Sobolev spaces in one dimension and absolutely continuous functions Jordan Bell [email protected] Department of Mathematics, University of Toronto October 21, 2015 1 Locally integrable functions and distributions 1 Let λ be Lebesgue measure on R. We denote by Lloc (λ) the collection of Borel measurable functions f : R → R such that for each compact subset K of R, Z Z NK (f ) = |f |dλ = 1K |f |dλ < ∞. K R 1 We denote by L1loc (λ) the collection of equivalence classes of elements of Lloc (λ) where f ∼ g when f = g almost everywhere. 1 Write B(x, r) = {y ∈ R : |y − x| < r} = (x − r, x + r). For f ∈ Lloc (λ) and x ∈ R, we say that x is a Lebesgue point of f if Z 1 lim |f (y) − f (x)|dλ(y) = 0. r→0 λ(B(x, r)) B(x,r) It is immediate that if f is continuous at x then x is a Lebesgue point of f . 1 The Lebesgue differentiation theorem1 states that for f ∈ Lloc (λ), almost every x ∈ R 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 rn → 0 such that En ⊂ B(x, rn ) and λ(En ) ≥ α · λ(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 1 f ∈ Lloc (λ) and for each x ∈ R, En (x) is a sequence that shrinks nicely to x, then Z 1 f (x) = lim f dλ n→∞ λ(En ) E (x) n at each Lebesgue point of f .2 1 Walter 2 Walter Rudin, Real and Complex Analysis, third ed., p. 138, Theorem 7.7. Rudin, Real and Complex Analysis, third ed., p. 140, Theorem 7.10. 1 For a nonempty open set Ω in R, we denote by Cck (Ω) the collection of C k functions φ : R → R such that supp φ = {x ∈ R : φ(x) 6= 0} 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.3 R 1 Theorem 1. If f ∈ Lloc (λ) and R f φdλ = 0 for all φ ∈ D(R), then f = 0 almost everywhere. R Proof. There is some η ∈ D(−1, 1) with R ηdλ = 1. We can explicitly write this out: ( |x| < 1 c−1 exp x21−1 η(x) = 0 |x| ≥ 1, where Z 1 c= exp −1 1 2 y −1 dλ(y) = 0.443994 . . . . For x a Lebesgue point of f and for 0 < r < 1, Z f (x) = f (x) · η(y)dλ(y) R Z y 1 η dλ(y) = f (x) · r R r Z x−y 1 = f (x) · η dλ(y) r R r Z Z 1 x−y 1 x−y = (f (x) − f (y))η dλ(y) + f (y)η dλ(y) r R r r R r Z x−y 1 = (f (x) − f (y))η dλ(y) r R r Z 1 x−y = (f (x) − f (y))η dλ(y). r (x−r,x+r) r Then |f (x)| ≤ kηk∞ · 1 r Z |f (y) − f (x)|dλ(y) → 0, r → 0, (x−r,x+r) meaning that f (x) = 0. This is true for almost all x ∈ R, showing that f = 0 almost everywhere. 3 Lars Hörmander, The Analysis of Linear Partial Differential Operators I, second ed., p. 15, Theorem 1.2.5. 2 1 For f ∈ Lloc (λ), define Λf : D(R) → R by Z Λf (φ) = f φdλ. R D(R) is a locally convex space, and one proves that Λf is continuous and thus belongs to the dual space D 0 (R), whose elements are called distributions.4 We 1 say that a distribution Λ is induced by f ∈ Lloc (λ) if Λ = Λf . For Λ ∈ D 0 (R), we define DΛ : D(R) → R by (DΛ)(φ) = −Λ(φ0 ). It is proved that DΛ ∈ D 0 (R).5 1 Let f, g ∈ Lloc (λ). If DΛf = Λg , we call g a distributional derivative 1 of f . In other words, for f ∈ Lloc (λ) to have a distributional derivative means 1 that there is some g ∈ Lloc (λ) such that for all φ ∈ D(R), Z Z 0 − f φ dλ = gφdλ. R R R 1 If g1 , g2 ∈ Lloc (λ) are distributional derivatives of f then R (g1 −g2 )φdλ = 0 for all φ ∈ D(R), 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 L1loc (λ), and is denoted Df ∈ L1loc (λ): Z Z − f φ0 dλ = (Df ) · φdλ, φ ∈ D(R). R 2 R The Sobolev space H 1 (R) We denoteR by L 2 (λ) the collection of Borel measurable functions f : R → R such that R |f |2 dλ < ∞, and we denote by L2 (λ) the collection of equivalence classes of elements of L 2 (λ) where f ∼ g when f = g almost everywhere, and write Z hf, giL2 = f gdλ. R 2 It is a fact that L (λ) is a Hilbert space. We define the Sobolev space H 1 (R) to be the set of f ∈ L2 (λ) that have a distributional derivative that satisfies Df ∈ L2 (λ). We remark that the elements of H 1 (R) are equivalence classes of elements of L 2 (λ). We define hf, giH 1 = hf, giL2 + hDf, DgiL2 . 4 Walter 5 Walter Rudin, Functional Analysis, second ed., p. 157, §6.11. Rudin, Functional Analysis, second ed., p. 158, §6.12. 3 Let f, g ∈ H 1 (R) and let φ ∈ D(R). Because f, g have distributional derivatives Df, Dg, Z Z Z 0 0 − (f + g)φ dλ = − f φ dλ − gφ0 dλ R R R Z Z = Df · φdλ + Dg · φdλ R R Z = (Df + Dg)φdλ. R This means that f + g has a distributional derivative, D(f + g) = Df + Dg. R Thus H 1 (R) is a linear space. If hf, f iH 1 = 0 then R |f |2 dλ = 0, which implies that f = 0 as an element of L2 (λ). Therefore h·, ·iH 1 is an inner product on H 1 (R). If fn is a Cauchy sequence in H 1 (R), 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, g ∈ L2 (λ). For φ ∈ D(R), Z Z 0 fn φ0 dλ − f φ dλ = − lim n→∞ R R Z = lim (Dfn ) · φdλ n→∞ R Z = gφdλ. R This means that f has distributional derivative, Df = g. Because f, Df ∈ L2 (λ) it is the case that f ∈ H 1 (R). Furthermore, 2 2 2 2 2 kfn − f kH 1 = kfn − f kL2 + kDfn − Df kL2 = kfn − f kL2 + kDfn − gkL2 → 0, meaning that fn → f in H 1 (R), which shows that H 1 (R) 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.6 1 Lemma 2. If f ∈ Lloc (λ) and Z f φ0 dλ = 0, φ ∈ D(R), R then there is some c ∈ R such that f = c almost everywhere. 6 Haim Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, p. 204, Lemma 8.1. 4 Proof. Fix η ∈ D(R) with R R ηdλ = 1. Let w ∈ D(R) and define Z h = w − η · wdλ, R which belongs to D(R) and satisfies R hdλ = 0. Define φ : R → R by Z x φ(x) = hdλ. R −∞ 0 R Using φ (x) = h(x) for all x and → R hdλ = 0 as x → ∞, check that R φ(x) φ ∈ D(R). Then by hypothesis, R f φ0 dλ = 0, i.e. Z 0= f hdλ R Z Z = f w − f η · wdλ dλ R R Z Z = f− f ηdλ · wdλ. R R Because this is true for all w ∈ D(R), by Theorem 1 we get that f = almost everywhere. R 1 (λ), let a ∈ R, and define f : R → R by Lemma 3. Let g ∈ Lloc Z x f (x) = g(y)dλ(y). a Then Z f φ0 dλ = − R Z gφdλ R for all φ ∈ D(R). Proof. Using Fubini’s theorem, Z Z a Z a 0 f (x)φ (x)dλ(x) = − g(y)dλ(y) φ0 (x)dλ(x) R −∞ x Z ∞ Z x + g(y)dλ(y) φ0 (x)dλ(x) a a Z a Z y 0 =− φ (x)dλ(x) g(y)dλ(y) −∞ −∞ Z ∞ Z ∞ 0 + φ (x)dλ(x) g(y)dλ(y) a y Z a =− Z φ(y)g(y)dλ(y) − Z−∞ = − g(y)φ(y)dλ(y). R 5 ∞ φ(y)g(y)dλ(y) a R f ηdλ For real numbers a, b with a < b, we say that a function f : [a, b] → R is absolutely continuous if for all > 0 there is some δ > 0 such that whenever (a1 , b1 ), . . . , (an , bn ) are P P disjoint intervals each contained in [a, b] with (bk − ak ) < δ it holds that |f (bk ) − f (ak )| < . We say that a function f : R → R 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 (R). Let f ∈ H 1 (R), let a ∈ R, and define h : R → R by Z x h(x) = Df dλ. a By Lemma 3 and by the definition of a distributional derivative, Z Z Z hφ0 dλ = − (Df ) · φdλ = f φ0 dλ, φ ∈ D(R). R R R Hence R (f − h)φ dλ = 0 for all φ ∈ D(R), which by Lemma 2 implies that there is some c ∈ R such that f − h = c almost everywhere. Let fe = c + h. On the one hand, the fact that Df ∈ L1loc (λ) implies that h ∈ ACloc (R) and so fe ∈ ACloc (R). On the other hand, fe = f almost everywhere. Furthermore, because fe is locally absolutely continuous, integration by parts yields Z Z 0 e f φ dλ = − fe0 φdλ, R 0 R R and by definition of a distributional derivative, Z Z feφ0 dλ = − (Dfe)φdλ. R R e0 Therefore by Theorem 1, f = Dfe almost everywhere. But the fact that fe = f almost everywhere implies that Dfe = Df almost everywhere, so fe0 = Df almost everywhere. In particular, fe0 ∈ L2 (λ). Theorem 4. For f ∈ H 1 (R), there is a function fe ∈ ACloc (R) such that fe = f almost everywhere and fe0 = Df almost everywhere. The function fe is 21 -Hölder continuous. Proof. For x, y ∈ R,7 x Z fe0 dλ, fe(x) − fe(y) = y and using the Cauchy-Schwarz inequality, Z x |fe(x) − fe(y)| ≤ |fe0 |dλ y ≤ |x − y|1/2 Z x |fe0 |2 dλ 1/2 y ≤ kDf kL2 |x − y|1/2 . 7 cf. Giovanni Leoni, A First Course in Sobolev Spaces, p. 222, Theorem 7.13. 6 7