequality In mathematics, the Poincaré inequality is a result in the theory of Sobolev spaces, named after the French mathematician Henri Poincaré. The inequality allows one to obtain bounds on a function using bounds on its derivatives and the geometry ...
.
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
Leonard Gross (born February 24, 1931) is an American mathematician and Professor Emeritus of Mathematics at Cornell University.
Gross has made fundamental contributions to mathematics and the mathematically rigorous study of quantum field theo ...
() 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
*.
*
*.
*
*
*
*
* ,
MAA review*, Translated from the Russian by T. O. Shaposhnikova.
*.
*.
*
*
*
{{Functional analysis
Inequalities
Sobolev spaces
Compactness theorems