33.3. MIHLIN’S THEOREM 1125

||g||2L2(E) =∫

E|φ (x)|2 dx≤ α

∫E|φ (x)|dx = α ||φ ||1.

Now consider the second of the inequalities in 33.3.13.

||g||1 =∫

E|g(x)|dx+

∫Ω

|g(x)|dx

=∫

E|φ (x)|dx+∑

i

∫Qi

|g|dx

≤∫

E|φ (x)|dx+∑

i

∫Qi

1m(Qi)

∫Qi

|φ (x)|dm(x)dm

=∫

E|φ (x)|dx+∑

i

∫Qi

|φ (x)|dm(x) = ||φ ||1

This proves the claim. From the claim, it follows that b ∈ L2 (Rn)∩L1 (Rn) .Because of 33.3.13, g ∈ L1 (Rn) and so F−1ρ ∗ g ∈ L2 (Rn). (Since ρ ∈ L2, it follows

F−1ρ ∈ L2 and so this convolution is indeed in L2.) By Plancherel’s theorem,∣∣∣∣F−1ρ ∗g

∣∣∣∣2 =

∣∣∣∣F (F−1ρ ∗g

)∣∣∣∣2.

By Corollary 32.3.19 on Page 1110, the expression on the right equals

(2π)n/2 ||ρFg||2

and so ∣∣∣∣F−1ρ ∗g

∣∣∣∣2 = (2π)n/2 ||ρFg||2 ≤Cn ||ρ||∞ ||g||2.

From this and 33.3.13m([∣∣F−1

ρ ∗g∣∣≥ α/2

])≤ Cn ||ρ||2∞

α2 α (1+4n) ||φ ||1 =Cnα−1 ||φ ||1. (33.3.14)

This is what is wanted so far as g is concerned. Next it is required to estimate

m([∣∣F−1

ρ ∗b∣∣≥ α/2

]).

If Q is one of the cubes whose union is Ω, let Q∗ be the cube with the same center as Qbut whose sides are 2

√n times as long.

Qi

Q∗i

yi