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MATH 601.101 - Assignment #1 Due on Friday October 9, 2009 Name —————————————– Student number ————————— 1 Problem 1: Given a linear continuous operator satisfying A : C0∞ (X) → C0∞ (X) supp(Au) ⊂ supp(u), u ∈ C0∞ (X) then for any subdomain X " ⊂ X, with compact closure in X, we get a linear differential operator by restricting A to C0∞ (X " ). 2 Problem 2: Verify that the pseudodifferential operator A is properly supprted if and only if its adjoint A∗ is. 3 Problem 3: Given ΨDO A with kernel KA , Prove that KA ∈ C ∞ (U × V ) then A : E " (V ) → C ∞ (U ). 4 Problem 4: m−!|α|+δ|β| m (i) Prove that a ∈ S!,δ (X × Rn ) then ∂ξα ∂xβ a ∈ S!,δ ! (X × Rn ). m m (ii) Prove that a ∈ S!,δ (X × Rn ) and b ∈ S!,δ (X × Rn ) then m+m! a × b ∈ S!,δ (X × Rn ). 5 Problem 5: !n 2 n Given the Laplacian ∆ = j=1 ∂xj and open U ⊂ R and two distributions u, v ∈ D" such that ∆u = v prove that sing supp(u) = sing supp(v). 6 Problem 6: (i) Show that the Dirac measure at zero is given by δ = the sense of oscillatory integrals. " dξ eixξ (2π) n in (ii) Write the operator Dα δ = i−|α| ∂xα δ as an oscillatory integral and prove that sing supp(Dα δ) = {0}.