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}.