33.3. MIHLIN’S THEOREM 1127

Thus, by 33.3.13, ∫E∗

∣∣F−1ρ ∗b(x)

∣∣dx ≤ C1 ||b||1≤ C1 [||φ ||1 + ||g||1]≤ C1 [||φ ||1 + ||φ ||1]≤ 2C1 ||φ ||1 .

Consequently,

m([∣∣F−1

ρ ∗b∣∣≥ α

2

]∩E∗

)≤ 4C1

α||φ ||1 .

From 33.3.10, 33.3.14, and 33.3.9,

m[∣∣F−1

ρ ∗φ∣∣> α

]≤ m

[∣∣F−1ρ ∗g

∣∣≥ α

2

]+m

[∣∣F−1ρ ∗b

∣∣≥ α

2

]≤ Cn

α||φ ||1 +m

([∣∣F−1ρ ∗b

∣∣≥ α

2

]∩E∗

)+m(Ω∗)

≤ Cn

α||φ ||1 +

4C1

α||φ ||1 +Cnm(Ω)≤ A

α||φ ||1

becausem(Ω)≤ α

−1 ||φ ||1

by 33.3.9. This proves the lemma.The next lemma extends this lemma by giving a weak (2,2) estimate and a (2,2) esti-

mate.

Lemma 33.3.2 Suppose ρ ∈ L∞ (Rn)∩L2 (Rn) and suppose also that there exists a con-stant C1 such that ∫

|x|>2|y|

∣∣F−1ρ (x−y)−F−1

ρ (x)∣∣dx≤C1. (33.3.20)

Then F−1ρ∗ maps L1 (Rn)+L2 (Rn) to measurable functions and there exists a constant Adepending only on C1,n, ||ρ||∞ such that

m([∣∣F−1

ρ ∗ f∣∣> α

])≤ A|| f ||1

αif f ∈ L1 (Rn), (33.3.21)

m([∣∣F−1

ρ ∗ f∣∣> α

])≤(

A|| f ||2

α

)2

if f ∈ L2 (Rn). (33.3.22)

Thus, F−1ρ∗ is weak type (1,1) and weak type (2,2). Also∣∣∣∣F−1ρ ∗ f

∣∣∣∣2 ≤ A || f ||2 if f ∈ L2 (Rn). (33.3.23)