252 CHAPTER 10. CHANGE OF VARIABLES
Notation 10.4.2 Often this is written differently. Note that from the spherical coordinateformulas, f
(h(⃗
φ ,θ ,ρ))
= f (ρω) where |ω| = 1. Letting Sp−1 denote the unit sphere,{ω ∈ Rp : |ω|= 1} , the inside integral in the above formula is sometimes written as∫
Sp−1f (ρω)dσ
where σ is a measure on Sp−1. See [27] for another description of this measure. It isn’t animportant issue here. Either 10.10 or the formula∫
∞
0ρ
p−1(∫
Sp−1f (ρω)dσ
)dρ
will be referred to as polar coordinates and is very useful in establishing estimates. Hereσ(Sp−1
)≡∫
A Φ
(⃗φ ,θ
)dmp−1.
Example 10.4.3 For what values of s is the integral∫
B(0,R)
(1+ |x|2
)sdy bounded inde-
pendent of R? Here B(0,R) is the ball, {x ∈ Rp : |x| ≤ R} .
I think you can see immediately that s must be negative but exactly how negative? Itturns out it depends on p and using polar coordinates, you can find just exactly what isneeded. From the polar coordinates formula above,∫
B(0,R)
(1+ |x|2
)sdy =
∫ R
0
∫Sp−1
(1+ρ
2)sρ
p−1dσdρ
= Cp
∫ R
0
(1+ρ
2)sρ
p−1dρ
Now the very hard problem has been reduced to considering an easy one variable prob-lem of finding when
∫ R0 ρ p−1
(1+ρ2
)s dρ is bounded independent of R. You need 2s+(p−1)<−1 so you need s <−p/2.
10.5 Approximation with Smooth FunctionsIt is very important to be able to approximate measurable and integrable functions withcontinuous functions having compact support. Recall Theorem 9.4.2. This implies thefollowing.
Theorem 10.5.1 Let f ≥ 0 be Fn measurable and let∫
f dmn < ∞. Then thereexists a sequence of continuous functions {hn} which are zero off a compact set such thatlimn→∞
∫| f −hn|p dmn = 0.
Definition 10.5.2 Let U be an open subset of Rn. C∞c (U) is the vector space of
all infinitely differentiable functions which equal zero for all x outside of some compact setcontained in U. Similarly, Cm
c (U) is the vector space of all functions which are m timescontinuously differentiable and whose support is a compact subset of U.
Corollary 10.5.3 Let U be a nonempty open set in Rn and let f ∈ Lp (U,mn) . Thenthere exists g ∈Cc (U) such that ∫
| f −g|p dmn < ε