1146 CHAPTER 33. FOURIER ANALYSIS IN Rn

≤ 2n |y|C (n) |x|n−1

|x|2n =C (n)2n |y||x|n+1 .

Thus ∫|x|>2|y|

∣∣∣∣Ω(x)(

1|x−y|n

− 1|x|n

)∣∣∣∣dx

≤ C (n)∫|x|>2|y|

|y||x|n+1 dx

≤C (n)∫|u|>2

1

|u|n+1 du. (33.4.63)

From 33.4.62 and 33.4.63,∫|x|>2|y|

|K (x−y)−K (x)|dx≤C (n,LipΩ).

This proves the theorem.

33.5 Helmholtz DecompositionsIt turns out that every vector field which has its components in Lp can be written as a sumof a gradient and a vector field which has zero divergence. This is a very remarkable result,especially when applied to vector fields which are only in Lp. Recall that for u a functionof n variables, ∆u = ∑

ni=1

∂ 2u∂x2

i.

Definition 33.5.1 Define

Φ(y)≡

{− 1

a1ln |y| , if n = 2,

1(n−2)an−1

|y|2−n , if n > 2.

where ak denotes the area of the unit sphere, Sk.

Then it is routine to verify ∆Φ = 0 away from 0. In fact, if n > 2,

Φ,ii (y) =Cn

[1|y|n−n

y2i

|y|n+2

], Φ,i j (y) =Cn

yiy j

|y|n+2 , (33.5.64)

while if n = 2,

Φ,22 (y) =C2y2

1− y22(

y21 + y2

2

)2 , Φ,11 (y) =C2y2

2− y21(

y21 + y2

2

)2 ,

Φ,i j (y) =C2y1y2(

y21 + y2

2

)2 .

Also,∇Φ(y) =

−yan−1 |y|n

. (33.5.65)

In the above the subscripts following a comma denote partial derivatives.