1142 CHAPTER 33. FOURIER ANALYSIS IN Rn

= α (n)B|y|n

(2π)n1|y|n

((4π)n− (3π)n) =C (n)B.

Returning to 33.4.49, the terms involving x−y have now been estimated. Thus, col-lecting the terms which have not yet been estimated along with those that have,

B =

∣∣∣∣∣∣∣∫

3π|y|−1<|x|≤R

Kε (x)e−ix·ydx

∣∣∣∣∣∣∣≤ 1

2

∣∣∣∣∣∣∣∫|x|<R

Kε (x)e−ix·ydx−∫|x|<R

Kε (x− z)e−ix·ydx

+∫

|x|<3π|y|−1

Kε (x− z)e−ix·ydx−∫

|x|<3π|y|−1

Kε (x)e−ix·ydx

∣∣∣∣∣∣∣+C (n)B+g(R)

where g(R)→ 0 as R→ ∞. Using |z|= π/ |y| again,

B≤12

∫3|z|<|x|<R

|Kε (x)−Kε (x− z)|dx+C (n)B+g(R).

But the integral in the above is dominated by C (n)B by 33.4.44 which was establishedearlier. Therefore, from 33.4.47,

|FKεR| ≤C (n)B+g(R)

where g(R)→ 0.Now KεR→ Kε in L2 (Rn) because

||KεR−Kε ||L2(Rn) ≤ B∫|x|>R

1

|x|2n dx

= B∫

Sn−1

∫∞

R

1ρn+1 dρdσ ,

which converges to 0 as R→ ∞ and so FKεR→ FKε in L2 (Rn) by Plancherel’s theorem.Therefore, by taking a subsequence, still denoted by R, FKεR (y)→ FKε (y) a.e. whichshows

|FKε (y)| ≤C (n)B a.e.

This proves the lemma.

Corollary 33.4.4 Suppose 33.4.40 - 33.4.42 hold. Then if g ∈ C1c (Rn), Kε ∗ g converges

uniformly and in Lp (Rn) as ε → 0.