952 CHAPTER 26. INTEGRALS AND DERIVATIVES

This is a picture of two balls of radius r in Rp, U and V having centers at x and yrespectively, which intersect in the set W. The center of U is on the boundary of V and thecenter of V is on the boundary of U as shown in the picture. There exists a constant, C,independent of r depending only on p such that

m(W )

m(U)=

m(W )

m(V )=

1C.

You could compute this constant if you desired but it is not important here.Then

|u(x)−u(y)| =1

m(W )

∫W|u(x)−u(y)|dz

≤ 1m(W )

∫W|u(x)−u(z)|dz+

1m(W )

∫W|u(z)−u(y)|dz

=C

m(U)

[∫W|u(x)−u(z)|dz+

∫W|u(z)−u(y)|dz

]≤ C

m(U)

[∫U|u(x)−u(z)|dz+

∫V|u(y)−u(z)|dz

]Now consider these two terms. Let q > p

Using spherical coordinates and letting U0 denote the ball of the same radius as U butwith center at 0,

1m(U)

∫U|u(x)−u(z)|dz

=1

m(U0)

∫U0

|u(x)−u(z+x)|dz

Now using spherical coordinates, Section 13.9, and letting C denote a generic constantwhich depends on p,

=1

m(U0)

∫ r

0ρ

p−1∫

Sp−1|u(x)−u(ρw+x)|dσ (w)dρ

≤ 1m(U0)

∫ r

0ρ

p−1∫

Sp−1

∫ρ

0|Dwu(x+ tw)|dtdσ (w)dρ

=1

m(U0)

∫ r

0ρ

p−1∫

Sp−1

∫ρ

0|∇u(x+ tw) ·w|dtdσ (w)dρ

≤ 1m(U0)

∫ r

0ρ

p−1∫

Sp−1

∫ r

0|∇u(x+ tw) ·w|dtdσ (w)dρ

=1

m(U0)

∫Sp−1

∫ r

0|∇u(x+ tw) ·w|

∫ r

0ρ

p−1dρdtdσ (w)

=C∫ r

0

∫Sp−1|∇u(x+ tw)|dσ (w)dt =C

∫ r

0

∫Sp−1

|∇u(x+ tw)|t p−1 t p−1dσ (w)dt