Exercises
7.3 Nontangential Maximal Functions and Carleson Measures
7.3.1 Definition and Basic Properties of Carleson Measures
Definition 7.3.1. Let F be a measurable function on Rn+1+ . For x in RnletΓ(x)be the cone with vertex x defined by
Γ(x) ={(y,t)∈Rn×R+: |y−x|<t}.
A picture of this cone is shown in Figure 7.1. The nontangential maximal function of F is the function
F∗(x) = sup
(y,t)∈Γ(x)|F(y,t)|
defined on Rn. This function is obtained by taking the supremum of the values of F inside the coneΓ(x).
We remark that if F∗(x) =0 for almost all x∈Rn, then F is identically equal to zero on Rn+1+ .
Fig. 7.1 The coneΓ(x)trun- cated at height t.
x
|y - x| < t
Definition 7.3.2. Given a ball B=B(x0,r)in Rnwe define the tent or cylindrical tent over B to be the “cylindrical set”
T(B) ={(x,t)∈Rn+1+ : x∈B, 0<t≤r}. For a cube Q in Rnwe define the tent over Q to be the cube
T(Q) =Q×(0, (Q)].
A tent over a ball and over a cube are shown in Figure 7.2. A positive measureμon Rn+1+ is called a Carleson measure if
μC =sup
Q
1
|Q|μ(T(Q))<∞, (7.3.1) where the supremum in (7.3.1) is taken over all cubes Q in Rn. The Carleson func- tion of the measureμis defined as
C(μ)(x) =sup
Qx
1
|Q|μ(T(Q)), (7.3.2) where the supremum in (7.3.2) is taken over all cubes in Rncontaining the point x.
Observe thatC(μ)L∞=μC. We also define
μcylinderC =sup
B
1
|B|μ(T(B)), (7.3.3)
where the supremum is taken over all balls B in Rn. One should verify that there exist dimensional constants cnand Cnsuch that
cnμC ≤μcylinderC ≤CnμC
for all measuresμon Rn+1+ , that is, a measure satisfies the Carleson condition (7.3.1) with respect to cubes if and only if it satisfies the analogous condition (7.3.3) with respect to balls. Likewise, the Carleson functionC(μ)defined with respect to cubes is comparable toCcylinder(μ)defined with respect to cylinders over balls.
B(x0,r) Q
r
r
Fig. 7.2 The tents over the ball B(x0,r)and over a cube Q in R2.
Examples 7.3.3. The Lebesgue measure on Rn+1+ is not a Carleson measure. Indeed, it is not difficult to see that condition (7.3.1) cannot hold for large balls.
Let L be a line in R2. For A measurable subsets of R2+defineμ(A)to be the linear Lebesgue measure of the set L∩A. Thenμ is a Carleson measure on R2+. Indeed,
the linear measure of the part of a line inside the box[x0−r,x0+r]×(0,r]is at most equal to the diagonal of the box, that is,√
5r.
Likewise, let P be an affine plane in Rn+1 and define a measureν by setting ν(A)to be the n-dimensional Lebesgue measure of the set A∩P for any A⊆Rn+1+ . A similar idea shows thatνis a Carleson measure on Rn+1+ .
We now turn to the study of some interesting boundedness properties of functions on Rn+1+ with respect to Carleson measures.
A useful tool in this study is the Whitney decomposition of an open set in Rn. This is a decomposition of a general open setΩ in Rnas a union of disjoint cubes whose lengths are proportional to their distance from the boundary of the open set.
For a given cube Q in Rn, we denote by(Q)its length.
Proposition 7.3.4. (Whitney decomposition) LetΩ be an open nonempty proper subset of Rn. Then there exists a family of closed cubes{Qj}jsuch that
(a)(jQj=Ω and the Qj’s have disjoint interiors;
(b)√
n(Qj)≤dist(Qj,Ωc)≤4√ n(Qj);
(c) if the boundaries of two cubes Qjand Qktouch, then 1
4 ≤(Qj) (Qk)≤4 ; (d) for a given Qjthere exist at most 12nQk’s that touch it.
The proof of Proposition 7.3.4 is given in Appendix J.
Theorem 7.3.5. There exists a dimensional constant Cnsuch that for allα >0, all measuresμ≥0 on Rn+1+ , and allμ-measurable functions F on Rn+1+ , we have
μ{(x,t)∈Rn+1+ : |F(x,t)|>α}
≤Cn
{F∗>α}C(μ)(x)dx. (7.3.4)
In particular, ifμis a Carleson measure, then μ{|F|>α}
≤CnμC|{F∗>α}|.
Proof. We prove this theorem by working with the equivalent definition of Carleson measures and Carleson functions using balls and cylinders over balls. As observed earlier, these quantities are comparable to the corresponding quantities using cubes.
We begin by observing that for any function F the setΩα={F∗>α}is open, and in particular, F∗is Lebesgue measurable. Indeed, if x0∈Ωα, then there is a (y0,t0)∈Rn+1+ such that|F(y0,t0)|>α. If d0is the distance from y0to the sphere formed by the intersection of the hyperplane t0+Rnwith the boundary of the cone Γ(x0), then|x0−y0| ≤t0−d0. It follows that the open ball B(x0,d0)is contained in Ωα, since for z∈B(x0,d0)we have|z−y0|<t0, hence F∗(z)≥ |F(y0,t0)|>α.
Let {Qk} be the Whitney decomposition of the setΩα. For each x∈Ωα, set δα(x) =dist(x,Ωαc). Then for z∈Qkwe have
δα(z)≤√
n(Qk) +dist(Qk,Ωαc)≤5√
n(Qk) (7.3.5)
in view of Proposition 7.3.4 (b). For each Qk, let Bkbe the smallest ball that contains Qk. Then the radius of Bkis√
n(Qk)/2. Combine this observation with (7.3.5) to obtain that
z∈Qk =⇒ B(z,δα(z))⊆12 Bk. This implies that ,
z∈Ωα
T
B(z,δα(z))
⊆,
k
T(12 Bk). (7.3.6) Next we claim that
{|F|>α} ⊆ ,
z∈Ωα
T
B(z,δα(z))
. (7.3.7)
Indeed, let (x,t)∈Rn+1+ such that|F(x,t)|>α. Then by the definition of F∗we have that F∗(y)>α for all y∈Rnsatisfying|x−y|<t. Thus B(x,t)⊆Ωα and so δα(x)≥t. This gives that(x,t)∈T
B(x,δα(x))
, which proves (7.3.7).
Combining (7.3.6) and (7.3.7) we obtain {|F|>α} ⊆,
k
T(12 Bk).
Applying the measureμand using the definition of the Carleson function, we obtain μ{|F|>α}
≤
∑
k
μT(12 Bk)
≤
∑
k
|12 Bk| inf
x∈Qk
Ccylinder(μ)(x)
≤ 12n
∑
k
|Bk|
|Qk|
QkCcylinder(μ)(x)dx
≤ Cn
ΩαC(μ)(x)dx,
since|Bk|=2−nnn/2vn|Qk|. This proves (7.3.4).
Corollary 7.3.6. For any Carleson measureμand everyμ-measurable function F on Rn+1+ we have
Rn+1+ |F(x,t)|pdμ(x,t)≤CnμC
Rn
(F∗(x))pdx for all 0<p<∞.
Proof. Simply use Proposition 1.1.4 and the previous theorem.
A particular example of this situation arises when F(x,t) = f∗Φt(x)for some nice integrable function Φ. Here and in the sequel, Φt(x) =t−nΦ(t−1x). For in- stance one may takeΦtto be the Poisson kernel Pt.
Theorem 7.3.7. LetΦbe a function on Rnthat satisfies for some 0<C,δ <∞,
|Φ(x)| ≤ C
(1+|x|)n+δ . (7.3.8)
Letμbe a Carleson measure on Rn+1+ . Then for every 1<p<∞there is a constant Cp,n(μ)such that for all f ∈Lp(Rn)we have
Rn+1+ |(Φt∗f)(x)|pdμ(x,t)≤Cp,n(μ)
Rn|f(x)|pdx, (7.3.9) where Cp,n(μ)≤C(p,n)μC. Conversely, suppose thatΦ is nonnegative and sat- isfies (7.3.8) and|x|≤1Φ(x)dx>0. Ifμis a measure on Rn+1+ such that for some 1<p<∞there is a constant Cp,n(μ)such that (7.3.9) holds for all f ∈Lp(Rn), thenμis a Carleson measure with norm at most a multiple of Cp,n(μ).
Proof. Ifμis a Carleson measure, we may obtain (7.3.9) as a consequence of Corol- lary 7.3.6. Indeed, for F(x,t) = (Φt∗f)(x)we have
F∗(x) =sup
t>0
sup
y∈Rn
|y−x|<t
|(Φt∗f)(y)|.
Using (7.3.8) and Corollary 2.1.12, this is easily seen to be pointwise controlled by the Hardy–Littlewood maximal operator, which is Lp bounded. See also Exercise 7.3.4.
Conversely, if (7.3.9) holds, then we fix a ball B=B(x0,r)in Rnwith center x0
and radius r>0. Then for(x,t)in T(B)we have (Φt∗χ2B)(x) =
x−2B
Φt(y)dy≥
B(0,t)
Φt(y)dy= B(0,1)
Φ(y)dy=cn>0,
since B(0,t)⊆x−2B(x0,r)whenever t≤r. Therefore, we have μ(T(B)) ≤ 1
cnp
Rn+1+ |(Φt∗χ2B)(x)|pdμ(x,t)
≤ Cp,n(μ) cnp
Rn|χ2B(x)|pdx
= 2nCp,n(μ) cnp |B|.
This proves thatμis a Carleson measure withμC ≤2nc−pn Cp,n(μ).