26.6. RADEMACHER’S THEOREM 957
Define the average of a function over a set, E ⊆ Rn as follows.∫−
Ef dx≡ 1
m(E)
∫E
f dx.
Then
|u(x)−u(y)| =∫−
W|u(x)−u(y)|dz
≤∫−
W|u(x)−u(z)|dz+
∫−
W|u(z)−u(y)|dz
=C
m(U)
[∫W|u(x)−u(z)|dz+
∫W|u(z)−u(y)|dz
]≤ C
[∫−
U|u(x)−u(z)|dz+
∫−
V|u(y)−u(z)|dz
]Now consider these two terms. Using spherical coordinates and letting U0 denote the ballof the same radius as U but with center at 0,∫
−U|u(x)−u(z)|dz
=1
m(U0)
∫U0
|u(x)−u(z+x)|dz
=1
m(U0)
∫ r
0ρ
n−1∫
Sn−1|u(x)−u(ρw+x)|dσ (w)dρ
≤ 1m(U0)
∫ r
0ρ
n−1∫
Sn−1
∫ρ
0|∇u(x+ tw) ·w|dtdσdρ
≤ 1m(U0)
∫ r
0ρ
n−1∫
Sn−1
∫ρ
0|∇u(x+ tw)|dtdσdρ
≤ C1r
∫ r
0
∫Sn−1
∫ r
0|∇u(x+ tw)|dtdσdρ
= C1r
∫ r
0
∫Sn−1
∫ r
0
|∇u(x+ tw)|tn−1 tn−1dtdσdρ
= C∫
Sn−1
∫ r
0
|∇u(x+ tw)|tn−1 tn−1dtdσ
= C∫
U0
|∇u(x+ z)||z|n−1 dz
≤ C(∫
U0
|∇u(x+ z)|p dz)1/p(∫
U|z|p
′−np′)1/p′
26.6. RADEMACHER’S THEOREM 957Define the average of a function over a set, E C R” as follows.f fac= wb IeThenlu(x)—u(y)] =f u(x) —u(y)|de< f, u(x)—n( ‘ovaeef lu (2) —u(y)|dz~ =F gaa [whe<C if, ju (x) <u(a)|dc+f \u(y) —u(a)|deNow consider these two terms. Using spherical coordinates and letting Up denote the ballof the same radius as U but with center at 0,i \u(x) —u(z)|dzJ, wx) etx) |dzUoat_ aun hh ? n— fe ) —u(pw+x)|do(w)dp—_ nl V -w|dtdodon; p an |Vu(x--tw)-w|dtdodpIA1 rd p< n Vv7 m (Uo) [ p Lf | u(x+tw)|dtdodp1 r rcf | | \Vu (x +tw)|dtdodpsnVu(x-+t= Cc “fff "Wu tI nt HagdpgnIAnl_ c/ [pee "ladegn-l pn-lc/ [Vu(x+a)lUojz|""\/p , \1/Pc(| Vu(x-t2))?ds) (/ i")Up U/IA