33.3. MIHLIN’S THEOREM 1135

Proof: F−1ρ = limm→∞ F−1ρm in L2 (Rn). Let mk → ∞ be such that convergence ispointwise a.e. Then if |y| ≤ t, Fatou’s lemma implies∫

|x|≥2t

∣∣F−1ρ (x−y)−F−1

ρ (x)∣∣dx≤

lim infl→∞

∫|x|≥2t

∣∣∣F−1ρml

(x−y)−F−1ρml

(x)∣∣∣dx

≤ lim infl→∞

ml

∑k=−ml

∫|x|≥2t

∣∣F−1γk (x−y)−F−1

γk (x)∣∣dx

≤C (L,n,φ ,C0)∞

∑k=−∞

min(

t2−k,(

2−kt)n/2−L

). (33.3.36)

Now consider the sum in 33.3.36,∞

∑k=−∞

min(

t2−k,(

2−kt)n/2−L

). (33.3.37)

t2 j = min(

t2 j,(2 jt)n/2−L

)exactly when t2 j ≤ 1. This occurs if and only if

j ≤− ln(t)/ ln(2)

Therefore 33.3.37 is no larger than

∑j≤− ln(t)/ ln(2)

2 jt + ∑j≥− ln(t)/ ln(2)

(2 jt)n/2−L

.

Letting a = L−n/2, this equals

t ∑k≥ln(t)/ ln(2)

2−k + t−α∑

j≥− ln(t)/ ln(2)

(2−a) j

≤ 2t(

12

)ln(t)/ ln(2)

+ t−a(

12a

)− ln(t)/ ln(2)

= 2t(

12

)log2(t)

+ t−a(

12a

)− log2(t)

= 2+1 = 3.

Similarly, 33.3.35 holds. This proves the lemma.Now it is possible to prove Mihlin’s theorem.

Theorem 33.3.7 (Mihlin’s theorem) Suppose ρ satisfies

C0 ≥ sup{|x||α| |Dαρ (x)| : |α| ≤ L, x ∈ Rn \{0}},

where L is an integer greater than n/2 and ρ ∈CL (Rn \{0}). Then for every p > 1, thereexists a constant Ap depending only on p, C0, φ , n, and L, such that for all ψ ∈ G ,∣∣∣∣F−1

ρ ∗ψ∣∣∣∣

p ≤ Ap ||ψ||p.