Kenneth Kuttler

1144 CHAPTER 33. FOURIER ANALYSIS IN Rn

∣∣∣∣Kε ∗ f −Kη ∗ f∣∣∣∣

p ≤ ||Kε ∗ ( f −g)||p +∣∣∣∣Kε ∗g−Kη ∗g

∣∣∣∣p

+∣∣∣∣Kη ∗ ( f −g)

∣∣∣∣p

≤ 2A(p,n,B) || f −g||p +∣∣∣∣Kε ∗g−Kη ∗g

∣∣∣∣p .

Choose g such that 2A(p,n,B) || f −g||p ≤ δ/2. Then if ε,η are small enough, Corollary33.4.4 implies the last term is also less than δ/2. Thus, limε→0 Kε ∗ f exists in Lp (Rn).Let T f be the element of Lp (Rn) to which it converges. Then 33.4.54 follows and T isobviously linear because

T (a f +bg) = limε→0

Kε ∗ (a f +bg) = limε→0

(aKε ∗ f +bKε ∗g)

= aT f +bT g.

This proves the theorem.When do conditions 33.4.40-33.4.42 hold? It turns out this happens for K given by the

following.

K (x)≡ Ω(x)|x|n

, (33.4.55)

whereΩ(λx) = Ω(x) for all λ > 0, (33.4.56)

Ω is Lipschitz on Sn−1,∫Sn−1

Ω(x)dσ = 0. (33.4.57)

Theorem 33.4.6 For K given by 33.4.55 - 33.4.57, it follows there exists a constant B suchthat

|K (x)| ≤ B |x|−n, (33.4.58)∫a<|x|<b

K (x)dx = 0, (33.4.59)∫|x|>2|y|

|K (x−y)−K (x)|dx≤ B. (33.4.60)

Consequently, the conclusions of Theorem 33.4.5 hold also.

Proof: 33.4.58 is obvious. To verify 33.4.59,∫a<|x|<b

K (x)dx =∫ b

∫Sn−1

Ω(ρw)

ρn ρn−1dσdρ

=∫ b

1ρ

∫Sn−1

Ω(w)dσdρ = 0.

It remains to show 33.4.60.

K (x−y)−K (x) = |x−y|−n(

Ω

(x−y|x−y|

)−Ω

(x|x|

))+Ω(x)

|x−y|n− 1|x|n

)(33.4.61)

1144 CHAPTER 33. FOURIER ANALYSIS IN R"||Kexf—Ky*f\|, < ||Ke*(f—g)|l,+||Ke*8—Kn *8l|,+||Kn*(f—8)||,< 2A (p,n,B)||f —g||,+||Ke*8 — Kn *8\|,,-Choose g such that 2A (p,n,B) ||f —g||, < 6/2. Then if €,7 are small enough, Corollary33.4.4 implies the last term is also less than 6/2. Thus, lime_,9 Ke * f exists in L? (IR").Let Tf be the element of L? (R") to which it converges. Then 33.4.54 follows and T isobviously linear becauseT(af+bg) = lim Ke * (af +bg) = lim (aKe * f + bKe *8)= aTf+bTeg.This proves the theorem.When do conditions 33.4.40-33.4.42 hold? It turns out this happens for K given by thefollowing.K(x)= xl” (33.4.55)whereQ (Ax) = Q(x) for all A > 0, (33.4.56)Q is Lipschitz on S’~,; Q(x)do =0. (33.4.57)snTheorem 33.4.6 For K given by 33.4.55 - 33.4.57, it follows there exists a constant B suchthat|K (x)| < Blx|", (33.4.58)| K (x)dx=0, (33.4.59)a<|x|<b| IK (x—y) —K(x)|dx <B. (33.4.60)x|>2ly|Consequently, the conclusions of Theorem 33.4.5 hold also.Proof: 33.4.58 is obvious. To verify 33.4.59,b| K(x)dx = | | 2 (pw) p” 'dodpa<|x|<b a Jsr-l— pp”by_ Q(w)dodp =0.Loh. (w)dodpIt remains to show 33.4.60.K(x—y)—K(x) = |x—y|” (o( ==) o(5))+(x) ( ! ! ) (33.4.61)x—yl" |x|"