Sobolev spaces in one dimension and absolutely continuous functions
1 Locally integrable functions and distributions
Let be Lebesgue measure on . We denote by the collection of Borel measurable functions such that for each compact subset of ,
We denote by the collection of equivalence classes of elements of where when almost everywhere.
Write . For and , we say that is a Lebesgue point of if
It is immediate that if is continuous at then is a Lebesgue point of . The Lebesgue differentiation theorem11 1 Walter Rudin, Real and Complex Analysis, third ed., p. 138, Theorem 7.7. states that for , almost every is a Lebesgue point of . A sequence of Borel sets is said to shrink nicely to if there is some and a sequence such that and . The sequence shrinks nicely to , the sequence shrinks nicely to , and the sequence shrinks nicely to . It is proved that if and for each , is a sequence that shrinks nicely to , then
at each Lebesgue point of .22 2 Walter Rudin, Real and Complex Analysis, third ed., p. 140, Theorem 7.10.
For a nonempty open set in , we denote by the collection of functions such that
is compact and is contained in . We write , whose elements are called called test functions. The following statement is called the fundamental lemma of the calculus of variations or the Du Bois-Reymond Lemma.33 3 Lars Hörmander, The Analysis of Linear Partial Differential Operators I, second ed., p. 15, Theorem 1.2.5.
If and for all , then almost everywhere.
There is some with . We can explicitly write this out:
For a Lebesgue point of and for ,
meaning that . This is true for almost all , showing that almost everywhere. ∎
For , define by
is a locally convex space, and one proves that is continuous and thus belongs to the dual space , whose elements are called distributions.44 4 Walter Rudin, Functional Analysis, second ed., p. 157, §6.11. We say that a distribution is induced by if . For , we define by
It is proved that .55 5 Walter Rudin, Functional Analysis, second ed., p. 158, §6.12.
Let . If , we call a distributional derivative of . In other words, for to have a distributional derivative means that there is some such that for all ,
If are distributional derivatives of then for all , which by Theorem 1 implies that almost everywhere. It follows that if has a distributional derivative then the distributional derivative is unique in , and is denoted :
2 The Sobolev space
We denote by the collection of Borel measurable functions such that , and we denote by the collection of equivalence classes of elements of where when almost everywhere, and write
It is a fact that is a Hilbert space.
We define the Sobolev space to be the set of that have a distributional derivative that satisfies . We remark that the elements of are equivalence classes of elements of . We define
Let and let . Because have distributional derivatives ,
This means that has a distributional derivative, . Thus is a linear space. If then , which implies that as an element of . Therefore is an inner product on .
If is a Cauchy sequence in , then is a Cauchy sequence in and is a Cauchy sequence in , and hence these sequences have limits . For ,
This means that has distributional derivative, . Because it is the case that . Furthermore,
meaning that in , which shows that is a Hilbert space.
3 Absolutely continuous functions
We prove a lemma that gives conditions under which a function, for which integration by parts needs not make sense, is equal to a particular constant almost everywhere.66 6 Haim Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, p. 204, Lemma 8.1.
then there is some such that almost everywhere.
Fix with . Let and define
which belongs to and satisfies . Define by
Using for all and as , check that . Then by hypothesis, , i.e.
Because this is true for all , by Theorem 1 we get that almost everywhere. ∎
Let , let , and define by
for all .
Using Fubini’s theorem,
For real numbers with , we say that a function is absolutely continuous if for all there is some such that whenever are disjoint intervals each contained in with it holds that . We say that a function is locally absolutely continuous if for each nonempty compact interval , the restriction of to is absolutely continuous. We denote the collection of locally absolutely continuous by .
Let , let , and define by
By Lemma 3 and by the definition of a distributional derivative,
Hence for all , which by Lemma 2 implies that there is some such that almost everywhere. Let . On the one hand, the fact that implies that and so . On the other hand, almost everywhere. Furthermore, because is locally absolutely continuous, integration by parts yields
and by definition of a distributional derivative,
Therefore by Theorem 1, almost everywhere. But the fact that almost everywhere implies that almost everywhere, so almost everywhere. In particular, .
For , there is a function such that almost everywhere and almost everywhere. The function is -Hölder continuous.
For ,77 7 cf. Giovanni Leoni, A First Course in Sobolev Spaces, p. 222, Theorem 7.13.
and using the Cauchy-Schwarz inequality,