1486 CHAPTER 45. TRACES OF SOBOLEV SPACES
Using the repeated index summation convention to save space, we obtain that in terms ofweak derivatives,
Rg,n(x′,xn
)=
∫Rn−1
[ψ (xn) f
(y′,ψ (xn)
)−∫
ψ(xn)
0(ψ f ),n
(y′, t)
dt]·[
φ ,k
(x′−y′
xn
)(yk− xk
xnn
)+φ
(x′−y′
xn
)(1−n)
xnn
]dy′
=∫Rn−1
[ψ (xn) f
(x′− xnz′,ψ (xn)
)−∫
ψ(xn)
0(ψ f ),n
(x′− xnz′, t
)dt]·[
φ ,k(z′)(yk− xk
xnn
)zk +φ
(z′) (1−n)
xnn
]xn
ndz′
and so ∣∣Rg,n(x′,xn
)∣∣ ≤ C (φ)
∣∣∣∣∫B(0,1)
[ψ (xn) f
(x′− xnz′,ψ (xn)
)−∫
ψ(xn)
0(ψ f ),n
(x′− xnz′, t
)dt]∣∣∣∣
≤ C (φ)
xn−1n
{∫B(0,xn)
∣∣ψ (xn) f(x′+y′,ψ (xn)
)∣∣dy′
+∫
B(0,xn)
∫ψ(xn)
0
∣∣∣(ψ f ),n(x′+y′, t
)∣∣∣dtdy′}
Therefore,(∫∞
0
∫Rn−1
∣∣Rg,n(x′,xn
)∣∣p dx′dxn
)1/p
≤
C (φ)
(∫∞
0
∫Rn−1
(1
xn−1n
∫B(0,xn)
∣∣ψ (xn) f(x′+y′,ψ (xn)
)∣∣dy′)p
dx′dxn
)1/p
+C (φ)
(∫∞
0
∫Rn−1
(1
xn−1n
∫B(0,xn)
∫ψ(xn)
0
∣∣∣(ψ f ),n(x′+y′, t
)∣∣∣dtdy′)p
dx′dxn
)1/p
(45.2.3)Consider the first term on the right. We change variables, letting y′ = z′xn. Then this termbecomes
C (φ)
(∫ 1
0
∫Rn−1
(∫B(0,1)
∣∣ψ (xn) f(x′+ xnz′,ψ (xn)
)∣∣dz′)p
dx′dxn
)1/p
≤C (φ)∫
B(0,1)
(∫ 1
0
∫Rn−1
∣∣ψ (xn) f(x′+ xnz′,ψ (xn)
)∣∣p dx′dxn
)1/p
dz′