33.4. SINGULAR INTEGRALS 1145

where 33.4.56 was used to write Ω

(z|z|

)= Ω(z). The first group of terms in 33.4.61 is

dominated by

|x−y|−n Lip(Ω)

∣∣∣∣ x−y|x−y|

− x|x|

∣∣∣∣and an estimate is required for |x|> 2 |y|. Since |x|> 2 |y|,

|x−y|−n ≤ (|x|− |y|)−n ≤ 2n

|x|n.

Also ∣∣∣∣ x−y|x−y|

− x|x|

∣∣∣∣= ∣∣∣∣ (x−y) |x|−x |x−y||x| |x−y|

∣∣∣∣≤∣∣∣∣ (x−y) |x|−x |x−y|

|x|(|x|− |y|)

∣∣∣∣≤ ∣∣∣∣ (x−y) |x|−x |x−y||x|(|x|/2)

∣∣∣∣=

2

|x|2|x |x|−y |x|−x |x−y||= 2

|x|2|x(|x|− |x−y|)−y |x||

≤ 2

|x|2|x| ||x|− |x−y||+ |y| |x| ≤ 2

|x|2(|x| |x−(x−y)|+ |y| |x|)

≤ 4

|x|2|x| |y|= 4

|y||x|

.

Therefore, ∫|x|>2|y|

|x−y|−n∣∣∣∣Ω( x−y

|x−y|

)−Ω

(x|x|

)∣∣∣∣dx

≤ 4(2n)∫|x|>2|y|

1|x|n|y||x|

dxLip(Ω)

=C (n,LipΩ)∫|x|>2|y|

|y||x|n+1 dx

=C (n,LipΩ)∫|u|>2

1

|u|n+1 du. (33.4.62)

It remains to consider the second group of terms in 33.4.61 when |x|> 2 |y|.∣∣∣∣ 1|x−y|n

− 1|x|n

∣∣∣∣= ∣∣∣∣ |x|n−|x−y|n

|x−y|n |x|n∣∣∣∣

≤ 2n

|x|2n ||x|n−|x−y|n|

≤ 2n

|x|2n |y|[|x|n−1 + |x|n−2 |x−y|+

· · ·+ |x| |x−y|n−2 + |x−y|n−1]