PROBLEM SET 3SEMICLASSICALMICROLOCALANALYSISDUE:DEC.07,2020(1) [Schur's test for Lp(Rn)]Wecan extend Schur'stesttoprovetheboundednessof an integral operator onthespace LP(Rn). More precisely, let K : R" ×Rn → C be a continuous function satisfyingCi= sup[K(r, y)]dy <+00and|K(z,y)]dr<+00,C2 = sup3Let Ak be the integral operator with Schwartz kernel K.Thenfor any1≤p≤+oo,IIAk ll/(R") ≤Cl/Pc/PIllp(R"),where as usual, + = 1.[Ref: T.Tao, Interpolation, Schur's test, Young's inequality,Hausdorff-Young, Chris-Kiselev,https://www.math.ucla.edu/~tao/247a.1.06f/notes2.pdf,in$5threediffer-entproofsaregiven.(2)[Pseudolocality]Suppose a E S(i), X1, X2 E C(R") such that suppxo n suppx1 = 0. Prove:IIxiaW xllc(L2(R") = O(h).(3)[Application of Beals theorem: Solving an operator equation]Fix h = 1 in this problem.Suppose ct E S(1) is a family of symbols which is contin-uous in t for |t < to. For any go E S(1), consider the operator equation(αt + c(t)")Q(t) = 0, Q(0) =%W :According to some advanced version of Picard theorem in ODE (in suitable Banachspace), there exists a unique solution Q(t) E C(L?(Rn)) for [tl ≤ to. Prove: Thesolution Q(t) has the form Q(t) = qtw with qt E S(1).[Ref:Zworski,SemiclassicalAnalysis.Lemma8.5](4)[Square rootof positiveh-PsDO]Suppose a E S(m) is elliptic and positive-valued. Prove: There exists b E S(Vm)such thatb+b=a+o(h~).As a consequence, (6W)? = aW + O(h).Hint:Usethemethod intheproof of Theorem1.3inLecture141
PROBLEM SET 3 SEMICLASSICAL MICROLOCAL ANALYSIS DUE: DEC. 07, 2020 (1) [Schur’s test for L p (R n )] We can extend Schur’s test to prove the boundedness of an integral operator on the space L p (R n ). More precisely, let K : R n×R n → C be a continuous function satisfying C1 = sup x Z Rn |K(x, y)|dy < +∞ and C2 = sup y Z Rn |K(x, y)|dx < +∞, Let AK be the integral operator with Schwartz kernel K. Then for any 1 ≤ p ≤ +∞, kAKfkLp(Rn) ≤ C 1/p 1 C 1/p0 2 kfkLp(Rn) , where as usual, 1 p + 1 p 0 = 1. [Ref: T. Tao, Interpolation, Schur’s test, Young’s inequality, Hausdorff-Young, ChrisKiselev, https://www.math.ucla.edu/∼tao/247a.1.06f/notes2.pdf, in §5 three different proofs are given.] (2) [Pseudolocality] Suppose a ∈ S(1), χ1, χ2 ∈ C∞ 0 (R n ) such that suppχ0 ∩ suppχ1 = ∅. Prove: kχ1ba W χ2kL(L2(Rn)) = O(~ ∞). (3) [Application of Beals theorem: Solving an operator equation] Fix ~ = 1 in this problem. Suppose ct ∈ S(1) is a family of symbols which is continuous in t for |t| ≤ t0. For any q0 ∈ S(1), consider the operator equation ( (∂t + cd(t) W )Q(t) = 0, Q(0) = qb0 W . According to some advanced version of Picard theorem in ODE (in suitable Banach space), there exists a unique solution Q(t) ∈ L(L 2 (R n )) for |t| ≤ t0. Prove: The solution Q(t) has the form Q(t) = qbt W with qt ∈ S(1). [ Ref: Zworski, Semiclassical Analysis. Lemma 8.5] (4) [Square root of positive ~-PsDO] Suppose a ∈ S(m) is elliptic and positive-valued. Prove: There exists b ∈ S( √ m) such that b ? b = a + O(~ ∞). As a consequence, (bbW ) 2 = baW + O(~∞). Hint: Use the method in the proof of Theorem 1.3 in Lecture 14. 1
2PROBLEMSET3SEMICLASSICALMICROLOCALANALYSISDUE:DEC.07,2020(5)[Trace class h-PsDOs]Prove Proposition 2.6 in Lecture 13.[Ref:Dimassi-Sjostrand, Spectral Asymptotics in the Semi-Classical Limit, s9(6)[Anti-Wick quantization]Let 4o(r) = (πh)-n/4er/2h. Note that [0oll2 = 1. Let Po : L?(Rn) -→ L?(Rn) bethe orthogonal projection on thevector @o, namelydo(u) = (u, do)do.(a) Prove Po is a semiclassical pseudodifferential operator and find its Weyl symbol.(b) For any (ro, So) E Rn, define a unitary operator Uro.5o : L2(Rn) -→ L?(Rn) byUro.5ou(r) =eir-sou(r - ro).DefineProoUrPUProveProiaositiveerato(Pro,ou, u) ≥0, VuE9.(c) For any a E S(m), define the anti-Wick quantization of a to the the operatoranti-Wicku(y) :=a(r,s)Pr,eu(y)drdsProve: the anti-Wick quantization is a positive quantization, i.e. if a ≥ O, thenaanti-Wick is a positive operator.(d) Prove: The aanti-Wick is a semiclassical pseudodifferential operator whose Weylsymbol isa(y, n)e-[(r-w)+(-n)/ndydn.a(a,s) =(πh)-n(e) Prove: If ae s(1), then a e s(1) and a- a e hs(1)( Prove: For ae S(i) lantwiek ale(2(r") = 0(h),(g) Prove the Sharp Garding inequality (Theorem 2.1 in Lecture 15) via the anti-Wick quantization.[Ref:Shubin, Pseudodifferential operators and spectral theory,g24][Ref:Wong,An introduction to semiclassical and microlocal analysis,Problem 22 onpage 67]
2 PROBLEM SET 3 SEMICLASSICAL MICROLOCAL ANALYSIS DUE: DEC. 07, 2020 (5) [Trace class ~-PsDOs] Prove Proposition 2.6 in Lecture 13. [Ref: Dimassi-Sjostrand, Spectral Asymptotics in the Semi-Classical Limit, §9] (6) [Anti-Wick quantization] Let Φ0(x) = (π~) −n/4 e x 2/2~ . Note that kΦ0kL2 = 1. Let P0 : L 2 (R n ) → L 2 (R n ) be the orthogonal projection on the vector Φ0, namely Φ0(u) = hu, Φ0iΦ0. (a) Prove P0 is a semiclassical pseudodifferential operator and find its Weyl symbol. (b) For any (x0, ξ0) ∈ R n , define a unitary operator Ux0,ξ0 : L 2 (R n ) → L 2 (R n ) by Ux0,ξ0 u(x) = e ix·ξ0 u(x − x0). Define Px0,ξ0 := Ux0,ξ0P0U ∗ x0,ξ0 . Prove: Px0,ξ0 is a positive operator, i.e. hPx0,ξ0 u, ui ≥ 0, ∀u ∈ S . (c) For any a ∈ S(m), define the anti-Wick quantization of a to the the operator ba anti-Wicku(y) := Z R2n a(x, ξ)Px,ξu(y)dxdξ, Prove: the anti-Wick quantization is a positive quantization, i.e. if a ≥ 0, then ba anti-Wick is a positive operator. (d) Prove: The ba anti-Wick is a semiclassical pseudodifferential operator whose Weyl symbol is a˜(x, ξ) = (π~) −n Z R2n a(y, η)e −[(x−y) 2+(ξ−η) 2 ]/~ dydη. (e) Prove: If a ∈ S(1), then ˜a ∈ S(1) and ˜a − a ∈ ~S(1). (f) Prove: For a ∈ S(1), kba anti-Wick − baW kL(L2(Rn)) = O(~). (g) Prove the Sharp Garding inequality (Theorem 2.1 in Lecture 15) via the antiWick quantization. [Ref: Shubin, Pseudodifferential operators and spectral theory, §24] [Ref: Wong, An introduction to semiclassical and microlocal analysis, Problem 22 on page 67]