326 CHAPTER 11. REGULAR MEASURES

Definition 11.4.5 Let E be a measurable set. Then x ∈ E is called a point of den-sity ifx /∈Z and limr→0

µ(B(x,r)∩E)µ(B(x,r)) = 1. Recall Z is the set of pointsxwhere µ (B(x,r))= 0

for some r > 0.

Proposition 11.4.6 Let E be a measurable set. Then µ a.e. x ∈ E is a point of density.

Proof: This follows from letting f (x) = XE (x) in Theorem 11.4.4. ■From Theorem 11.4.2, µ ([M f > λ ])≤ Np

λ∥ f∥L1 .

M f ≡ supr≤1

1µ (B(x,r))

∫B(x,r)

| f |dµ ≤ supr≤1

1µ (B(x,r))

∫B(x,r)

(| f |X[| f |> λ

2 ]+

λ

2

)dµ

= M(

f X[| f |> λ2 ]

)+

λ

2

Therefore, [M f > λ ]⊆[M(

f X[| f |> λ2 ]

)> λ/2

]so

[M f > 2λ ]⊆[M(

f X[| f |>λ ]

)> λ

]≤

Np

λ

∫[| f |>λ ]

| f |dµ.

This shows the following modified weak estimate.

Corollary 11.4.7 Let f be in L1 (Rp,µ) . Then µ ([M f > 2λ ])≤ Npλ

∫[| f |>λ ] | f |dµ .

11.5 Strong Estimates for Maximal FunctionHere p > 1, not the dimension. Let ∥ f∥p

Lp(Rn)≡∫Rn | f |p dµ. Let λ

1/p ≡ 2η so λ = 2pη p

and dλ = 2p pη p−1dη . Then use Corollary 11.4.7 so

∫|M f |p dµ =

∫∞

0µ

([M f > λ

1/p])

dλ =∫

∞

0µ ([M f > 2η ])2p pη

p−1dη

≤∫

∞

0

Nη

∫[| f |>η ]

| f |2p pηp−1dµdη = N2p p

∫ ∫ | f |0| f |η p−2dηdµ =Cp

∫| f |p dµ

Of course this is all assuming that M f is measurable. This is most easily shown ifµ = mn Lebesgue measure and this is the case of most interest to me. Consider x→∫

B(x,r) | f (y)|dm ≡ fr (x). This is continuous assuming f ∈ L1loc thanks to continuity of

translation of Lebesgue measure. Thus x→ 1B(x,r)

∫B(x,r) | f (y)|dm is continuous. Now it

follows that M f (x) = sup0<r<1 fr (x) = sup0<r<1,r∈Q fr (x) is Borel measurable. There-fore, we can state the following corollary.

Corollary 11.5.1 Let ∥ f∥Lp <∞ where µ =mn inRn. Then ∥M f∥Lp ≤Cp ∥ f∥Lp whereC depends only on p> 1 and the dimension. This is called a strong estimate for the maximalfunction as opposed to the one from Theorem 11.4.2 for p = 1 which is called a weakestimate.