Kenneth Kuttler

1134 CHAPTER 33. FOURIER ANALYSIS IN Rn

=C (n,L) ∑|α|≤L

2−2k|α|∫

∣∣Dα (z jγk (z))∣∣2 dz (33.3.33)

where Sk is given in 33.3.28. Now∣∣Dα (z jγk (z))∣∣ = 2−k

∣∣∣Dα

(ρ (z)z j2k

(2kz))∣∣∣

= 2−k∣∣∣Dα

(ρ (z)ψ j

(2kz))∣∣∣

where ψ j (z)≡ z jφ (z). By Lemma 33.3.4, this is dominated by

2−kC (α,n,φ , j,C0) |z|−|α| .

Therefore, 33.3.33 is dominated by

C (L,n,φ , j,C0) ∑|α|≤L

2−2k|α|∫

2−2k |z|−2|α| dz

≤ C (L,n,φ , j,C0) ∑|α|≤L

2−2k|α|2−2k(

2−2−k)(−2|α|)(

22−k)n

≤ C (L,n,φ , j,C0) ∑|α|≤L

2−kn−2k

≤C (L,n,φ , j,C0)2−kn2−2k.

It follows that 33.3.31 is no larger than

C (L,n,φ ,C0) t2kn/22−kn/22−k =C (L,n,φ ,C0) t2−k. (33.3.34)

It follows from 33.3.34 and 33.3.30 that if |y| ≤ t,∫|x|≥2t

∣∣F−1γk (x−y)−F−1

γk (x)∣∣dx≤

C (L,n,φ ,C0)min(

t2−k,(

2−kt)n/2−L

With this inequality, the next lemma which is the desired result can be obtained.

Lemma 33.3.6 There exists a constant depending only on the indicated objects, C1 =C (L,n,φ ,C0) such that when |y| ≤ t,∫

|x|≥2t

∣∣F−1ρ (x−y)−F−1

ρ (x)∣∣dx≤C1

∫|x|≥2t

∣∣F−1ρm (x−y)−F−1

ρm (x)∣∣dx≤C1. (33.3.35)

1134 CHAPTER 33. FOURIER ANALYSIS IN R"=C(n,L) ¥ gael | |D% (zj7% (2) | dz (33.3.33)k|a|<Lwhere S; is given in 33.3.28. Now>" (pts (2%))0 (ptows(2%))|where y ;(z) =z;@ (z). By Lemma 33.3.4, this is dominated by|D* (z)% (2))|2-*C(a,n,@, j,Co) |z| I.Therefore, 33.3.33 is dominated byC(L,n,o,j,Co) Ye gael | 272k |g|-2lel gekla] <L< C(L,n,6,),Co) Y 2-72 Qe (22-*)"|ayeL< C(L,n,, j,Co) y? 2k 2k|o|<L<C(L,n,, j,Cy) 22>It follows that 33.3.31 is no larger thanC(L,n, Co) t2k"/22-*"/?2-* = C(L,n,@,Co) 12. (33.3.34)It follows from 33.3.34 and 33.3.30 that if |y| <r,[ |Poine-9)-F nde|x|>2r/2-LC(L,n,@,Co) min (2 (2%) )With this inequality, the next lemma which is the desired result can be obtained.Lemma 33.3.6 There exists a constant depending only on the indicated objects, C) =C(L,n,@,Co) such that when |y| <t,lon |F'p(x-y)—F |p (x)|ax <QI. [FP (XY) —F'P yy (®) [dx < Ci. (33.3.35)