Sobolev spaces in one dimension and absolutely continuous functions

Jordan Bell
October 21, 2015

1 Locally integrable functions and distributions

Let λ be Lebesgue measure on . We denote by loc1(λ) the collection of Borel measurable functions f: such that for each compact subset K of ,


We denote by Lloc1(λ) the collection of equivalence classes of elements of loc1(λ) where fg when f=g almost everywhere.

Write B(x,r)={y:|y-x|<r}=(x-r,x+r). For floc1(λ) and x, we say that x is a Lebesgue point of f if


It is immediate that if f is continuous at x then x is a Lebesgue point of f. The Lebesgue differentiation theorem11 1 Walter Rudin, Real and Complex Analysis, third ed., p. 138, Theorem 7.7. states that for floc1(λ), almost every x is a Lebesgue point of f. A sequence of Borel sets En is said to shrink nicely to x if there is some α>0 and a sequence rn0 such that EnB(x,rn) and λ(En)αλ(B(x,rn)). The sequence B(x,n-1)=(x-n-1,x+n-1) shrinks nicely to x, the sequence [x,x+n-1] shrinks nicely to x, and the sequence [x-n-1,x] shrinks nicely to x. It is proved that if floc1(λ) and for each x, En(x) is a sequence that shrinks nicely to x, then


at each Lebesgue point of f.22 2 Walter Rudin, Real and Complex Analysis, third ed., p. 140, Theorem 7.10.

For a nonempty open set Ω in , we denote by Cck(Ω) the collection of Ck functions ϕ: such that


is compact and is contained in Ω. We write 𝒟(Ω)=Cc(Ω), 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.

Theorem 1.

If fLloc1(λ) and Rfϕ𝑑λ=0 for all ϕD(R), then f=0 almost everywhere.


There is some η𝒟(-1,1) with η𝑑λ=1. We can explicitly write this out:




For x a Lebesgue point of f and for 0<r<1,

f(x) =f(x)η(y)𝑑λ(y)



meaning that f(x)=0. This is true for almost all x, showing that f=0 almost everywhere. ∎

For floc1(λ), define Λf:𝒟() by


𝒟() is a locally convex space, and one proves that Λf is continuous and thus belongs to the dual space 𝒟(), whose elements are called distributions.44 4 Walter Rudin, Functional Analysis, second ed., p. 157, §6.11. We say that a distribution Λ is induced by floc1(λ) if Λ=Λf. For Λ𝒟(), we define DΛ:𝒟() by


It is proved that DΛ𝒟().55 5 Walter Rudin, Functional Analysis, second ed., p. 158, §6.12.

Let f,gloc1(λ). If DΛf=Λg, we call g a distributional derivative of f. In other words, for floc1(λ) to have a distributional derivative means that there is some gloc1(λ) such that for all ϕ𝒟(),


If g1,g2loc1(λ) are distributional derivatives of f then (g1-g2)ϕ𝑑λ=0 for all ϕ𝒟(), which by Theorem 1 implies that g1=g2 almost everywhere. It follows that if f has a distributional derivative then the distributional derivative is unique in Lloc1(λ), and is denoted DfLloc1(λ):


2 The Sobolev space H1()

We denote by 2(λ) the collection of Borel measurable functions f: such that |f|2𝑑λ<, and we denote by L2(λ) the collection of equivalence classes of elements of 2(λ) where fg when f=g almost everywhere, and write


It is a fact that L2(λ) is a Hilbert space.

We define the Sobolev space H1() to be the set of fL2(λ) that have a distributional derivative that satisfies DfL2(λ). We remark that the elements of H1() are equivalence classes of elements of 2(λ). We define


Let f,gH1() and let ϕ𝒟(). Because f,g have distributional derivatives Df,Dg,

-(f+g)ϕ𝑑λ =-fϕ𝑑λ-gϕ𝑑λ

This means that f+g has a distributional derivative, D(f+g)=Df+Dg. Thus H1() is a linear space. If f,fH1=0 then |f|2𝑑λ=0, which implies that f=0 as an element of L2(λ). Therefore ,H1 is an inner product on H1().

If fn is a Cauchy sequence in H1(), then fn is a Cauchy sequence in L2(λ) and Dfn is a Cauchy sequence in L2(λ), and hence these sequences have limits f,gL2(λ). For ϕ𝒟(),

-fϕ𝑑λ =-limnfnϕ𝑑λ

This means that f has distributional derivative, Df=g. Because f,DfL2(λ) it is the case that fH1(). Furthermore,


meaning that fnf in H1(), which shows that H1() 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.

Lemma 2.

If fLloc1(λ) and


then there is some cR such that f=c almost everywhere.


Fix η𝒟() with η𝑑λ=1. Let w𝒟() and define


which belongs to 𝒟() and satisfies h𝑑λ=0. Define ϕ: by


Using ϕ(x)=h(x) for all x and ϕ(x)h𝑑λ=0 as x, check that ϕ𝒟(). Then by hypothesis, fϕ𝑑λ=0, i.e.

0 =fh𝑑λ

Because this is true for all w𝒟(), by Theorem 1 we get that f=fη𝑑λ almost everywhere. ∎

Lemma 3.

Let gLloc1(λ), let aR, and define f:RR by




for all ϕD(R).


Using Fubini’s theorem,

f(x)ϕ(x)𝑑λ(x) =--a(xag(y)𝑑λ(y))ϕ(x)𝑑λ(x)

For real numbers a,b with a<b, we say that a function f:[a,b] is absolutely continuous if for all ϵ>0 there is some δ>0 such that whenever (a1,b1),,(an,bn) are disjoint intervals each contained in [a,b] with (bk-ak)<δ it holds that |f(bk)-f(ak)|<ϵ. We say that a function f: is locally absolutely continuous if for each nonempty compact interval [a,b], the restriction of f to [a,b] is absolutely continuous. We denote the collection of locally absolutely continuous by ACloc().

Let fH1(), let a, and define h: by


By Lemma 3 and by the definition of a distributional derivative,


Hence (f-h)ϕ𝑑λ=0 for all ϕ𝒟(), which by Lemma 2 implies that there is some c such that f-h=c almost everywhere. Let f~=c+h. On the one hand, the fact that DfLloc1(λ) implies that hACloc() and so f~ACloc(). On the other hand, f~=f almost everywhere. Furthermore, because f~ is locally absolutely continuous, integration by parts yields


and by definition of a distributional derivative,


Therefore by Theorem 1, f~=Df~ almost everywhere. But the fact that f~=f almost everywhere implies that Df~=Df almost everywhere, so f~=Df almost everywhere. In particular, f~L2(λ).

Theorem 4.

For fH1(R), there is a function f~ACloc(R) such that f~=f almost everywhere and f~=Df almost everywhere. The function f~ is 12-Hölder continuous.


For x,y,77 7 cf. Giovanni Leoni, A First Course in Sobolev Spaces, p. 222, Theorem 7.13.


and using the Cauchy-Schwarz inequality,

|f~(x)-f~(y)| yx|f~|𝑑λ