The Sobolev embedding theorem is a result in functional analysis which proves that certain Sobolev spaces 
can be embedded in various spaces including 
, 
, and 
for various domains 
, 
in 
and for miscellaneous values of 
, 
, 
, 
, 
, 
, and 
(usually depending on properties of the domains 
and 
).
Because numerous such embeddings are possible, many individual results may be termed
"the" Sobolev embedding theorem, whereas in actuality the phrase "Sobolev
embedding theorem" is best thought of as an umbrella term encompassing all such
results.
To proceed, let 
be a domain (i.e., a bounded,
connected open set)
in 
and let 
be the intersection of 
with a hyperplane of dimension
in 
for 
.
Let 
,
be integers and let 
. Under these constructions, one has a number
of function space embeddings, the collection of
which will be referred to as the Sobolev embedding theorem.
For example, if 
satisfies a so-called "cone condition" (i.e.,
if there exists a finite cone 
such that each 
is the vertex of a finite cone 
contained in 
and congruent to 
), then
 |
(1)
|
if either 
or if 
and 
.
For such 
,
,
and 
,
one also has that
 |
(2)
|
and
 |
(3)
|
for 
.
If instead 
,
then
 |
(4)
|
and
 |
(5)
|
for 
.
Finally, if 
and if either 
or if 
and 
,
then
 |
(6)
|
and
 |
(7)
|
for 
.
Note that the above embeddings are all due essentially to Sobolev and depend only
on 
,
,
,
,
,
,
and the dimension of the cone 
in the cone condition.
Other types of domains also provide a number of embeddings. If 
satisfies the so-called "strong local Lipschitz condition"
(i.e., if each point 
on the boundary 
of 
has a neighborhood 
whose intersection with 
is the graph of a Lipschitz function), for example, then the target
space 
of (1) can be replaced with the smaller space 
. Moreover, if 
, then
 |
(8)
|
for 
.
If instead 
,
then
 |
(9)
|
with (9) holding also for 
provided that 
and 
.
A number of the above-stated results can be proved either entirely or almost so by way of the so-called Sobolev embedding inequalities. These inequalities follow from
the Littlewood-Paley decomposition
of a function 
in L-p for 
. Indeed, in this context, Minkowski's
inequality combined with the Littlewood-Paley decomposition of such a function
implies a number of inequalities, e.g.,
 |
(10)
|
when 
satisfies 
.
In the event that 
and 
, there is an analogous inequality:
 |
(11)
|
where 
satisfies 
.
Inequalities (10) and (11) are true whenever the respective right-hand sides are
finite and can be expanded even further using fractional
differentiation and integration operators
to yield many of the embedding results stated previously.
The above results can be further altered to allow for even more general embeddings. For example, if the 
-spaces being embedded above are replaced with the 
Sobolev spaces (i.e., the Sobolev spaces of functions, the trace
of whose 
-order
derivatives vanishes for all 
), then the resulting embeddings hold for arbitrary domains
in 
.
Moreover, it can be shown that the embeddings associated to the above-mentioned cone
condition still hold for domains 
which satisfy only a "weakened cone condition."
