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RE:THE NECESSITY OFTHE CARLEMAN WEIGHT CONDITIONMICHAEL VANVALKENBURGHmvanvalk@Abstract.These are my notes-to-self regarding the proof of Theorem 8.1.1from H¨o rmander’s old book [2].The proof uses a family of wave packets withparabolic scaling,which we would like to relate to more recent developmentsin wavepacketology.The only novelty here is that there are some added com-ments;it is basically an exercise in L A T E X.1.PreliminariesLet Ωbe a bounded open set in R n .Let P (x,D )be a differential operator of degree m on Ωwith bounded coefficients,whose principal part P m (x,D )has C 1(Ω)coefficients.Let ϕ∈C 2(Ω)be real-valued.Let P m,ϕ(x,ξ):=P m (x,ξ+iσN ).We consider the Carleman estimate (1)τ |α|≤m −1m −1αe 2τϕ|D αu |2dx ≤K 1 e 2τϕ|P u |2dx for all u ∈C ∞0(Ω)and for all τsufficiently large.We will prove Theorem 8.1.1of H¨o rmander’s old book [2],which gives a necessarycondition for (1)to hold.We use the original proof but add some comments.In what follows,we use the notation P (j )m :=∂ξj P m and P m,j :=∂x j P m .2.Theorem 8.1.1of H ¨ormander’s old book Theorem 1.Fix x ∈Ω,and define N:=∇ϕ(x ).Let σ∈R \{0}and ξ∈R n be such that P m (x,ζ)=0,where ζ=ξ+iσN .If (1)holds,then|ζ|2(m −1)≤K 1iσ{P m,ϕ,P m,ϕ}≡2K 1 j,k P (j )m (x,ζ)P (k )m (x,ζ)∂x j ∂x k ϕ+σ−1Im jP (j )m (x,ζ)P m,j (x,ζ)Proof.We may assume x =0and ϕ(0)=0.Then the idea is to use a certain family of wave packets with parabolic scaling 1and complex phase.We will see that the choice of the wave packet is natural from an analytic standpoint.Date :July 16,2006.1Ok,I’m looking back on this from the future:it is now December 3,2007.These are not parabolically scaling in the traditional sense–which would mean √τscaling in one variable and τscaling in another variable.Here we are looking at √τscaling in x in one factor in a product,and τscaling in x in the other factor in the product.(This handout is not meant to be too serious...)12MICHAEL V ANV ALKENBURGH mvanvalk@Let ψ∈C ∞0(Ω)and define˜u τ(x ):=e iτσ−1 x,ζ ψ(x √τ)=e −τ x,N e iτσ−1 x,ξ ψ(x √τ)We think of σas determining the scale of ξ.For all of our musical friends who may be reading this,if σ=2k ,for k ∈Z ,then changing k changes our octave reference point.This is very nearly the parabolically scaling family of wave packets that Cand`e s and Donoho have mentioned in comparison to curvelets [1].The only two differences are:(a)Cand`e s and Donoho normalize by multiplying by √τ,which is irrelevant for (1);and (b)the crucial presence of e −τ x,N ,which acts as a counterweight to the weighted norm in (1).Jealous of the successes of the curvelet transform and the FBI transform,we would like to explore the above “one-sided”wave packets,for possible further use in unique continuation problems.We note thatϕ(x )− x,ξ =A (x )+o (|x |2)where A is a quadratic form,so that A (τ−1/2x )=τ−1A (x ).This is the reason for choosing the wave packets to scale parabolically.While this choice is natural from an analytic standpoint,it seems magical from a geometric standpoint that we should choose a wave packet with O (√τ)oscillations.For technical reasons which will become clear toward the end of the proof,we perturb the phase:Letw (x ):= x,ζ +O (|x |2)andu τ(x ):=e iτσ−1w (x )ψ(x √τ)We compute:(2)P (x,D )u τ(x )=(τσ)m P m (x,∇w )e iτσ−1w ψ(x √τ)+(τσ)m −1n j =1P (j )m (x,∇w )√τe iτσ−1w (D j ψ)(x √τ)+lower order=(τσ)m −1e iτσ−1w (τσ)P m (x,∇w )ψ(x √τ)+n j =1P (j )m (x,∇w )√τ(D j ψ)(x √τ)+O (1) where O (1)is uniform in large τ.Also,for some a ,(3)P m (x,∇ϕ)= a,x +o (|x |),as x →0,RE:THE NECESSITY OF THE CARLEMAN WEIGHT CONDITION3 so we have(στ)2(m−1)τ−1τn2e2τϕ|P uτ|2dx=e2A(x)+o(|x|2)|σ−1x,a +o(|x|)ψ(x)+nj=1P(j)m(x√τ),ζ+o(|x|))+τ−12O(1)|2dxτ→∞−−−−→e2A|σ−1 x,a ψ+nj=1P(j)m(0,ζ)D jψ|2dx(4)Similarly,(5)(στ)2(m−1)τn2|α|≤m−1m−1αe2τϕ|Dαuτ|2dx=|α|≤m−1m−1α|ζα|2e2A|φ|2dx+lower orderτ→∞−−−−→|ζ|2(m−1)e2A|ψ|2dxSo,(1)together with the above gives(6)|ζ|2(m−1)e2A|ψ|2dx≤K1e2A|σ−1 x,a ψ+nj=1P(j)m(0,ζ)D jψ|2dxIf we can arrange to have a=0,then we will have reduced to the simplest Carle-man estimate–the Carleman estimate for operators of the form b·∇and constant quadratic weight.And it is not so hard to see that such a Carleman estimate is equivalent to the weight A being positive definite on the vector b(this is Lemma 8.1.3of H¨o rmander’s old book[2]).Because we allowed a perturbation of our phase w,we are in fact able to arrange a=0.From the definition(3)of a,we see that a=0is equivalent to(7)P m,k(0,ζ)+nj=1P(j)m(0,ζ)w jk(0)=0for k=1,2,···,nwhere we use the notation w jk=∂2w/∂x j∂x k.We are able tofind a w that satisfies this equation,provided that P(j)m(0,ζ)=0for some j;but we see that this is case, by taking an appropriateψin(6).With this choice of w,(6)implies that(8)|ζ|2(m−1)≤2K1nj,k=1[ϕjk(0)−σ−1Im w jk(0)]P(j)m(0,ζ)P(k)m(0,ζ)and from here we just use(7)tofinish the proof.3.Additional commentsA Carleman estimate is fundamentally a relation between quantum mechanics and classical mechanics;that is,it is a relation between a differential operator and its principal symbol.From this point of view it is perhaps not surprising that4MICHAEL V ANV ALKENBURGH mvanvalk@wave packets should play a fundamental role,and,considering the most basic wave packet,e ixξe−12x2,it seems natural that we should use parabolic scaling.However, it seems that“one-side”wave packets could play a larger role in Carleman estimates and,more generally,in unique continuation problems.One objection to this viewpoint is that the necessity of the Carleman condition is “the easy direction”for two reasons.First,in practice in PDE,it seems ridiculous that we would know a Carleman estimate before we knew the principal symbol. Second,the proof is easier—classical mechanics should be a limiting case of quantum mechanics,whereas to get results in the other direction we would expect to rely on some heavy machinery,eg of microlocal analysis.(I’m also typing up notes for the other direction,but that will take much longer.)But we can still wildly speculate that a”one-sided wave packet decomposition”could exist and be useful.References[1]Cand`e s,Emmanuel J.;Donoho,David L.Continuous curvelet transform.I.Resolutionof the wavefront put.Harmon.Anal.19(2005),no.2,162–197.[2]H¨o rmander,Lars.Linear Partial Differential Operators.Springer-Verlag,Berlin,1963.。