330 BRUCE K. DRIVER† 17. Weak and Strong Derivatives and Sobolev Spaces For this section, let Ω be an open subset of Rd, p,q,r [1, ],Lp(Ω)= p p p ∈ ∞ L (Ω, Ω,m) and Lloc(Ω)=Lloc(Ω, Ω,m), where m is Lebesgue measure on Rd B B d p Bp and Ω is the Borel σ —algebraonΩ. If Ω = R , we will simply write L and Lloc Bp d p d for L (R ) and Lloc(R ) respectively. Also let f,g := fgdm h i ZΩ for any pair of measurable functions f,g : Ω C such that fg L1(Ω). For example, by Hölder’s inequality, if f,g is defined→ for f Lp(Ω) and∈ g Lq(Ω) p h i ∈ ∈ when q = p 1 . The following simple but useful remark will be used (typically without further− comment) in the sequel. 1 1 1 Remark 17.1. Suppose r, p, q [1, ] are such that r− = p− + q− and ft f p q ∈ ∞ r → in L (Ω) and gt g in L (Ω) as t 0, then ftgt fg in L (Ω). Indeed, → → → ftgt fg = (ft f) gt + f (gt g) k − kr k − − kr ft f gt + f gt g 0 as t 0 ≤ k − kp k kq k kp k − kq → → d p p Definition 17.2 (Weak Differentiability). Let v R and f L (Ω)(f Lloc(Ω)) p ∈p ∈ ∈ then ∂vf is said to exist weakly in L (Ω)(Lloc(Ω)) if there exists a function g Lp(Ω)(g Lp (Ω)) such that ∈ ∈ loc (17.1) f,∂vφ = g, φ for all φ C∞(Ω). h i −h i ∈ c (w) The function g if it exists will be denoted by ∂v f. (By Corollary 9.27, there is at 1 (w) most one g L (Ω) such that Eq. (17.1) holds, so ∂v f is well defined.) ∈ loc 1 (w) 1 Lemma 17.3. Suppose f L (Ω) and ∂v f exists weakly in L (Ω). Then ∈ loc loc (w) (1) suppm(∂v f) suppm(f), where suppm(f) is the essential support of f relative to Lebesgue⊂ measure, see Definition 9.14. (w) (2) If f is continuously differentiable on U o Ω, then ∂v f = ∂vf a.e. on U. ⊂ Proof. (1) Since (w) ∂ f,φ = f,∂vφ =0for all φ C∞(Ω supp (f)), h v i −h i ∈ c \ m (w) and application of Corollary 9.27 shows ∂v f =0a.e. on Ω suppm(f). (w) \ (w) So by Lemma 9.15, Ω supp (f) Ω supp (∂v f), i.e. supp (∂v f) \ m ⊂ \ m m ⊂ suppm(f). 1 (2) Suppose that f U is C and let ψ Cc∞(U) which we view as a function d | ∈d in Cc∞(R ) by setting ψ 0 on R U. By Corollary 9.24, there exists ≡ \ γ Cc∞(Ω) such that 0 γ 1 and γ =1in a neighborhood of supp(ψ). ∈ ≤ ≤d 1 d Then by setting γf =0on R supp(γ) we may view γf Cc (R ) and so by standard integration by parts\ (see Lemma 9.25) and∈ the ordinary product rule, (w) ∂ f,ψ = f,∂vψ = γf,∂vψ h v i −h i −h i (17.2) = ∂v (γf) ,ψ = ∂vγ f + γ∂vf,ψ = ∂vf,ψ h i h · i h i REAL ANALYSIS LECTURE NOTES 331 wherein the last equality we have used ψ∂vγ =0and ψγ = ψ. Since Eq. (17.2) is true for all ψ Cc∞(U), an application of Corollary 9.27 with (w) ∈ (w) h = ∂v f(x) ∂vf(x) and µ = m shows ∂v f(x)=∂vf(x) for m —a.e. x U. − ∈ 1 d (w) Lemma 17.4 (Product Rule). Let f Lloc(Ω),v R and φ C∞(Ω). If ∂v f 1 (w) ∈ 1 ∈ ∈ exists in Lloc(Ω), then ∂v (φf) exists in Lloc(Ω) and (w) (w) ∂ (φf)=∂vφ f + φ∂ f a.e. v · v d 1 d Moreover if φ Cc∞(R ) and F := φf L (herewedefine F on R by setting d ∈ (w) ∈ (w) 1 d F =0on R Ω ), then ∂ F = ∂vφ f + φ∂v f exists weakly in L (R ). \ · Proof. Let ψ C∞(Ω), then ∈ c (w) φf, ∂vψ = f,φ∂vψ = f,∂v (φψ) ∂vφ ψ = ∂ f,φψ + ∂vφ f,ψ −h i −h i −h − · i h v i h · i (w) = φ∂ f,ψ + ∂vφ f,ψ . h v i h · i This proves the first assertion. To prove the second assertion let γ Cc∞(Ω) such ∈ d that 0 γ 1 and γ =1on a neighborhood of supp(φ). So for ψ Cc∞(R ), using ≤ ≤ ∈ ∂vγ =0on supp(φ) and γψ C∞(Ω), we find ∈ c F, ∂vψ = γF,∂vψ = F, γ∂vψ = (φf) ,∂v (γψ) ∂vγ ψ h i h i h i h − · i (w) = (φf) ,∂v (γψ) = ∂ (φf) , (γψ) h i −h v i (w) (w) = ∂vφ f + φ∂ f,γψ = ∂vφ f + φ∂ f,ψ . −h · v i −h · v i (w) (w) This show ∂v F = ∂vφ f + φ∂v f as desired. · p Lemma 17.5. Suppose p [1, ),v Rd and f L (Ω). ∈ ∞ ∈ ∈ loc p (w) p (1) If there exists fm m∞=1 Lloc(Ω) such that ∂v fm exists in Lloc(Ω) for { } ⊂ p all m and there exists g L (Ω) such that for all φ C∞(Ω), ∈ loc ∈ c lim fm,φ = f,φ and lim ∂vfm,φ = g,φ m m →∞h i h i →∞h i h i (w) p then ∂v f exists in Lloc(Ω) and ∂vf = g. (w) p (2) If ∂v f exists in Lloc(Ω) then there exists fn Cc∞(Ω) such that fn f p ∈ (w) p→ in L (K) (i.e. limn f fn Lp(K) =0)and ∂vfn ∂v f in L (K) →∞ k − k → for all K @@ Ω. Proof. (1) Since (w) f,∂vφ = lim fm,∂vφ = lim ∂ fm,φ = g, φ m m v h i →∞h i − →∞h i h i (w) p for all φ C∞(Ω),∂v f exists and is equal to g L (Ω). ∈ c ∈ loc (2) Let K0 := and ∅ c Kn := x Ω : x n and d(x, Ω ) 2/n { ∈ | | ≤ ≥ } o (so Kn Kn+1 Kn+1 for all n and Kn Ω as n or see Lemma ⊂ ⊂ o ↑ →∞ 8.10) and choose ψn C∞(K , [0, 1]) using Corollary 9.24 so that ψn =1 ∈ c n on a neighborhood of Kn 1. Given a compact set K Ω, for all sufficiently − ⊂ 332 BRUCE K. DRIVER† large m, ψmf = f on K and by Lemma 17.4 and item 1. of Lemma 17.3, we also have (w) (w) (w) ∂ (ψmf)=∂vψm f + ψm∂ f = ∂ f on K. v · v v This argument shows we may assume suppm(f) is a compact subset of Ω in which case we extend f to a function F on Rd by setting F = f on Ω c p d (w) (w) and F =0on Ω . This function F is in L (R ) and ∂v F = ∂v f. Indeed, d d if φ Cc∞(R ) and ψ Cc∞(R ) is chosen so that supp(ψ) Ω, 0 ψ 1 ∈ ∈ ⊂ ≤ ≤ and ψ =1in a neighborhood of suppm(f), then F, ∂vφ = F, ψ∂vφ = f,∂v (ψφ) h i h i h i = ∂(w)f,ψφ = ψ∂(w)f,φ −h v i −h v i (w) p d which shows ∂v F exist in L (R ) and (w) (w) (w) ∂v F = ψ∂v f =1Ω∂v f. d Let χ C∞(B0(1)) with d χdm =1and set δk(x)=n χ(nx). Then ∈ c R there exists N N and a compact subset K Ω such that fn := F δn ∈ R ⊂ ∗ ∈ Cc∞(Ω) and supp(fn) K for all n N. By Proposition 9.23 and the (w) ⊂ ≥ definition of ∂v F, ∂vfn(x)=F ∂vδn(x)= F (y)∂vδn(x y)dy ∗ d − ZR (w) (w) = F, ∂v [δn(x )] = ∂ F, δn(x ) = ∂ F δn(x). −h − · i h v − · i v ∗ (w) (w) p Hence by Theorem 9.20, fn F = f and ∂vfn ∂v F = ∂v f in L (Ω) as n . → → →∞ d p Definition 17.6 (Strong Differentiability). Let v R and f L , then ∂vf is p ∈ ∈p said to exist strongly in L if the limt 0 (τ tvf f) /t exists in L , where as above → − −(s) τvf(x):=f(x v). We will denote the limit by ∂v f. − p d (s) p (w) It is easily verified that if f L ,v R and ∂v f L exists then ∂v f exists (w) (s) ∈ ∈ ∈ d and ∂v f = ∂v f. To check this assertion, let φ Cc∞(R ) andthenusingRemark 17.1, ∈ (s) 1 τ tvf f ∂v f φ = L — lim − − φ. t 0 t · → Hence (s) τ tvf f τtvφ φ ∂v f φdm = lim − − φdm =lim f − dm d · t 0 d t t 0 d t ZR → ZR → ZR d d = 0 fτtvφdm = f 0τtvφdm = f ∂vφdm, dt| d d · dt| − d · ZR ZR ZR wherein we have used Corollary 5.43 to differentiate under the integral in the fourth (w) (s) equality. This shows ∂v f exists and is equal to ∂v f. What is somewhat more (w) (s) surprising is that the converse assertion that if ∂v f exists then so does ∂v f. The next theorem is a generalization of Theorem 10.36 from L2 to Lp.