1136 CHAPTER 33. FOURIER ANALYSIS IN Rn

Proof: Since ρm satisfies 33.3.35, and is obviously in L2 (Rn)∩ L∞ (Rn), Theorem33.3.3 implies there exists a constant Ap depending only on p,n, ||ρm||∞, and C1 such thatfor all ψ ∈ G and p ∈ (1,∞), ∣∣∣∣F−1

ρm ∗ψ∣∣∣∣

p ≤ Ap ||ψ||p.

Now ||ρm||∞ ≤ ||ρ||∞ because

|ρm (x)| ≤ |ρ (x)|m

∑k=−m

φ

(2kx)≤ |ρ (x)|. (33.3.38)

Therefore, since C1 =C1 (L,n,φ ,C0) and C0 ≥ ||ρ||∞,∣∣∣∣F−1ρm ∗ψ

∣∣∣∣p ≤ Ap (L,n,φ ,C0, p) ||ψ||p .

In particular, Ap does not depend on m. Now, by 33.3.38, the observation that ρ ∈ L∞ (Rn),limm→∞ ρm (y) = ρ (y) and the dominated convergence theorem, it follows that for θ ∈ G .∣∣(F−1

ρ ∗ψ)(θ)∣∣≡ ∣∣∣∣(2π)n/2

∫ρ (x)Fψ (x)F−1

θ (x)dx∣∣∣∣

= limm→∞

∣∣(F−1ρm ∗ψ

)(θ)∣∣≤ lim

m→∞sup∣∣∣∣F−1

ρm ∗ψ∣∣∣∣

p ||θ ||p′

≤ Ap (L,n,φ ,C0, p) ||ψ||p ||θ ||p′ .

Hence F−1ρ ∗ψ ∈ Lp (Rn) and∣∣∣∣F−1ρ ∗ψ

∣∣∣∣p ≤ Ap ||ψ||p. This proves the theorem.

33.4 Singular IntegralsIf K ∈ L1 (Rn) then when p > 1,

||K ∗ f ||p ≤ || f ||p .

It turns out that some meaning can be assigned to K ∗ f for some functions K which arenot in L1. This involves assuming a certain form for K and exploiting cancellation. Theresulting theory of singular integrals is very useful. To illustrate, an application will begiven to the Helmholtz decomposition of vector fields in the next section. Like Mihlin’stheorem, the theory presented here rests on Theorem 33.3.3, restated here for convenience.

Theorem 33.4.1 Let ρ ∈ L2 (Rn)∩L∞ (Rn) and suppose∫|x|≥2|y|

∣∣F−1ρ (x−y)−F−1

ρ (x)∣∣dx≤C1.

Then for each p ∈ (1,∞), there exists a constant, Ap, depending only on

p,n, ||ρ||∞,

and C1 such that for all φ ∈ G , ∣∣∣∣F−1ρ ∗φ

∣∣∣∣p ≤ Ap ||φ ||p .