33.4. SINGULAR INTEGRALS 1139

= A+B. (33.4.47)

Consider A. By 33.4.41 ∫ε<|x|<3π|y|−1

Kε (x)dx = 0

and so

A =

∣∣∣∣∣∣∣∫

ε<|x|<3π|y|−1

Kε (x)(e−ix·y−1

)dx

∣∣∣∣∣∣∣Now ∣∣e−ix·y−1

∣∣= |2−2cos(x ·y)|1/2 ≤ 2 |x ·y| ≤ 2 |x| |y|

so, using polar coordinates, this expression is no larger than

2B∫

ε<|x|<3π|y|−1

|x|−n |x| |y|dx≤C (n)B |y|∫ 3π/|y|

ε

dρ ≤ BC (n).

Next, consider B. This estimate is based on the trick which follows. Let

z≡ yπ/ |y|2

so that|z|= π/ |y| , z ·y =π.

Then ∫3π|y|−1<|x|≤R

Kε (x)e−ix·ydx = 12

∫3π|y|−1<|x|≤R

Kε (x)e−ix·ydx

− 12

∫3π|y|−1<|x|≤R

Kε (x)e−i(x+z)·ydx.(33.4.48)

Here is why. Note in the second of these integrals,

−12

∫3π|y|−1<|x|≤R

Kε (x)e−i(x+z)·ydx

= −12

∫3π|y|−1<|x|≤R

Kε (x)e−ix·ye−iz·ydx

= −12

∫3π|y|−1<|x|≤R

Kε (x)e−ix·ye−iπ dx

=12

∫3π|y|−1<|x|≤R

Kε (x)e−ix·ydx.