Math 608, Homework 10∗
There are a few optional problems on this assignment. You don’t have to do them unless you want
to (they are long), but you should make sure you know how to do them if necessary. You can use
their results in other exercises without proof.
(
1. Exercise 8.6. Hint: use the relation
1 1
−
p r
)
(
+
1 1
−
q r
)
+
1
= 1.
r
2. Let M = M (Rn ) be the space of all complex Borel measures.
(a) Show that the two definitions of convolution ∗ on M given in class coincide.
(b) Show that (M, ∗) is a commutative Banach algebra with a unit, where the normed space
structure on M is the one in Exercise 5.8. (Hint: you might want to use Exercise 3.21.)
(c) Optional problem. Show that for appropriate g, in particular for g ∈ C0 ,
∫
∫∫
g(x)d(µ ∗ ν)(x) =
g(x + y) dµ(x) dν(y).
3. For f ∈ Lp , 1 ≤ p ≤ ∞, and µ ∈ M , define the function f ∗ µ, and show that ∥f ∗ µ∥p ≤
∥f ∥p ∥µ∥, so that Lp ∗ M ⊂ Lp . Show that L∞ ∗ M ̸⊂ C, but C0 ∗ M ⊂ C0 .
4. Compute f ∗ g in L1 (R), where
(a)
f = χ[a,b] ,
(b)
f = e−x
2 /2a
g = χ[c,d] .
,
g = e−x
2 /2b
g=
1
+ b2
.
(c) Optional problem.
f=
Answer: π
∗
x2
1
,
+ a2
a+b
1
.
2
ab x + (a + b)2
c 2012 by Michael Anshelevich.
⃝
1
x2
5. Show that each of the following families of measures {µt : t ≥ 0} on R forms a convolution
semigroup, that is, for all s, t ≥ 0
µs ∗ µt = µs+t .
(a) Cauchy distributions:
µt =
(b) Gaussian distributions:
1
t
dx.
2
π x + t2
1 −x2 /2t
µt = √
e
dx.
2πt
(c) Poisson distributions:
µt = e−t
∞
∑
tn
n=0
n!
δn .
(Hint: first compute δa ∗ δb .)
6. Let
2
1
e−|x| /4t
n/2
(4πt)
n
be the Gauß-Weierstraß kernel on R , f ∈ C0 (Rn ), and w(x, t) = (f ∗ ϕt )(x). Show that w(x, t)
solves the Cauchy initial value problem for the heat equation on Rn , that is:
ϕt (x) =
(a) w extends to a continuous function w : Rn × R+ → C. (Hint: you may use the multidimensional result of Exercise 5 without proof.)
(b) On Rn × (0, ∞),
n
∂w ∑ ∂ 2 w
=
,
∂t
∂x2i
i=1
(c) w(x, 0) = f (x).
7. Optional problem. Let
2
t
σn (t2 + |x|2 ) n+1
2
n
n
be the Poisson kernel on R , f ∈ C0 (R ), and u(x, t) = (f ∗ ϕt )(x). Show that u(x, t) solves the
Dirichlet boundary problem for the Laplace’s equation for a half-space, that is
P (x, t) = ϕt (x) =
(a) u extends to a continuous function u : Rn × R+ → C. (Hint: you may use the multidimensional result of Exercise 5 without proof.)
(b) On Rn × (0, ∞),
n
∑
∂ 2u ∂ 2u
△u =
+ 2 = 0,
2
∂x
∂t
i
i=1
(c) u(x, 0) = f (x).
Quiz 10
∫
∫
(a) We say that µt → µ weakly if for any g ∈ C0 , g dµt → g dµ (in light of the results
from Chapter 7, this should really be called weak∗ convergence). Show that if µt → µ
weakly, then ν ∗ µt → ν ∗ µ weakly. (Hint: you might want to use the measure ν̌ defined by
ν̌(E) = ν(−E).)
(b) Let {ϕt : t > 0} be an approximate identity. Show that ϕt dx → δ0 weakly as t → 0.
(c) Show that (ϕt ∗ µ) dx → µ weakly (here we use the notation from Exercise 3).
(d) Let µ be a finite measure on R, and
∫
1
Gµ (z) =
dµ(u)
R z −u
its Stieltjes transform (also called Borel of Cauchy transform). It is not hard to show (you
may want to check this, but are not required to do so) that Gµ : C+ → C− , so that Gµ maps
the upper half-plane to the lower half-plane. Prove the Stieltjes inversion formula: µ can be
recovered from its Stieltjes transform as the weak limit
1
dµ(x) = − lim ℑGµ (x + iy) dx.
π y↓0
Hint: write out the last expression explicitly in terms of x and y.