Measure, Integration & Real Analysis, 2021a

358 Chapter 11 Fourier Analysis
11.38 L p -norm of a convolution
Suppose 1 ≤ p ≤ ∞, f ∈ L 1 (∂D), and g ∈ L p (∂D). Then
‖ f ∗ g‖ p ≤‖f ‖ 1 ‖g‖ p .
Proof
We use the following result to estimate the norm in L p (∂D):
11.39
If F : ∂D → C is measurable and 1 ≤ p ≤ ∞, then
{∫
}
‖F‖ p = sup |Fh| dσ : h ∈ L p′ (∂D) and ‖h‖ p ′ = 1 .
∂D
Hölder’s inequality (7.9) shows that the left side of the equation above is greater
than or equal to the right side. The inequality in the other direction almost follows
from 7.12, but7.12 would require the hypothesis that F ∈ L p (∂D) (and we want the
equation above to hold even if ‖F‖ p = ∞). To get around this problem, apply 7.12
to truncations of F and use the Monotone Convergence Theorem (3.11); the details
of verifying 11.39 are left to the reader.
Suppose h ∈ L p′ (∂D) and ‖h‖ p ′ = 1. Then
∫
∂D
∫
|( f ∗ g)(z)h(z)| dσ(z) ≤
∫
=
∫
≤
∂D
∂D
∂D
(∫
∂D
∫
| f (w)|
)
| f (w)g(zw)| dσ(w)|h(z)| dσ(z)
∂D
|g(zw)h(z)| dσ(z) dσ(w)
| f (w)|‖g‖ p ‖h‖ p ′ dσ(w)
11.40
= ‖ f ‖ 1 ‖g‖ p ,
where the second line above follows from Tonelli’s Theorem (5.28) and the third line
follows from Hölder’s inequality (7.9) and 11.17. Now11.39 (with F = f ∗ g) and
11.40 imply that ‖ f ∗ g‖ p ≤‖f ‖ 1 ‖g‖ p .
Order does not matter in convolutions, as we now prove.
11.41 convolution is commutative
Suppose f , g ∈ L 1 (∂D). Then f ∗ g = g ∗ f .
Proof
Suppose z ∈ ∂D is such that ( f ∗ g)(z) is defined. Then
∫
( f ∗ g)(z) =
∂D
∫
f (w)g(zw) dσ(w) =
∂D
f (zζ)g(ζ) dσ(ζ) =(g ∗ f )(z),
where the second equality follows from making the substitution ζ = zw (which
implies that w = zζ); the invariance of the integral under this substitution is explained
in connection with 11.17.
Measure, Integration & Real Analysis, by Sheldon Axler