33.4. SINGULAR INTEGRALS 1141

Look at the picture. Similarly, ∫|x−z|≤3π|y|−1

Kε (x− z)e−ix·ydx

=∫

|x|≤3π|y|−1Kε (x− z)e−ix·ydx

−∫

|x−z|>3π|y|−1,|x|<3π|y|−1Kε (x− z)e−ix·ydx+∫

|x−z|<3π|y|−1,|x|>3π|y|−1Kε (x− z)e−ix·ydx.

(33.4.51)

The last integral in 33.4.50 is taken over a set that is contained in

B(0,R+ |z|)\B(0,R)

illustrated in the following picture as the region between the small ball centered at 0 andthe big ball which surrounds the two small balls

0 z

and so this integral is dominated by

B(

1(R−|z|)n

)α (n)((R+ |z|)n−Rn),

an expression which converges to 0 as R→ ∞. Similarly, the second integral on the rightin 33.4.50 converges to zero as R→ ∞. Now consider the last two integrals in 33.4.51.Letting 3π |y|−1 play the role of R and using |z| = π/ |y|, these are each dominated by anexpression of the form

B

 1(3π |y|−1−|z|

)n

α (n)((

3π |y|−1 + |z|)n−(

3π |y|−1)n)

= B

 1(3π |y|−1−π |y|−1

)n

α (n) ·

((3π |y|−1 +π |y|−1

)n−(

3π |y|−1)n)