Exercises in Global Analysis II
University of Bonn, Summer Semester 2017
Professor: Prof. Matthias Lesch
Assistant: Saskia Roos
Sheet 7
1. Schur Test [5 points]
Let (X, A, µ), (Y, B, ν) be (σ-finite) measure spaces and let k : X × Y → C be an A ⊗ Bmeasurable function. Assume that
Z
Z
C1 := sup
|k(x, y)|dν(y) and C2 := sup
|k(x, y)|dµ(x)
x∈X
y∈Y
Y
X
are finite.
R
Prove that for 1 ≤ p ≤ ∞ the integral operator Kf (x) := Y k(x, y)f (y)dν(y) maps
1
0
1
Lp (Y, ν) continuously into Lp (X, µ) and its norm can be estimated by C1p · C2p , where
1
+ p10 = 1.
p
Remark: This can be proved straight-forward using Hölder and duality for Lp -spaces.
Alternatively, you may show that you need the finiteness of C2 to show L1 -continuity and
C1 to show L∞ -continuity and then conclude the general case by applying the Ries-Thorin
interpolation theorem.
2. An iterated integral as distribution [5 points]
Let a ∈ C ∞ (Rn × Rn ). Assume that there exist m ∈ R, 0 ≤ δ < ρ ≤ 1 such that for all
multiindices α, β ∈ Nn ,
|∂xα ∂ξβ a(x, ξ)| ≤ Cα,β (1 + |ξ|)m−ρ·|β|+δ·|α| .
For a Schwartz function u ∈ S(Rn ) put
Z Z
hI(a), ui :=
Rn
ihx,ξi
e
a(x, ξ)u(x)dx d¯ξ.
Rn
Prove:
a ) The iterated integral defining I(a) exists and I(a) ∈ S 0 (Rn ) is a tempered distribution.
b ) singsupp I(a) ⊂ {0}. That is, I(a) is a smooth function on Rn \ {0}. Moreover, for
|x| ≥ δ > 0 there are estimates
|∂xα I(a)(x)| ≤ Cα,N (1 + |x|)−N , α ∈ Nn , N ∈ N.
d be the Fourier transform of I(a). Then σ ∈ C ∞ (Rn ) and for all
c ) Let σ := I(a)
multiindices α ∈ N
|∂ξα σ(ξ)| ≤ Cα (1 + |ξ|)m−ρ·|α| .
Furthermore, I(a) is given by
Z
Z
hI(a), ui :=
ihx,ξi
e
Rn
σ(ξ)u(x)dx d¯ξ.
Rn
Note: as opposed to a the function σ has no x-dependance.
3. Derivatives of homogenous functions [5 points]
Let Ω ⊂ Rn be open and let a ∈ C ∞ (Ω × RN ) be positively homogeneous of degree m for
|ξ| ≥ 1, i.e.
a(x, λ × ξ) = λm · a(x, ξ) for λ ≥ 1, |ξ| ≥ 1.
Prove that for each pair of multiindices α, β and each compact subset K ⊂ Ω we have
|∂xβ ∂ξα a(x, ξ)| ≤ C(α, β, K) · (1 + |ξ|)m−|α| .
4. A little bit Analysis [5 points]
Let f ∈ C 2 [−ε, ε]. Prove
|f 0 (0)|2 ≤
4
kf k2∞ + 4kf k∞ kf 00 k∞ .
2
ε
Due to Tuesday, June 13.
Homepage of the lecture: http://guests.mpim-bonn.mpg.de/saroos/teaching.html