Three observations on commutators of Singular Integral Operators with BMO functions

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by  Carlos Pérez

THREE OBSERVATIONS ON COMMUTATORS OF SINGULAR
INTEGRAL OPERATORS WITH BMO FUNCTIONS
CARLOS PÉREZ AND ISRAEL P. RIVERA-RÍOS
6
1
0
2 0. Introduction

n
The purpose of this paper is to present some observations concerning commutators
a
J of singular integral operators with BMO functions. These operators were introduced

3 by Coifman, Rochberg and Weiss in [6] as a tool to extend the classical factorization
1 theorem for Hardy spaces in the unit circle to Rn. These operators are defined by the

] expression
A
C (1) T f(x) = (b(x) b(y))K(x,y)f(y)dy,
h. b ˆRn −
t where K is a kernel satisfying the standard Calderón-Zygmund estimates and where
a
m b, the “symbol” of the operator, is a locally integrable function. Of course, these are
[ special cases of the more general commutators given by the expression

1 T = [b,T] = M T T M
b b b
v ◦ − ◦
3 where T is any operator and M is the multiplication operator M f = b f.
b b
9 The classical well known result from [6] establishes that [b,T] is a bou·nded operator
1
3 on Lp(Rn), 1 < p < , when the symbol b is a BMO function. We state this result.
∞
0
. Theorem 1. Let T be a singular integral operator and b a BMO function. The com-
1
0 mutator Tb is bounded on Lp(Rn) for every 1 < p < .
∞
6
1 In the same paper it is shown that b BMO is also a necessary condition namely, if
: the commutators [b,R ], j = 1, ,n o∈f b with the Riesz transforms R are bounded
v j j
i on Lp(Rn) for some p (1, )·a·n·d every j 1,2,...,n then b BMO.
X
∈ ∞ ∈ { } ∈
None of the different proofs of this result follows the usual scheme of the classical
r
a Calderón-Zygmund theory for proving the Lp(Rn) boundedness of singular integral op-
erators T. Two proofs of Theorem 1 can be found in [6]. The first and main one in
that paper is based on methods involving techniques similar to those used in [5] to un-
derstand the Calderón commutator. As far as we know this approach has not been so
influential. However, the second proof, based on the so called conjugation method from
operator theory, has been widely used. In fact, it is quite surprising that this proof was
postponedto theendofthepaper since itturnsouttobehighlyinteresting. Indeed, the
method shows the intimate connection between these commutators and the A theory
p
2010 Mathematics Subject Classification. 42B35,46E30.
The first author was supported by Severo Ochoa Excellence Programme and the Spanish Gov-
ernment grant MTM2014-53850-P and the second author was supported by Grant MTM2012-30748,
Spanish Government.
1
2 CARLOS PÉREZ AND ISRAELP. RIVERA-RÍOS
of weights. Furthermore, this proof can be applied to general linear operators, not only
for Singular Integral Operators. As a sample we will point out the following particular
L2 case:
Theorem 2. Suppose that T is a linear operator such that
T : L2(w) L2(w)
−→
for every w A . Then for every b BMO,
2
∈ ∈
[b,T] : L2(Rn) L2(Rn).
−→
The method of proof can be carried out in more generality as shown in [1]. The key
initial argument of the proof is that we can write [b,T] as a complex integral operator
using the Cauchy integral theorem as follows
d 1 T (f)
[b,T]f = ezbT(fe zb) = z dz, ε > 0
−
dz 2πi ˆ z2
(cid:12)z=0 z=ε
(cid:12) | |
where (cid:12)
(cid:12) f
z T (f) := ezbT z C.
z
→ ezb ∈
(cid:18) (cid:19)
This is called the “conjugation” of T by ezb and the terminology comes most probably
from group theory. Now, if is a norm we can apply Minkowski inequality:
k·k
1
[b,T]f sup T (f) ε > 0.
k k ≤ 2πε k z k
z=ε
| |
The effectiveness of the method can be checked in the modern context of weighted Lp
estimates. Indeed, the method produces very optimal bounds of the operator norm as
shown in [4] (see also [15]).
This method reveals the role played by the following operation:
f
f T (f) := wT
→ w w
(cid:18) (cid:19)
where w is a weight which, in this context, is an A weight. Indeed, this is the case by
p
the well known key property of the BMO class, if p > 1 and b BMO then there is a
∈
small ε , such that etb A , for any real number t such that t < ε . These operators
0 p 0
∈ | |
were already studied by B. Muckenhoupt and R. Wheeden in the 70’s and by E. Sawyer
in the 80’s. Some of the problems they left open were solved in [8]. A consequence of
the main result of [8] is that if w A then T is of weak type (1,1), namely
1 w
∈
T <
k wkL1(Rn)→L1,∞(Rn) ∞
with bound depending upon the A constant of w. However, we will exhibit examples
1
of weights w A A in Section 3 for which T is not of weak type (1,1), namely
p 1 w
∈ \
T = .
k wkL1(Rn)→L1,∞(Rn) ∞
This shows that the case w A is specially relevant. Perhaps, this phenomenon can be
1
∈
explained by the fact that the conjugation method is closely attached to commutators
with BMO functions which are not of weak type (1,1) as observed in [25]. Indeed, the
conjugation method works due to the property, already mentioned, that if p > 1 and
THREE OBSERVATIONS ON COMMUTATORS OF SIO WITH BMO FUNCTIONS 3
b BMO then etb A for small values of t. However, this property turns out to be
p
∈ ∈
false in the case p = 1. The lack of the weak type (1,1) property for commutators is
replaced by a LlogL inequality like (5) below and not better.
There is another proof of Theorem 1 based on the use of the sharp maximal function
of C. Fefferman and E. Stein which has also been very influential. It seems that it was
first discovered by J. O. Strömberg as mentioned by S. Janson in [17] (see also [28] pp.
417-419) The proof relies on combining the following key pointwise estimate
(2) M♯([b,T]f) c b (M (Tf)+M (f))
BMO r s
≤ k k
where 1 < r,s < and M (f) = M( f r)1/r together with the classical Fefferman-Stein
r
∞ | |
inequality:
M(f) c M♯(f) .
Lp Lp
k k ≤ k k
Here we use standard notation, M is the Hardy-Littlewood maximal function and M♯
is the sharp maximal function. The Lp boundedness of M and T yields the alternative
proof of Theorem 1. Proceeding in the same way we obtain the corresponding estimates
for A weights.
p
This approach was considered by S. Bloom in ([2]) extending in an interesting way
Theorem 1 but only on the real line.
Theorem 3. Let µ,λ A and let H be the Hilbert transform:
p
∈
[b,H] : Lp(µ) Lp(λ)
−→
1 1
where ν = µpλ−p if and only if
1
(3) b = sup b b < .
k kBMO(ν) ν(Q) ˆ | − Q| ∞
Q Q
The power of the pointwise estimate (2) is reflected in many situations, for instance
in [12], where similar results were derived for commutators of strongly singular integral
with symbol in the new BMO class (3) (see also [13, 14] for an alternative approach
based on dyadic shifts).
However, estimate (2) is not sharp enoughfor many purposes and much better results
can be obtained with the following variation:
(4) M♯([b,T]f) c b M (Tf)+M2(f) 0 < δ < ε < 1
δ ≤ k kBMO ε
whereM2 stands forM M (see [25]).(cid:0)Here, thekey differ(cid:1)ence isthatweareconsidering
◦
small parameters δ and ε. The estimate is sharp since M2 cannot be replaced by the
(pointwise) smaller operatorM. Indeed, otherwisethese commutatorswouldbeofweak
type (1,1) but, as we mentioned above, this is not the case [25] where it is shown that
commutators satisfy the following “LlogL” type estimate,
f
(5) w( x Rn : [b,T]f(x) > λ ) c Φ | | b wdx λ > 0,
{ ∈ | | } ≤ ˆ λ k kBMO
Rn (cid:18) (cid:19)
where w A , Φ(t) = tlog(e+t) and where c depends upon the A constant. This
1 1
∈
shows that these commutators are “more singular” than Calderón-Zygmund operators.
Theoriginalproofof(5)followsfromthekey pointwise(4)combinedwithagood-λtype
argument, but an alternative proof was obtained by the first author and G. Pradolini
4 CARLOS PÉREZ AND ISRAELP. RIVERA-RÍOS
in [26] with the bonus that non A weights can be considered. This argument is based
∞
on a variation of the classical scheme used to prove the weak type (1,1) for Calderón-
Zygmund operators. The statement of the result is the following.
Theorem 4. Let T be a Calderón-Zygmund operator and b BMO. If w is an
∈
arbitrary weight the following inequality holds
f(x)
w( x Rn : [b,T]f(x) > λ ) C Φ b | | M w(x)dx
{ ∈ | | } ≤ ε,T ˆ k kBMO λ L(logL)1+ε
Rn (cid:18) (cid:19)
for every ε > 0.
Very recently (c.f. [27]) the authors have obtained a quantitative version of the
endpoint estimate for arbitrary weights, namely Theorem 4. This result is analogous
to the one obtained by the first author and T. Hytönen for singular integrals in [15].
Theorem 5. Let T be a Calderón-Zygmund operator and b BMO. If w 0 is a
∈ ≥
weight then, for every ε > 0
c f
w( x Rn : [b,T]f(x) > λ ) Φ b | | M wdx.
{ ∈ | | } ≤ ε2 ˆ k kBMO λ L(logL)1+ε
Rn (cid:18) (cid:19)
The main novelty here is the appearance of the sharp factor 1 reflecting again
ε2
the higher singularity of the operator. As a corollary of this result we can derive the
following result obtained previously by C. Ortiz-Caraballo in [23],
f
w( x Rn : [b,T]f(x) > λ ) CΦ([w] )2 Φ b | | wdx.
{ ∈ | | } ≤ A1 ˆ k kBMO λ
Rn (cid:18) (cid:19)
We remark that it seems that the conjugationmethod cannot be applied to prove this
estimate. Therefore, estimate (5) or Theorem 4 works, so far, for Calderón-Zygmund
operators not for general linear operators assuming a minimal appropriate weighted
weak type estimate.
Another interesting difference between Calderón-Zygmund operators and commuta-
tors concerns their local behavior. A very nice way of expessing this is by means of the
following estimate due to Karagulyan [18]: there exists a constant c such that for each
cube Q and for each function f supported on the cube Q
1
(6) x Q : Tf(x) > tMf(x) ce ct t > 0.
−
Q |{ ∈ | | }| ≤
| |
This result can be seen as an improvement of Buckley’s exponential decay theorem [3]
which is a very useful result. For instance, it allows to improve in a quantitative way
the classical good-λ inequality between T and M: if p (0, ) and w A
∈ ∞ ∈ ∞
Tf c p[w] M(f) .
Lp(w) T A Lp(w)
k k ≤ ∞k k
Motivated by this result of Karagulyan, Ortiz-Caraballo, Rela and the first author
developed a new method for proving (6) in [24]. This method is flexible enough to deal
with other operators including the commutators. In particular, we have the following
sub-gaussian estimate.
THREE OBSERVATIONS ON COMMUTATORS OF SIO WITH BMO FUNCTIONS 5
Theorem 6. Let T be a Calderón-Zygmund operator and b BMO, then there exists
∈
a constant c such that for each f
1
(7) sup x Q : [b,T](fχ )(x) > tM2f(x) ce √ct b BMO t > 0.
Q − k k
Q |{ ∈ | | }| ≤
Q
| |
We will show in Section 1 that this subexponential decay is fully sharp. In Section
2, we will provide a new proof of (6) based on the pointwise domination: if T is a
Calderón-Zygmund operator, then it is possible to find a finite set of η-sparse families
3n (see Section 2 for the definitions) contained in the same or in different dyadic
{Sj}j=1
lattices and depending on f such that
j
D
3n
(8) Tf(x) c A f(x)
| | ≤ T Sj
j=1
X
where
1
A f(x) = f χ (x).
Sj Q ˆ | | Q
QX∈Sj | | Q
See Section 2 for details, in particular Theorem 7.
In view of the interest of an estimate like (8) it would be relevant to produce a
counterpart for commutators. The “natural” sparse operator for these commutators
would be
B f(x) = f χ (x).
S k kLlogL,Q Q
Q
X∈S
The reason that leads to consider this sparse operator in terms of the average
is due to the intimate relationship of commutators and M2 which is an
k · kLlogL,Q
operator pointwise equivalent to M . In Section 2 we prove the impossibility of
LlogL
having a domination theorem for commutators by these “sparse” operators.
1. First observation: Sharpness of the subexponential local decay
We prove in this section that Theorem 6 is sharp, i.e., we can find a Calderón-
Zygmund operator T, a symbol b BMO a function f and a cube Q such that
∈
1
x Q : [b,T]f(x) > tM2f(x) ce √ct b BMO
− k k
Q |{ ∈ | | }| ≥
| |
for some constant c > 0. More precisely we have the following.
Observation 1. Let b(x) = log x , then we can find a constant c such that
| |
x (0,1) : [b,H](χ )(x) > t e √ct
(0,1) −
|{ ∈ | | }| ≥
where H stands for the Hilbert transform.
Proof. Let f(x) = χ (x). We are going to show that
(0,1)
x (0,1) : [b,H]f(x) > tM2f(x) = x (0,1) : [b,H]f(x) > t ce √αt t > 0.
−
|{ ∈ | | }| |{ ∈ | | }| ≥
6 CARLOS PÉREZ AND ISRAELP. RIVERA-RÍOS
For x (0,1) we have that
∈
1 log(x) log(y) 1 log(x) 1/x log(1)
[b,H]f(x) = − dy = y dy = t dt.
ˆ x y ˆ x y ˆ 1 t
0 − 0 − 0 −
Now we observe that
1/x log(1) 1 log(1) 1/x log(1)
t dt = t dt+ t dt
ˆ 1 t ˆ 1 t ˆ 1 t
0 − 0 − 1 −
log(1)
and since t is positive for (0,1) (1, ) we have for 0 < x < 1 that
1 t ∪ ∞
−
1/x log(1)
[b,H]f(x) > t dt.
| | ˆ 1 t
1 −
Finally, a computation shows that
1/x log(1) 1 2
t dt log x 0.
ˆ 1 t ≈ x →
1 − (cid:18) (cid:19)
Consequently, we have that for some x < 1
0
2
1
[b,H]f(x) > c log 0 < x < x .
| | x 0
(cid:18) (cid:19)
and then for some t > 0,
0
(9)
2
1
x (0,1) : [b,H]f(x) > t x (0,x ) : c log > t = e √t/c t > t
0 − 0
|{ ∈ | | }| ≥ ∈ x
(cid:12)( )(cid:12)
(cid:18) (cid:19)
(cid:12) (cid:12)
as we wanted to prove. (cid:12)(cid:12) (cid:12)(cid:12) (cid:3)
(cid:12) (cid:12)
2. Second observation: a “natural” but false sparse domination result
for commutators
Before stating the result we are going to prove in this section we need some notation.
We borrow it from [21].
Definition 1 (Dyadic child). Let Q be a cube (with sides parallel to the axis). We call
dyadic child any of the 2n cubes obtained by partitioning Q by n “median hyperplanes”
(planes parallel to the faces of Q and dividing each edge into 2 equal parts).
If we iterate the partition process of the preceding definition we obtain a standard
dyadic lattice (Q) of subcubes of Q which has the usual properties:
D
(1) For each k = 0,1,2,... cubes in the k-th generation have sidelength 2 k and
−
tile Q in a regular way.
(2) Each Q in the k-th generation has 2n children in the in the (k + 1)-th gener-
′
ation contained in it and one and only one parent in the (k 1)-th generation
−
containing it (unless it is Q itself).
(3) If Q,Q (Q), then Q Q = or Q Q or Q Q.
′ ′′ ′ ′′ ′ ′′ ′′ ′
∈ D ∩ ∅ ⊆ ⊆
(4) If Q (Q), then (Q) (Q).
′ ′
∈ D D ⊆ D
THREE OBSERVATIONS ON COMMUTATORS OF SIO WITH BMO FUNCTIONS 7
Definition 2 (Dyadic lattice). A dyadic lattice in Rn is any collection of cubes such
D
that
(DL-1) If Q then each dyadic child of Q is in as well.
∈ D D
(DL-2) If Q,Q there exists Q such that Q,Q (Q).
′ ′′ ′ ′′
(DL-3) If K is a∈coDmpact set of Rn t∈heDre exists Q suc∈hDthat K Q.
∈ D ⊆
There is an easy way to build a dyadic lattice by considering a increasing sequence of
dyadic cubes Q such that Q = Rn. Then
j ∪∞j=1 j
∞
= (Q )
j
D D
j=1
[
is a dyadic lattice.
Definition 3. Let η (0,1). We say that a family of cubes is η-sparse if for
∈ S ⊆ D
each Q we can find a measurable subset E(Q) Q such that:
∈ S ⊂
(1) E(Q)’s are pairwise disjoint.
(2) η Q E(Q)
| | ≤ | |
Definition 4. Let Λ > 1. We say a family of cubes is Λ-Carleson if for every cube
S
Q we have
∈ D
P Λ Q .
| | ≤ | |
P ,P Q
∈XS ⊂
There is an interesting relation between Carleson and sparse families that we sum-
marize in the following lemma
Lemma 1. If is a Λ-Carleson family of cubes then it is 1-sparse. Conversely if is
S Λ S
a η-sparse family of cubes then it is a 1-Carleson family of cubes.
η
Armed with all these definitions we can state the following pointwise domination
theorem.
Theorem 7. Let T be a Calderón-Zygmund operator. There is a finite set of η-sparse
families 3n contained in the same or in different dyadic lattices and depending
{Sj}j=1 Dj
on f such that
3n
(10) T f(x) c A f(x)
∗ ≤ T Sj
j=1
X
where A f(x) = 1 f χ (x).
Sj Q∈Sj |Q| ´Q| | Q
The proof of thPis result can be found in [21] and [7]. In [19] M. Lacey obtains the
same estimate for Calderón-Zygmund operators that satisfy a Dini condition. Recently
afully quantitative version of Lacey’s result was obtainedin [16]and even morerecently
this quantitative version has been simplified in [20].
As a sample of the interest of this result we give a different proof of the exponential
estimate (6): there exists a constant c such that for each cube Q and for each f
8 CARLOS PÉREZ AND ISRAELP. RIVERA-RÍOS
supported on the cube Q
1
(11) x Q : T f(x) > tMf(x) ce ct t > 0,
∗ −
Q |{ ∈ }| ≤
| |
To prove this result we will use the classical vector-valued extension of the maximal
function introduced by Fefferman and Stein in (see [11]) can be written as follows:
∞ 1/q
M f(x) = (Mf (x))q = Mf(x) ,
q j q
| |
(cid:16)Xj=1 (cid:17)
where f = f is a vector–valued function.
{ j}∞j=1
Indeed, consider the distribution set
S := x Q : T f(x) > tMf(x) .
Q ∗
{ ∈ }
Then, by Theorem 7 and for appropriate c we have that
3n
S x Q : c A f(x) > tMf(x)
| Q| ≤ (cid:12)( ∈ T Sj )(cid:12)
(cid:12) Xj=1 (cid:12)
(cid:12) (cid:12)
(cid:12)3n (cid:12)
(cid:12) x Q : A f(x) > tc Mf(x) (cid:12)
≤ |{ ∈ Sj T,n }|
j=1
X
3n
x Q : χ (x) > tc
Q T,n
≤ (cid:12) ∈ (cid:12)
Xj=1 (cid:12)(cid:12) QX∈Sj (cid:12)(cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
 
(cid:12) (cid:12)
since trivially,
A f(x) Mf(x) χ (x).
Sj ≤ Q
QX∈Sj
We now use one of the key estimates from [24]. Indeed, let E(Q) be the family of
{ }
sets from Definition 3. We have then for some c > 0
q
1
χ (x) = Q χ (x)
Q Q | | Q
QX∈Sj QX∈Sj(cid:18)| | (cid:19)
q
1
c E(Q) χ (x)
≤ Q | | Q
QX∈Sj(cid:18)| | (cid:19)
q
1
c χ (x)dx χ (x)
≤ Q ˆ E(Q) Q
QX∈Sj(cid:18)| | Q (cid:19)
q
c M χ (x)
≤ q E(Q) Q
(cid:16) (cid:16) q (cid:17) (cid:17)
c M g(x(cid:8)) , (cid:9)
q
≤
(cid:0) (cid:1)
THREE OBSERVATIONS ON COMMUTATORS OF SIO WITH BMO FUNCTIONS 9
where g = χ . Now, since E(Q) is a pairwise disjoint family of subsets,
j E(Q) Q j { }
we have that for any j∈S
(cid:8) (cid:9)
1/q
q
(12) g (x) = χ (x) 1.
j ℓq E(Q)
k k   ≤
QX∈Sj(cid:0) (cid:1)
 
q
We finish our proof recalling that if g L , then M g(x) ExpL (see [11]) from
ℓq ∞ q
| | ∈ ∈
which we conclude the desired inequality (11):
(cid:0) (cid:1)
S ce ct Q , t > 0.
Q −
| | ≤ | |
Observation 2. Let T be a Calderón-Zygmund operator and b BMO. It is not
∈
possible to find a finite set of η-sparse families N , with N dimensional, contained
{Sj}j=1
in the same or in different dyadic lattices and depending on f such that
j
D
N
(13) [b,T]f(x) c B f(x) a.e. x Rn
| | ≤ b,T Sj ∈
j=1
X
where B f(x) = f χ (x).
Sj Q∈Sj k kLlogL,Q Q
We are going toPgive two proofs of this result. The first one is based on the Rubio
de Francia algorithm.
Proof 1. Supposethatequation(13)holds,thenwecanprovethefollowingL1 inequality
(14) [b,T]f c[w] M2f .
k kL1(w) ≤ A1k kL1(w)
Indeed,
N N
[b,T]f c B f c B f
k kL1(w) ≤ b,T k Sj kL1(w) ≤ b,T k Sj kL1(w)
j=1 j=1
X X
N
w(Q)
c f Q
≤ b,T k kLlogL,Q Q | |
Xj=1 QX∈Sj | |
N
c w(Q)
b,T f E(Q)
≤ η k kLlogL,Q Q | |
Xj=1 QX∈Sj | |
N
c
b,T M f(x)Mw(x)dx
≤ η ˆ LlogL
E(Q)
Xj=1 QX∈Sj
c
N b,T[w] M2f ,
≤ η A1k kL1(w)
since M2 M . We claim now the Lp version,
≈ LlogL
(15) [b,T]f c p M2f p > 1.
Lp(Rn) n Lp(Rn)
k k ≤ k k
10 CARLOS PÉREZ AND ISRAELP. RIVERA-RÍOS
Indeed, by duality we can find g 0 in Lp′(Rn) with unit norm such that
≥
[b,T]f = [b,T]f(x) g(x)dx.
k kLp(Rn) ˆ | |
Rn
We consider the Rubio de Francia algorithm
∞ Mk(g)
Rg = .
M k
Xk=0 k kLp′(Rn)
It’s a straightforward computation that R(g) is an A weight with constant
1
[Rg] 2 M c p
A1 ≤ k kLp′ ≤ n
and also that g Rg and Rg 2 g = 2. Then have that
≤ k kLp′ ≤ k kLp′(Rn)
[b,T]f(x) g(x)dx [b,T]f(x) Rg(x)dx
ˆ | | ≤ ˆ | |
Rn Rn
and using equation (14) and Hölder inequality
[b,T]f(x) Rg(x)dx c[Rg] M2f(x)Rg(x)dx
ˆ | | ≤ A1ˆ
Rn Rn
cp M2f(x)Rg(x)dx cp M2f Rg
≤ ˆ ≤ k kLp(Rn)k kLp′(Rn)
Rn
cp M2f .
Lp(Rn)
≤ k k
Hence equation (15) is established. Now since
M2 Lp(Rn) cn (p′)2 p > 1
k k ≤
we have that
(16) [b,T] Lp(Rn) cp(p′)2 p > 1
k k ≤
Now let us observe that if we take [b,H]f with b(x) = log x and f(x) = χ (x) then
(0,1)
| |
[b,H]f Lp(R) cp2 p > 1,
k k ≥
and this leads to a contradiction when p . To prove this lower estimate we use
→ ∞
estimate (9) from Theorem 1. Indeed, for some t > 0
0
[b,H]f Lp(R) [b,H]f Lp, (R) = supt x R : [b,H]f(x) > t p1
k k ≥ k k ∞ |{ ∈ | | }|
t>0
1
1 2 p
supt x (0,x ) : c log > t
≥ t>t0 (cid:12)( ∈ 0 (cid:18) x(cid:19) )(cid:12)
(cid:12) (cid:12)
≥ suptc(cid:12)(cid:12)(cid:12)e−p√t ≥ cp2t0e−√t0 (cid:12)(cid:12)(cid:12)
t>t0
and this concludes the first proof. (cid:3)
For the second proof we will rely on the sharpness result that was settled in the
previous section.