Exercises
7.2 Duality between H 1 and BMO
The next main result we discuss about BMO is a certain remarkable duality rela- tionship with the Hardy space H1. We show that BMO is the dual space of H1. This means that every continuous linear functional on the Hardy space H1can be real- ized as integration against a fixed BMO function, where integration in this context is an abstract operation, not necessarily given by an absolutely convergent integral.
Restricting our attention, however, to a dense subspace of H1such as the space of all finite sums of atoms, the use of the word integration is well justified. Indeed, for an L2atom for H1a and a BMO function b, the integral
|a(x)b(x)|dx<∞
converges absolutely, since a(x)is compactly supported and bounded and b(x)is locally (square) integrable.
Definition 7.2.1. Denote by H01(Rn)the set of all finite linear combinations of L2 atoms for H1(Rn). For b∈BMO(Rn)we define a linear functional Lbon H01(Rn) by setting
Lb(g) =
Rng(x)b(x)dx, g∈H01. (7.2.1) Certainly the integral in (7.2.1) converges absolutely and is well defined in this case.
This definition is also valid for general functions g in H1(Rn)if the BMO function b is bounded. Note that (7.2.1) remains unchanged if b is replaced by b+c, where c is an additive constant; this makes this integral unambiguously defined for b∈BMO.
To extend the definition of Lbon the entire H1 for all functions b in BMO we need to know that
Lb
H1→C≤Cnb
BMO, whenever b is bounded, (7.2.2) a fact that will be proved momentarily. Assuming (7.2.2), take b∈BMO and let bM(x) =bχ|b|≤Mfor M=1,2,3, . . .. SincebM
BMO≤3b
BMO, the sequence of linear functionals{LbM}M lies in a multiple of the unit ball of(H1)∗and there is a subsequence LMj that converges weakly to a bounded linear functionalLbon H1. This means that for all f in H1(Rn)we have
LbM j(f)→Lb(f) as j→∞. Observe that for g∈H01we also have
LbM j(g)→Lb(g),
and since each LbM j satisfies (7.2.2), Lbis a bounded linear functional on H01. Since H01is dense in H1,Lbis the unique bounded extension ofLbon H1.
Having set the definition of Lb, we proceed by showing the validity of (7.2.2).
Let b be a bounded BMO function. Given f in H1, find a sequence akof L2atoms for H1supported in cubes Qksuch that
f=
∑
∞ k=1λkak (7.2.3)
and ∞
k=1
∑
|λk| ≤2f
H1.
Since the series in (7.2.3) converges in H1, it must converge in L1, and then we have
|Lb(f)| =
Rn f(x)b(x)dx
=
∑
∞k=1
λk
Qkak(x)
b(x)−Avg
Qk
b dx
≤
∑
∞k=1
|λk|ak
L2|Qk|12 1
|Qk|
Qk
b(x)−Avg
Qk
b 2dx 12
≤ 2f
H1B2,nb
BMO,
where in the last step we used Corollary 7.1.8 and the fact that L2 atoms for H1 satisfyak
L2≤ |Qk|−12. This proves (7.2.2) for bounded functions b in BMO.
We have proved that every BMO function b gives rise to a bounded linear func- tionalLbon H1(Rn)(from now on denoted by Lb) that satisfies
Lb
H1→C≤Cnb
BMO. (7.2.4)
The fact that every bounded linear functional on H1arises in this way is the gist of the equivalence of the next theorem.
Theorem 7.2.2. There exist finite constants Cnand Cnsuch that the following state- ments are valid:
(a) Given b∈BMO(Rn), the linear functional Lbis bounded on H1(Rn)with norm at most Cnb
BMO.
(b) For every bounded linear functional L on H1there exists a BMO function b such that for all f ∈H01we have L(f) =Lb(f)and also
b
BMO≤CnLb
H1→C.
Proof. We have already proved (a) and so it suffices to prove (b). Fix a bounded linear functional L on H1(Rn)and also fix a cube Q. Consider the space L2(Q)of all square integrable functions supported in Q with norm
g
L2(Q)=
Q|g(x)|2dx 12
.
We denote by L20(Q)the closed subspace of L2(Q) consisting of all functions in L2(Q)with mean value zero. We show that every element in L20(Q)is in H1(Rn) and we have the inequality
g
H1≤cn|Q|12g
L2. (7.2.5)
To prove (7.2.5) we use the square function characterization of H1. We fix a Schwartz functionΨ on Rnwhose Fourier transform is supported in the annulus
1
2≤ |ξ| ≤2 and that satisfies (6.2.6) for allξ =0 and we letΔj(g) =Ψ2−j∗g. To estimate the L1norm of
∑j|Δj(g)|21/2
over Rn, consider the part of the integral
over 3√
n Q and the integral over(3√
n Q)c. First we use H¨older’s inequality and an L2estimate to prove that
3√
n Q
∑
j
|Δj(g)(x)|21
2dx≤cn|Q|12g
L2.
Now for x∈/3√
n Q we use the mean value property of g to obtain
|Δj(g)(x)| ≤ cng
L22n j+j|Q|1n+12
(1+2j|x−cQ|)n+2 , (7.2.6) where cQis the center of Q. Estimate (7.2.6) is obtained in a way similar to that we obtained the corresponding estimate for one atom; see Theorem 6.6.9 for details.
Now (7.2.6) implies that
(3√ n Q)c
∑
j
|Δj(g)(x)|21
2dx≤cn|Q|12g
L2,
which proves (7.2.5).
Since L20(Q)is a subspace of H1, it follows from (7.2.5) that the linear functional L : H1→C is also a bounded linear functional on L20(Q)with norm
L
L20(Q)→C≤cn|Q|1/2L
H1→C. (7.2.7)
By the Riesz representation theorem for the Hilbert space L20(Q), there is an element FQin(L20(Q))∗=L2(Q)/{constants}such that
L(g) =
Q
FQ(x)g(x)dx, (7.2.8)
for all g∈L20(Q), and this FQsatisfies FQ
L2(Q)≤L
L20(Q)→C. (7.2.9)
Thus for any cube Q in Rn, there is square integrable function FQsupported in Q such that (7.2.8) is satisfied. We observe that if a cube Q is contained in another cube Q, then FQdiffers from FQ by a constant on Q. Indeed, for all g∈L20(Q)we
have
Q
FQ(x)g(x)dx=L(g) =
Q
FQ(x)g(x)dx
and thus
Q(FQ(x)−FQ(x))g(x)dx=0. Consequently,
g→
Q(FQ(x)−FQ(x))g(x)dx
is the zero functional on L20(Q); hence FQ −FQmust be the zero function in the space(L20(Q))∗, i.e., FQ −FQis a constant on Q.
Let Qm= [−m,m]nfor m=1,2, . . .. We define a locally integrable function b(x) on Rnby setting
b(x) =FQm(x)− 1
|Q1|
Q1
FQm(t)dt (7.2.10)
whenever x∈Qm. We check that this definition is unambiguous. Let 1≤ <m.
Then for x∈Q, b(x)is also defined as in (7.2.10) within the place of m. The difference of these two functions is
FQm−FQ−Avg
Q1
(FQm−FQ) =0,
since the function FQm−FQis constant in the cube Q(which is contained in Qm), as indicated earlier.
Next we claim that there is a locally integrable function b on Rnsuch that for any cube Q there is a constant CQsuch that
FQ=b−CQ on Q. (7.2.11)
Indeed, given a cube Q pick the smallest m such that Q is contained in Qm and let CQ=−AvgQ1(FQm) +D(Q,Qm), where D(Q,Qm)is the constant value of the function FQm−FQon Q.
We have now found a locally integrable function b such that for all cubes Q and all g∈L20(Q)we have
Qb(x)g(x)dx=
Q(FQ(x) +CQ)g(x)dx=
QFQ(x)g(x)dx=L(g), (7.2.12) as follows from (7.2.8) and (7.2.11). We conclude the proof by showing that b∈ BMO(Rn). By (7.2.11), (7.2.9), and (7.2.7) we have
sup
Q
1
|Q|
Q|b(x)−CQ|dx = sup
Q
1
|Q|
Q|FQ(x)|dx
≤ sup
Q
|Q|−1|Q|12FQ
L2(Q)
≤ sup
Q
|Q|−12L
L20(Q)→C
≤ cnL
H1→C<∞.
Using Proposition 7.1.2 (3), we deduce that b∈BMO andb
BMO≤2cnL
H1→C. Finally, (7.2.12) implies that
L(g) =
Rnb(x)g(x)dx=Lb(g)
for all g∈H01(Rn), proving that the linear functional L coincides with Lbon a dense subspace of H1. Consequently, L=Lb, and this concludes the proof of part (b).
Exercises
7.2.1. Use Exercise 1.4.12(a) and (b) to deduce that b
BMO ≈ sup
fH1≤1
Lb(f) , f
H1 ≈ sup
bBMO≤1
Lb(f) .
7.2.2. Suppose that a locally integrable function u is supported in a cube Q in Rn
and satisfies
Qu(x)g(x)dx=0
for all square integrable functions g on Q with mean value zero. Show that u is almost everywhere equal to a constant.