1124 CHAPTER 33. FOURIER ANALYSIS IN Rn

verify which imply condition 33.3.8. I think the approach used here is due to Hormander[69] and is found in Berg and Lofstrom [16]. For many more references and generaliza-tions, you might look in Triebel [124]. A different proof based on singular integrals isin Stein [122]. Functions, ρ which yield an inequality of the sort in 33.3.7 are called Lp

multipliers.

Lemma 33.3.1 Suppose ρ ∈ L∞ (Rn)∩L2 (Rn) and suppose also there exists a constant C1such that ∫

|x|≥2|y|

∣∣F−1ρ (x−y)−F−1

ρ (x)∣∣dx≤C1. (33.3.8)

Then there exists a constant A depending only on C1, ||ρ||∞, and n such that

m([

x :∣∣F−1

ρ∗φ (x)∣∣> α

])≤ A

α||φ ||1

for all φ ∈ G .

Proof: Let φ ∈ G and use the Calderon decomposition to write Rn = E ∪Ω where Ω isa union of cubes, {Qi} with disjoint interiors such that

αm(Qi)≤∫

Qi

|φ (x)|dx≤ 2nαm(Qi) , |φ (x)| ≤ α a.e. on E. (33.3.9)

The proof is accomplished by writing φ as the sum of a good function and a bad func-tion and establishing a similar weak inequality for these two functions separately. Thenthis information is used to obtain the desired conclusion.

g(x) ={

φ (x) if x ∈ E1

m(Qi)

∫Qi

φ (x)dx if x ∈ Qi ⊆Ω, g(x)+b(x) = φ (x). (33.3.10)

Thus ∫Qi

b(x)dx =∫

Qi

(φ (x)−g(x))dx =∫

Qi

φ (x)dx−∫

Qi

φ (x)dx = 0, (33.3.11)

b(x) = 0 if x /∈Ω. (33.3.12)

Claim:||g||22 ≤ α (1+4n) ||φ ||1 , ||g||1 ≤ ||φ ||1. (33.3.13)

Proof of claim:||g||22 = ||g||

2L2(E)+ ||g||

2L2(Ω).

Thus

||g||2L2(Ω) = ∑i

∫Qi

|g(x)|2 dx

≤ ∑i

∫Qi

(1

m(Qi)

∫Qi

|φ (y)|dy)2

dx

≤ ∑i

∫Qi

(2nα)2 dx≤ 4n

α2∑

im(Qi)

≤ 4nα

2 1α

∑i

∫Qi

|φ (x)|dx≤ 4nα ||φ ||1.