SEVENTH ASSIGNMENT, DUE OCTOBER 30 IN CLASS18.155 FALL 2001RICHARD MELROSEProblem 1. Distributions of compact support.i) Recall the definition of the support of a distribution, defined in terms of itscomplementRn\ supp(u) =p ∈ Rn; ∃ U ⊂ Rn, open, with p ∈ U such that uU= 0ii) Show that if u ∈ C−∞(Rn) and φ ∈ C∞c(Rn) satisfysupp(u) ∩ supp(φ) = ∅then u(φ) = 0.iii) Consider the space C∞(Rn) of all smooth functions on Rn, without restric-tion on supports. Show that for each Nkfk(N)= sup|α|≤N, |x|≤N|Dαf(x)|is a seminorn on C∞(Rn) (meaning it satisfies kfk ≥ 0, kcf k = |c|kf k forc ∈ C and the triangle inequality but that kfk = 0 do es not necessarilyimply that f = 0.iv) Show that C∞c(Rn) ⊂ C∞(Rn) is dense in the sense that for each f ∈C∞(Rn) there is a sequence fnin C∞c(Rn) such that kf − fnk(N)→ 0 foreach N.v) Let E0(Rn) temporarily (or permanantly if you prefer) denote the dual spaceof C∞(Rn) (which is also written E(Rn)), that is, v ∈ E0(Rn) is a linear mapv : C∞(Rn) −→ C which is continuous in the sense that for some N(1) |v(f)| ≤ Ckfk(N)∀ f ∈ C∞(Rn).Show that such a v ‘is’ a distribution and that the map E0(Rn) −→ C−∞(Rn)is injective.vi) Show that if v ∈ E0(Rn) satisfies (1) and f ∈ C∞(Rn) has f = 0 in |x| <N + for some > 0 then v(f) = 0.vii) Conclude that each element of E0(Rn) has compact support when consideredas an element of C−∞(Rn).viii) Show the converse, that each element of C−∞(Rn) with compact support isan element of E0(Rn) ⊂ C−∞(Rn) and hence conclude that E0(Rn) ‘is’ thespace of distributions of compact support.I will denote the space of distributions of compact support by C−∞c(R).Problem 2. Hypoellipticity of the heat operator H = iDt+ ∆ = iDt+nPj=1D2xjonRn+1.12 RICHARD MELROSE(1) Using τ to denote the ‘dual variable’ to t and ξ ∈ Rnto denote the dualvariables to x ∈ Rnobserve that H = p(Dt, Dx) where p = iτ + |ξ|2.(2) Show that |p(τ, ξ)| >12|τ| + |ξ|2.(3) Use an inductive argument to show that, in (τ, ξ) 6= 0 where it makes sense,(2) DkτDαξ1p(τ, ξ)=|α|Xj=1qk,α,j(ξ)p(τ, ξ)k+j+1where qk,α,j(ξ) is a polynomial of degree (at most) 2j − |α|.(4) Conclude that if φ ∈ C∞c(Rn+1) is identically equal to 1 in a neighbourhoodof 0 then the functiong(τ, ξ) =1 − φ(τ, ξ)iτ + |ξ|2is the Fourier transform of a distribution F ∈ S0(Rn) with sing supp(F ) ⊂{0}. [Remember that sing supp(F ) is the c omplement of the largest opensubset of Rnthe restriction of F to which is smooth].(5) Show that F is a parametrix for the heat operator.(6) Deduce that iDt+ ∆ is hypoelliptic – that is, if U ⊂ Rnis an open set andu ∈ C−∞(U) satisfies (iDt+ ∆)u ∈ C∞(U) then u ∈ C∞(U).(7) Show that iDt− ∆ is also hypo elliptic.Department of Mathematics, Massachusetts Institute of TechnologyE-mail address:
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