SIXTH ASSIGNMENT, DUE OCTOBER 23 IN CLASS18.155 FALL 2001RICHARD MELROSEProblem 1. Hilbert space and the Riesz representation theorem. If you need helpwith this, it can be found in lots of places – for instance [1] has a nice treatment.i) A pre-Hilbert space is a vector space V (over C) with a ‘positive definitesesquilinear inner pro duct’ i.e. a functionV × V 3 (v, w) 7→ hv, wi ∈ Csatisfying• hw, vi = hv, wi• ha1v1+ a2v2, wi = a1hv1, wi + a2hv2, wi• hv, vi ≥ 0• hv, vi = 0 ⇒ v = 0.Prove Schwarz’ inequality, that|hu, vi| ≤ hui12hvi12∀ u, v ∈ V.Hint: Reduce to the case hv, vi = 1 and then expandhu − hu, viv , u − hu, vivi ≥ 0.ii) Show that kvk = hv, vi1/2is a norm and that it satisfies the parallelogramlaw:(1) kv1+ v2k2+ kv1− v2k2= 2kv1k2+ 2kv2k2∀ v1, v2∈ V.iii) Conversely, suppose that V is a linear space over C with a norm whichsatisfies (1). Show that4hv, wi = kv + wk2− kv − wk2+ ikv + iwk2− ikv − iwk2defines a pre-Hilbert inner product which gives the original norm.iv) Let V be a Hilbert space, so as in (i) but complete as well. Let C ⊂ V be aclosed non-empty convex subset, meaning v, w ∈ C ⇒ (v + w)/2 ∈ C. Showthat there exists a unique v ∈ C minimizing the norm, i.e. such thatkvk = infw∈Ckwk.Hint: Use the parallelogram law to show that a norm minimizing se-quence is Cauchy.v) Let u : H → C be a continuous linear functional on a Hilbert space, so|u(ϕ)| ≤ Ckϕk ∀ ϕ ∈ H. Show that N = {ϕ ∈ H; u(ϕ) = 0} is closed andthat if v0∈ H has u(v0) 6= 0 then each v ∈ H can be written uniquely inthe formv = cv0+ w, c ∈ C, w ∈ N.12 RICHARD MELROSEvi) With u as in v), not the zero functional, show that there exists a uniquef ∈ H with u(f) = 1 and hw, fi = 0 for all w ∈ N.Hint: Apply iv) to C = {g ∈ V ; u(g) = 1}.vii) Prove the Riesz Representation theorem, that every continuous linear func-tional on a Hilb e rt space is of the formuf: H 3 ϕ 7→ hϕ, f i for a unique f ∈ H.Problem 2. Density of C∞c(Rn) in Lp(Rn).i) Recall in a few words why simple integrable functions are dense in L1(Rn)with respect to the norm kfkL1=RRn|f(x)|dx.ii) Show that simple functionsPNj=1cjχ(Uj) where the Ujare open and boundedare also dense in L1(Rn).iii) Show that if U is open and bounded then F (y) = v(U ∩ Uy), where Uy={z ∈ Rn; z = y + y0, y0∈ U } is continuous in y ∈ Rnand thatv(U ∩ U{y) + v(U{∩ Uy) → 0 as y → 0.Solution. To prove continuity as y → 0 consider the setUδ= {p ∈ U; |p − q| ≤ δ =⇒ q ∈ U}, δ > 0.Since the balls are closed, this is an open set andUδ⊂ Uy∩ U ⊂ U if |y| ≤ δ.Furthermore, the Uδincrease as δ decreases and[δ >0Uδ= U.If we set Vn= U1/(n+1)\ U1/nthe Vnare measurable and disjoint so∞Xn=1v(Vn) = v(U) =⇒ v(Uδ) → v(U) as δ → 0.The same argument applies to continuity at a general point ¯y. SimplytakeU¯y , δ= {p ∈ U; p + y ∈ U if |y − ¯y| ≤ δ} .These open sets increas as δ decre ases with the union being U ∩ U¯y. iv) If U is open and bounded and ϕ ∈ C∞c(Rn) show thatf(x) =ZUϕ(x − y)dy ∈ C∞c(Rn).v) Show that if U is open and bounded thensup|y |≤δZ|χU(x) − χU(x − y)|dx → 0 as δ ↓ 0.vi) If U is open and bounded and ϕ ∈ C∞c(Rn), ϕ ≥ 0,Rϕ = 1 thenfδ→ χUin L1(Rn) as δ ↓ 0wherefδ(x) = δ−nZϕyδχU(x − y)dy.Hint: Write χU(x) = δ−nRϕyδχU(x) and use v).3vii) Conclude that C∞c(Rn) is dense in L1(Rn).viii) Show that C∞c(Rn) is dense in Lp(Rn) for any 1 ≤ p < ∞.Problem 3. Schwartz representation theorem. Here we (well you) come to gripswith the general structure of a tempered distribution.i) Recall briefly the proof of the Sobolev embedding theorem and the c orre-sponding estimatesupx∈Rn|φ(x)| ≤ CkφkHm,n2< m ∈ R.ii) For m = n + 1 write down a(n equivalent) norm on the right in a form thatdoes not involve the Fourier transform.iii) Show that for any α ∈ N0|Dα(1 + |x|2)Nφ| ≤ Cα,NXβ≤α(1 + |x|2)N|Dβφ|.iv) Deduce the general estimatessup|α|≤Nx∈Rn(1 + |x|2)N|Dαφ(x)| ≤ CNk(1 + |x|2)NφkHN +n+1.v) Conclude that for each tempered distribution u ∈ S0(Rn) there is an integerN and a constant C such that|u(φ)| ≤ Ck(1 + |x|2)NφkH2N∀ φ ∈ S(Rn).vi) Show that v = (1 + |x|2)−Nu ∈ S0(Rn) satisfies|v(φ)| ≤ Ck(1 + |D|2)NφkL2∀ φ ∈ S(Rn).vi) Recall (from class or just show it) that if v is a tempered distribution thenthere is a unique w ∈ S0(Rn) such that (1 + |D|2)Nw = v.vii) Use the Riesz Representation Theorem to conclude that for each tempereddistribution u there exists N and w ∈ L2(Rn) such that(2) u = (1 + |D|2)N(1 + |x|2)Nw.viii) Use the Fourier transform on S0(Rn) (and the fact that it is an isomorphismon L2(Rn)) to show that any tempered distribution can be written in theformu = (1 + |x|2)N(1 + |D|2)Nw for some N and some w ∈ L2(Rn).ix) Show that any tempered distribution can be written in the formu = (1 + |x|2)N(1 + |D|2)N+n+1˜w for some N and some ˜w ∈ H2(n+1)(Rn).x) Conclude that any tempered distribution can be written in the formu = (1 + |x|2)N(1 + |D|2)MU for some N, Mand a bounded continuous function UReferences[1] George F. Simmons, Introduction to topology and modern analysis, Robert E. Krieger Pub-lishing Co. Inc., Melbourne, Fla., 1983, Reprint of the 1963 original. MR 84b:54002Department of Mathematics, Massachusetts Institute of TechnologyE-mail address:
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