Exercises
7.4 The Sharp Maximal Function
7.4.4 Estimates for Singular Integrals Involving the Sharp FunctionFunction
We use the sharp function to obtain pointwise estimates for singular integrals. These enable us to recover previously obtained estimates for singular integrals, but also to deduce a new endpoint boundedness result from L∞to BMO.
Let us recall some facts from Chapter 4. Suppose that K is defined on Rn\ {0}
and satisfies
|K(x)| ≤ A1|x|−n, (7.4.11)
|K(x−y)−K(x)| ≤ A2|y|δ|x|−n−δ whenever|x| ≥2|y|>0, (7.4.12)
r<R<∞sup
r≤|x|≤RK(x)dx ≤ A3. (7.4.13)
Let W be a tempered distribution that coincides with K on Rn\ {0}and let T be the linear operator given by convolution with W .
Under these assumptions we have that T is L2 bounded with norm at most a constant multiple of A1+A2+A3(Theorem 4.4.1), and hence it is also Lpbounded with a similar norm on Lp for 1<p<∞ (Theorem 4.3.3). Furthermore, under
the preceding conditions, the maximal singular integral T(∗)is also bounded from Lp(Rn)to itself for 1<p<∞(Theorem 4.3.4).
Theorem 7.4.9. Let T be given by convolution with a distribution W that coincides with a function K on Rn\ {0}satisfying (7.4.12). Assume that T has an extension that is L2bounded with a norm B. Then there is a constant Cnsuch that for any s>1 the estimate
M#(T(f))(x)≤Cn(A2+B)max(s,(s−1)−1)M(|f|s)1s(x) (7.4.14) is valid for all f in(s≤p<∞Lpand almost all x∈Rn.
Proof. In view of Proposition 7.4.2 (2), given any cube Q, it suffices to find a con- stant a such that
1
|Q|
Q|T(f)(y)−a|dy≤Cs,n(A2+B)M(|f|s)1s(x) (7.4.15) for all x∈Q. To prove this estimate we employ a theme that we have seen several times before. We write f=fQ0+fQ∞, where fQ0=fχ6√
n Qand fQ∞=fχ(6√
n Q)c. Here 6√
n Q denotes the cube that is concentric with Q, has sides parallel to those of Q, and has side length 6√
n(Q), where(Q)is the side length of Q.
We now fix an f in(s≤p<∞Lpand we select a=T(fQ∞)(x). Then a is finite for almost all x∈Q. It follows that
1
|Q|
Q|T(f)(y)−a|dy
≤ 1
|Q|
Q|T(fQ0)(y)|dy+ 1
|Q|
Q|T(fQ∞)(y)−T(fQ∞)(x)|dy. (7.4.16) In view of Theorem 4.3.3, T maps Lsto Lswith norm at most a dimensional constant multiple of max(s,(s−1)−1)(B+A2). The first term in (7.4.16) is controlled by 1
|Q|
Q|T(fQ0)(y)|sdy 1
s ≤ Cnmax(s,(s−1)−1)(B+A2) 1
|Q|
Rn|fQ0(y)|sdy 1
s
≤ Cnmax(s,(s−1)−1)(B+A2)M(|f|s)1s(x). To estimate the second term in (7.4.16), we first note that
Q|T(fQ∞)(y)−T(fQ∞)(x)|dy≤
Q
(6√ nQ)c
K(y−z)−K(x−z) f(z)dz
dy.
We make a few geometric observations. Since both x and y are in Q, we have
|x−y| ≤√
n(Q). Also (see Figure 7.3), since z∈/6√
n Q and x∈Q, we must have
|x−z| ≥dist Q,(6√
n Q)c
≥(3√ n−1
2)(Q)≥2√
n(Q)≥2|x−y|.
Therefore, we have|x−z| ≥2|x−y|, and this allows us to conclude that K(y−z)−K(x−z) = K((x−z)−(x−y))−K(x−z) ≤A2 |x−y|δ
|x−z|n+δ using condition (7.4.12). Using these observations, we bound the second term in (7.4.16) by
1
|Q|
Q
(6√
n Q)c
A2|x−y|δ
|x−z|n+δ |f(z)|dz dy ≤ Cn
A2
|Q|
(6√
n Q)c
(Q)n+δ
|x−z|n+δ |f(z)|dz
≤ CnA2
Rn
(Q)δ
((Q) +|x−z|)n+δ|f(z)|dz
≤ CnA2M(f)(x)
≤ CnA2(M(|f|s)(x))1s,
where we used the fact that|x−z|is at least(Q)and Theorem 2.1.10. This proves
(7.4.15) and hence (7.4.14).
Fig. 7.3 The cubes Q and 6√
n Q. The distance d is equal to(3√n−12)(Q).
z
x Q y
6 n
d
Q
.
. .
The inequality (7.4.14) in Theorem 7.4.9 is noteworthy, since it provides a point- wise estimate for T(f)in terms of a maximal function. This clearly strengthens the Lpboundedness of T . As a consequence of this estimate, we deduce the following result.
Corollary 7.4.10. Let T be given by convolution with a distribution W that coin- cides with a function K on Rn\ {0}that satisfies (7.4.12). Assume that T has an extension that is L2bounded with a norm B. Then there is a constant Cnsuch that the estimate
T(f)
BMO≤Cn(A2+B)f
L∞ (7.4.17)
is valid for all f ∈L∞-(
1≤p<∞Lp .
Proof. We take s=2 in Theorem 7.4.9 and we observe that T(f)
BMO=M#(T(f))
L∞≤Cn(A2+B)M(|f|2)12
L∞, and the last expression is easily controlled by Cn(A2+B)f
L∞.
At this point we have not defined the action of T(f)when f lies merely in L∞; and for this reason we restricted the functions f in Corollary 7.4.10 to be also in some Lp. There is, however, a way to define T on L∞abstractly via duality. Theorem 6.7.1 gives that T and thus also its adjoint T∗map H1to L1. Then the adjoint operator of T∗(i.e., T ) maps L∞to BMO and is therefore well defined on L∞. In this way, however, T(f)is not defined explicitly when f is in L∞. Such an explicit definition is given in the next chapter in a slightly more general setting.
Remark 7.4.11. In the hypotheses of Theorem 7.4.9 we could have replaced the condition that T maps L2to L2by the condition that T maps Lrto Lr,∞with norm B for some 1<r<∞.
Exercises
7.4.1. Let 0<q<∞. Prove that for every p with q<p<∞there is a constant Cn,p,qsuch that for all functions f on Rnwith Md(f)∈Lq(Rn)we have
f
Lp≤Cn,p,qf1−θ
Lq fθ
BMO,
where 1p=1−θq .
7.4.2. Letμbe a positive Borel measure on Rn. (a) Show that the maximal operator
Mμd(f)(x) = sup
Qx Q dyadic cube
1 μ(Q)
Q|f(t)|dμ(t) maps L1(Rn,dμ)to L1,∞(Rn,dμ)with constant 1.
(b) For aμ-locally integrable function f , define the sharp maximal function with respect toμ,
M#μ(f)(x) =sup
Qx
1 μ(Q)
Q
f(t)−Avg
Q,μ f dμ(t),
where AvgQ,μ f denotes the average of f over Q with respect toμ. Assume that μ is a doubling measure with doubling constant C(μ)[this means thatμ(3Q)≤ C(μ)μ(Q)for all cubes Q]. Prove that for allγ >0, all λ >0, and allμ-locally integrable functions f on Rnwe have the estimate
μx : Mμd(f)(x)>2λ,M#μ(f)(x)≤γλ≤C(μ)γ μx : Mdμ(f)(x)>λ.
Hint: Part (a): For any x in the set{x∈Rn: Mμd(f)(x)>λ}, choose a maximal dyadic cube Q=Q(x)such thatQ|f(t)|dμ(t)>λ μ(Q). Part (b): Mimic the proof of Theorem 7.4.4.
7.4.3. Let 0<p0<∞and let Mμdand Mμ#be as in Exercise 7.4.2. Prove that for any p with p0≤p<∞there is a constant Cn(p,μ)such that for all locally integrable functions f with Mdμ(f)∈Lp0(Rn)we have
Mμd(f)
Lp(Rn,dμ)≤Cn(p,μ)M#μ(f)
Lp(Rn,dμ). 7.4.4. We say that a function f on Rnis in BMOd(or dyadic BMO) if
f
BMOd = sup
Q dyadic cube
1
|Q|
Q
f(x)−Avg
Q
f dx<∞.
(a) Show that BMO is a proper subset of BMOd.
(b) Suppose that A is a finite constant and that a function f in BMOdsatisfies Avg
Q1
f−Avg
Q2
f ≤A
for all adjacent dyadic cubes of the same length. Show that f is in BMO.
Hint: Consider first the case n=1. Given an interval I, find adjacent dyadic inter- vals of the same length I1and I2such that II1
(I2and|I1| ≤ |I|<2|I1|.
7.4.5. Suppose that K is a function on Rn\ {0}that satisfies (7.4.11), (7.4.12), and (7.4.13). Letη be a smooth function that vanishes in a neighborhood of the origin and is equal to 1 in a neighborhood of infinity. Forε>0 let Kη(ε)(x) =K(x)η(x/ε) and let Tη(ε) be the operator given by convolution with Kη(ε). Prove that for any 1<s<∞there is a constant Cn,ssuch that for all p with s<p<∞and f in Lpwe
have sup
ε>0
M#(Tη(ε)(f))
Lp(Rn)≤Cn,s(A1+A2+A3)f
Lp(Rn).
Hint: Observe that the kernels Kη(ε) satisfy (7.4.11), (7.4.12), and (7.4.13) uni- formly inε>0 and use Theorems 4.4.1 and 7.4.9.
7.4.6. Let 0<p0<∞and suppose that for some locally integrable function f we have that Md(f)lies in Lp0,∞(Rn). Show that for any p in (p0,∞)there exists a constant Cn(p)such that
f
Lp(Rn)≤Md(f)
Lp(Rn)≤Cn(p)M#(f)
Lp(Rn), where Cn(p)depends only on n and p.
Hint: With the same notation as in the proof of Theorem 7.4.5, use the hypothesis Md(f)
Lp0,∞ <∞to prove that IN <∞whenever p>p0. Then the arguments in the proofs of Theorem 7.4.5 and Corollary 7.4.6 remain unchanged.
7.4.7. Prove that the expressions ΣN(x) =
∑
N k=1sin(2πkx) k
are uniformly bounded in N and x. Then use Corollary 7.4.10 to prove that
N≥1sup
∑
Nk=1
e2πikx k
BMO
≤C<∞.
Deduce that the limit ofΣN(x)as N→∞can be defined as an element of BMO.
Hint: Use that the Hilbert transform of sin(2πkx)is cos(2πkx). Also note that the series∑∞k=1sin(2πkx)
k coincides with the periodic extension of the (bounded) function
=π(12−x)on[0,1).