Logarithmic bump conditions and the two weight boundedness of Calderón-Zygmund operators

0.28 MB·English

by  David Cruz-Uribe

LOGARITHMIC BUMP CONDITIONS AND THE TWO-WEIGHT
BOUNDEDNESS OF CALDERO´N–ZYGMUND OPERATORS
DAVID CRUZ-URIBE, SFO, ALEXANDER REZNIKOV, AND ALEXANDER VOLBERG
Abstract. We prove that if a pair of weights (u,v) satisfies a sharp A -bump
p
2 condition in the scale of all log bumps or certain loglog bumps, then Haar shifts
1 mapLp(v)intoLp(u)withaconstantquadraticinthecomplexityoftheshift. This
0
in turn implies the two weight boundedness for all Caldero´n-Zygmund operators.
2
This gives a partial answer to a long-standing conjecture. We also give a partial

n result for a related conjecture for weak-type inequalities. To prove our main
a
results we combine several different approaches to these problems; in particular
J
weusemanyoftheideasdevelopedtoprovetheA2 conjecture. Asabyproductof
3
our work we also disprove a conjecture by Muckenhoupt and Wheeden on weak-

] type inequalities for the Hilbert transform. This is closely related to the recent
P
counterexamples of Reguera, Scurry and Thiele.
A
.
h
t
a
m
1. Introduction
[

In this paper we prove several partial results related to a pair of long-standing
3
v conjectures in the theory of two-weight norm inequalities. To state the conjectures
6
and our results we recall a few facts about Orlicz spaces; see [3, Chapter 5] for
7
6 complete details. Given a Young function A, the complementary function A¯ is the
0 Young function that satisfies
.
2
1 t ≤ A−1(t)A¯−1(t) ≤ 2t, t > 0.
1
1 We will say that a Young function A¯ satisfies the B condition, 1 < p < ∞, if for
: p′
v some c > 0,
i
X ∞ A¯(t)dt
r < ∞.
a tp′ t
Zc
If A and A¯ are doubling (i.e., if A(2t) ≤ CA(t), and similarly for A¯), then A¯ ∈ B
p
if and only if
∞ tp p′−1 dt
< ∞.
A(t) t
Zc (cid:18) (cid:19)
Remark 1. As we will see with specific examples below, if A¯ ∈ B , then A¯(t) . tp′
p′
and A(t) & tp.
2010 Mathematics Subject Classification. 42B20,42B35, 47A30.
Key words and phrases. Caldero´n–Zygmund operators, Carleson embedding theorem, Bellman
function, stopping time, bump conditions, Orlicz norms.
The first author is supported by the Stewart-Dorwart faculty development fund at Trinity
College and by grant MTM2009-08934 from the Spanish Ministry of Science and Innovation; the
third author is supported by the NSF under the grant DMS-0758552.
1
2 DAVIDCRUZ-URIBE,SFO,ALEXANDERREZNIKOV,AND ALEXANDERVOLBERG
Given p, 1 < p < ∞, let A and B be Young functions such that A¯ ∈ B and
p′
B¯ ∈ B . We say that the pair of weights (u,v) satisfies an A bump condition with
p p
respect to A and B if
(1) supku1/pk kv−1/pk < ∞,
A,Q B,Q
Q
where the supremum is taken over all cubes Q in Rd, and the Luxemburg norm is
defined by
1
kfk = inf λ > 0 : A |f(x)|/λ dx ≤ 1 .
A,Q
|Q|
(cid:26) ZQ (cid:27)
(cid:0) (cid:1)
If (1) holds, then it is conjectured that
(2) T : Lp(v) → Lp(u).
Similarly, if the pair (u,v) satisfies the weaker condition
(3) supku1/pk kv−1/pk < ∞,
A,Q p′,Q
Q
then the conjecture is that
(4) T : Lp(v) → Lp,∞(u).
The conditions (1) and (3) are referred to as A bump conditions because they
p
may be thought of as the classical two-weight A condition with the localized Lp
p
and Lp′ norms “bumped up” in the scale of Orlicz spaces. These conditions have a
long history. They first appeared in connection with estimates for integral opera-
tors related to the spectral theory of Schro¨dinger operators: see Fefferman [8] and
Chang–Wilson–Wolff [1]. These papers demonstrate a very close connection with
uncertainty principles; forthisaspect alsosee thevery interesting paperofP´erezand
Wheeden [32]. The bump condition considered in [1, 8] was the Fefferman–Phong
condition that used “power” bumps: i.e., Young functions of the form A(t) = trp,
r > 1. Power bumps were independently introduced by Neugebauer [28]. Bump
conditions in full generality were introduced by P´erez [29, 30, 31].
The conjectured strongandweak-type inequalities forsingular integrals have been
studied extensively, but the full results have proved elusive. The strong-type con-
jecture is true for operators of bounded complexity (e.g., the Hilbert transform, the
Riesz transforms and the Buerling–Ahlfors operators): see [2].
Lerner[15]provedthatitholdsforanyCaldero´n-Zygmundoperatorifp > n. Very
recently, it was proved for p = 2 in any dimension and for any Caldero´n–Zygmund
operator using Bellman function techniques: see [19].
Theorem 1.1. Given p = 2, suppose the pair of weights (u,v) satisfies (1), where
A¯ ∈ B and B¯ ∈ B . Then every Calder´on-Zygmund singular integral operator T
2 2
satisfies kTfk ≤ Ckfk , where C depends only on T, the dimension d, and
L2(u) L2(v)
the suprema in (1).
Remark 2. It turns out that extending this theorem to p 6= 2, and especially
strengthening it by replacing two-side bump conditions (1) by weaker one-side con-
ditions (7) and (8) is difficult.
LOGARITHMIC BUMP CONDITIONS 3
Certain additional results are known in the special case that A and B are “log-
bumps”: that is, of the form
tp′
(5) A(t) = tplog(e+t)p−1+δ, A¯(t) ≈ ,
log(e+t)1+δ′
tp
(6) B(t) = tp′log(e+t)p′−1+δ, B¯(t) ≈ ,
log(e+t)1+δ′′
where δ > 0, δ′ = δ/(p − 1), δ′′ = δ/(p′ − 1). But even in this case the result for
all Caldero´n–Zygmund operators was unknown. The weak-type conjecture is only
known for log bumps: see [5]. For a complete history of both conjectures and these
partial results, we refer the reader to the work of P´erez, Cruz-Uribe, and Martell
[2, 3, 4, 6, 7], and Treil, Volberg, and Zheng [38], and the extensive references they
contain.
One can motivate the conjectures (1) ⇒ (2) and (3) ⇒ (4) (and the related con-
jectures we consider below) by considering a pair of conjectures due to Muckenhoupt
and Wheeden. First, they conjectured that a singular integral operator (in particu-
lar, theHilbert transform) satisfies (2) provided that theHardy-Littlewood maximal
operator satisfies
M : Lp(v) → Lp(u),
M : Lp′(u1−p′) → Lp′(v1−p′).
They also conjectured that (4) holds if the maximal operator satisfies the second,
Lp′ inequality. P´erez [31] (see also [3]) proved that a sufficient condition for each of
these estimates to hold for M is that the pair (u,v) satisfies
(7) supku1/pk kv−1/pk < ∞,
p,Q B,Q
Q
(8) supku1/pk kv−1/pk < ∞;
A,Q p′,Q
Q
in particular, both these conditions hold if (1) holds.
Though intuitively appealing, both of the Muckenhoupt-Wheeden conjectures
are false. A counter-example to the strong-type conjecture was recently found by
Reguera and Scurry [34]. The weak-type conjecture is an easy consequence of the
two-weight, weak (1,1) conjecture (also due to Muckenhoupt and Wheeden), but
this was recently proved false by Reguera and Thiele [35]. While this does not show
the conjecture false, it strongly suggests that it is. And as a byproduct of our ap-
proach to our main results we show that the weak-type conjecture is also false; as a
consequence we get another proof that their weak (1,1) conjecture is false.
Given the falsity of the Muckenhoupt-Wheeden conjectures (even for p = 2), the
A bump conjectures become even more interesting. And Theorem 1.1 and many
p
other results listed above strongly suggest that it should hold in the full range of p,
dimensions, and Caldero´n–Zygmund operators. Here we consider two even stronger
conjectures, motivated by the fact that the “separated” bump conditions (7) and (8)
are sufficient for the maximal operator inequalities that they posited.
Conjecture 1. Given p, 1 < p < ∞, suppose the pair of weights (u,v) satisfies
(7) and (8), where A¯ ∈ B and B¯ ∈ B . Then every Calder´on-Zygmund singular
p′ p
4 DAVIDCRUZ-URIBE,SFO,ALEXANDERREZNIKOV,AND ALEXANDERVOLBERG
integral operator T satisfies kTfk ≤ Ckfk , where C depends only on T, the
Lp(u) Lp(v)
dimension d, and the suprema in (7) and (8).
Conjecture 2. Given p, 1 < p < ∞, suppose the pair of weights (u,v) satisfies (8)
where A¯ ∈ B . Then every Calder´on-Zygmund singular integral operator T satisfies
p′
kTfk ≤ Ckfk , where C depends only on T, the dimension d, and the
Lp,∞(u) Lp(v)
supremum in (8).
We can prove Conjecture 1 in the special case when A, B are log bumps.
Theorem 1.2. Given p, 1 < p < ∞, suppose the pair of weights (u,v) satisfies
(7) and (8), where A and B are log bumps of the form (5) and (6). Then every
Calder´on-Zygmund singular integral operator T satisfies kTfk ≤ Ckfk ,
Lp(u) Lp(v)
where C depends only on T, the dimension d, and the suprema in (7) and (8).
Our techniques also immediately yield Conjecture 2 for log bumps. This gives a
new proof of the result originally proved in [5]; for completeness we include it here.
Theorem 1.3. Given p, 1 < p < ∞, suppose the pair of weights (u,v) satisfies
(8) where A is a log bump of the form (5). Then every Calder´on-Zygmund singular
integral operator T satisfies kTfk ≤ Ckfk , where C depends only on T,
Lp,∞(u) Lp(v)
the dimension d, and the supremum in (8).
Remark 3. Theorems 1.2 and 1.3 are both sharp, in the sense that if we take δ = 0
in the definition of A or B, then there exist pairs of weights that satisfy the bump
conditions but such that the corresponding norm inequalities are false. For details,
see [3].
Our proofof Theorems 1.2 and1.3 depends heavily onthe machinery developed to
prove the one-weight A conjecture [2, 9, 14]. In turn many of the techniques used to
2
prove the A conjecture have their genesis in nonhomogeneous Harmonic Analysis.
2
In particular, they go back to the random geometric constructions introduced in [21,
22, 23]. For a summary of these results, see [39].
The method of the proof allows us to extend it one step further to loglog-bumps
(some of them) and to prove
Theorem 1.4. Given p, 1 < p < ∞, suppose the pair of gauge functions A,B
satisfies loglog-bump condition (1) with sufficiently large positive δ, and the pair
of weights (u,σ) satisfies (37). Given any dyadic shift S of complexity (m,n), τ =
max(m,n)+1, kS(fσ)k ≤ Cτ2kfk , where C depends only on the dimension
Lp(u) Lp(σ)
d and the suprema in (37).
Theorem 1.5. Given p, 1 < p < ∞, suppose that the gauge function A sat-
isfies loglog-bump condition (9) with sufficiently large positive δ, and weights
(u,σ) satisfies condition (37). Given any dyadic shift S of complexity (m,n), τ =
max(m,n)+1, kS(fσ)k ≤ Cτ2kfk , where C depends only on the dimen-
Lp,∞(u) Lp(σ)
sion d and the supremum in (37).
The reader will see the dictionary between the languages of (u,v) and u,σ) in the
next Section.In particular, the condition (1) in terms of (u,v) becomes exactly (37)
in terms of the pair (u,σ).
LOGARITHMIC BUMP CONDITIONS 5
The remainder of this paper is organized as follows. In Section 2 we reformulate
our results andreduce theproblem to proving the corresponding results fora general
class of dyadic shift operators (Theorem 2.3 and 2.4). In Section 3 we prove Theo-
rem 2.3. It is important to note that in most of the proof we only need to assume
that A¯ ∈ B ,B¯ ∈ B ; only at one step are we forced to assume that A, B are log
p′ p
bumps. InSection 4 we describe the (minor) changes required to prove Theorem 2.4.
In Section 5 we show that the Muckenhoupt-Wheeden conjecture for the weak-type
inequality is false. Finally, in Section 6, we prove the loglog-bump theorems.
Acknowledgements. The authors are very grateful to the American Institute
of Mathematics, where this work was essentially done, for their hospitality. The
authors also want to thank Michael Lacey for his insightful remarks.
2. Preliminary Results
Hereafter, we will use the notation
1
hfi = f(x)dx.
Q |Q|
Z
Q
We also restate our weighted norm inequalities in an equivalent form. Let σ = v1−p′;
then we can rewrite (7) and (8) as
(9) suphui1/pkσ1/p′k < ∞,
B,Q
Q
Q
(10) supku1/pk hσi1/p′ < ∞.
A,Q
Q
Q
By the properties of the Luxemburg norm we have that either condition implies the
two-weight A condition:
p
(11) suphui1/phσi1/p′ < ∞.
Q Q
Q
Similarly, we can restate the conclusions of Theorems 1.2 and 1.3 as
kT(fσ)k ≤ Ckfk , kT(fσ)k ≤ Ckfk .
Lp(u) Lp(σ) Lp,∞(u) Lp(σ)
The B condition is closely connected to a generalization of the maximal operator.
p
Recall that the Hardy-Littlewood maximal operator is defined to be
Mf(x) = suph|f|i = supkfk .
Q 1,Q
Q∋x Q∋x
Given a Young function A, we define the Orlicz maximal operator M by
A
M f(x) := supkfk .
A A,Q
Q∋x
The following result is due to P´erez [31] (see also [3]).
Theorem 2.1. Fix p, 1 < p < ∞, and let A be a Young function such that A ∈ B .
p
Then M : Lp → Lp.
A
The B condition is also sufficient for a two-weight norm inequality for the Hardy-
p
Littlewood maximal operator. This result is also due to P´erez [31, 3].
6 DAVIDCRUZ-URIBE,SFO,ALEXANDERREZNIKOV,AND ALEXANDERVOLBERG
Theorem 2.2. Fix p, 1 < p < ∞, and let B be a Young function such that B¯ ∈ B .
p
If the pair of weights (u,σ) satisfies
(12) suphui1/pkσ1/p′k < ∞,
B,Q
Q
Q
then
(13) kM(fσ)k ≤ Ckfk .
Lp(u) Lp(σ)
Remark 4. The bump condition (12) is necessary in the following sense: suppose
that B is a function such that whenever (12) holds, the maximal operator satisfies
(13). Then B¯ ∈ B . See [31].
p
We now turn to the definition of the dyadic Haar shift operators that will replace
an arbitrary Caldero´n-Zygmund operator.
Definition 1. Given a dyadic cube Q, h is a (generalized) Haar function associated
Q
to a cube Q if
h (x) = c χ (x),
Q Q′ Q′
Q′∈ch(Q)
X
where ch(Q) is the set of dyadic children of Q and |c | ≤ 1.
Q′
Definition 2. We say that an operator S has a Haar shift kernel of complexity
(m,n) if
Sf(x) = S (f),
Q
Q
X
where
1
S (f) = (f,h )h
Q Q′ Q′′
|Q|
Q′,Q′′⊂Q
X
ℓ(Q′)=2−nℓ(Q)
ℓ(Q′′)=2−mℓ(Q)
and h and h are generalized Haar functions associated to the cubes Q′ and Q′′
Q′ Q′′
respectively. We say that S is a Haar shift of complexity (m,n) if it has a Haar shift
kernel of complexity (m,n), and it is bounded on L2(dx).
By the decomposition theorem of Hyto¨nen [9, 10], to prove Theorems 1.2 and 1.3
it will suffice to prove that they hold for Haar shift operators of complexity (m,n)
with a constant that grows polynomially in τ = max(m,n)+1. More precisely we
will prove the following.
Theorem 2.3. Given p, 1 < p < ∞, suppose the pair of gauge functions A,B
satisfies log-bump conditions 5, 6 and the pair of weights (u,v) satisfies (9) and (10).
Given any dyadic shift S of complexity (m,n), τ = max(m,n)+1, kS(fσ)k ≤
Lp(u)
Cτ2kfk , where C depends only on the dimension d and the suprema in (9) and
Lp(σ)
(10).
Theorem 2.4. Given p, 1 < p < ∞, suppose that the gauge function A satisfies
log-bump condition 5 and weights (u,v) satisfies condition (10). Given any dyadic
shift S of complexity (m,n), kS(fσ)k ≤ Cτ2kfk , where C depends only
Lp,∞(u) Lp(σ)
on the dimension d and the supremum in (10).
LOGARITHMIC BUMP CONDITIONS 7
3. Proof of Theorem 2.3
To prove the strong-type inequality we follow the argument used by Hyto¨nen
and Lacey [11] in the one-weight case, which in turn refines the proof given in [2].
Fix a function f that is bounded and has compact support. For each N > 0, let
Q = [−2N,2N]d. By Fatou’s lemma,
N
1/p
kS(fσ)k ≤ liminf |S(fσ)(x)−m |pu(x)dx ,
Lp(u) S(fσ)
N→∞
(cid:18)ZQN (cid:19)
where m is the median value of S(fσ) on Q . Fix N. Using the remarkable
S(fσ) N
decomposition theorem of Lerner [15], they show that there exists a family of dyadic
cubes L = {Qk} and pairwise disjoint sets {Ek} such that Ek ⊂ Qk, |Ek| ≥ 1|Qk|,
j j j j j 2 j
and
1/p
(14) |S(fσ)(x)−m |pu(x)dx
S(fσ)
(cid:18)ZQN (cid:19)
τ
≤ CτkM(fσ)k +Cτ k h|f|σi χ k .
Lp(u) (Qk)i Qkj Lp(u)
j
i=1 j,k
X X
(Here, given a dyadic cube Q, Qi denotes the i-th parent of Q.) The linear depen-
dence on τ in (14) can be found in [11]: see Lemma 2.4 and the discussion following
it. Alternatively, we can deduce it from the argument in [2] if: 1) we combine it
with the unweighted weak-type estimate with the right dependence on τ in [9, 14];
2) we precede the weak-type estimate of 1 S(1 f) by a careful pointwise estimate
Q Qτ
of this function on Q. This lets us reduce the weak-type estimate of this expression
to the weak-type estimate of 1 S(1 f) (with an error term that can be controlled).
Q Q
By Theorem 2.2, kM(fσ)k ≤ Ckfk . Therefore, it remains to estimate
Lp(u) Lp(σ)
the second term in (14). Again, following [11], we show that this reduces to a
two-weight estimate for a positive Haar shift operator.
We reorder the sum as follows: fix an integer i ∈ [1,τ] and sum over every cube
Q = (Qk)i and then over all cubes Qs ∈ L such that (Qs)i = Q. Then we have that
j r r
h|f|σi χ = h|f|σi χ = χi h|f|σi = Si(|f|σ),
(Qk)i Qkj Q R Q Q L
j
Xj,k XQ RXRi∈=LQ XQ
where the last sum is taken over all dyadic cubes Q and
χi = χ .
Q R
RXRi∈=LQ
Clearly, S (hereafter we omit the superscript i) is a positive operator. We claim
L
that it is in fact a positive Haar shift of complexity at most (0,τ − 1). From the
definition we have that
1
S f = χi f,
L |Q| Q
Q∈L Z
X Q
and so in the notation used above we have that
1
S = χi f,
Q |Q| Q
Z
Q
8 DAVIDCRUZ-URIBE,SFO,ALEXANDERREZNIKOV,AND ALEXANDERVOLBERG
Q′ = Q, h = χ , the Q′′ are all the (i−1)-children of Q, and h = c χ ,
Q′ Q Q′′ R R
R∈ch(Q′′)
where c = 1 if R ∈ L and c = 0 otherwise. Thus S has a Haar shiPft kernel of
R R L
complexity (0,i−1), i ≤ τ. To see that it is bounded on L2, we use the properties
of the cubes Qk. By duality, there exists g ∈ L2, kgk = 1, such that
j 2
kS fk2 = hfi χ (x)g(x)dx
L 2 (Qk)i Qkj
ZRd j,k j
X
≤ 2 hfi hgi |Ek| ≤ 2 Mf(x)Mg(x)dx.
(Qk)i Qk j
j,k j j ZRd
X
The last integral is bounded by kfk kgk by H¨older’s inequality and the unweighted
2 2
L2 inequality for the maximal operator.
Remark 5. It follows from this argument that both S and its adjoint S∗ are positive
L L
Haar shifts with uniform bounds.
Definition. Given a positive Haar shift operator S, define the associated maximal
singular integral operator by
S (x) := sup S (x) = sup S f(x).
♯ ǫ,v Q
0<ǫ≤v<∞ 0<ǫ≤v<∞
Q∈D,ǫ≤ℓ(Q)≤v
X
To prove that kS (fσ)k ≤ Cτkfk , we use the following result that is es-
L Lp(u) Lp(σ)
sentially due to Sawyer and can found in Hyto¨nen and Lacey [11] and Hyto¨nen, et
al. [12]. The precise statement below is gotten by combining Theorem 4.7 of [12]
with Corollary 3.2 and Lemma 3.3 of [11].
Theorem 3.1. Let S be a positive Haar shift of complexity (m,n). Then the asso-
ciated maximal singular integral S satisfies
♯
(15) kS (·σ)k
♯ Lp(σ)→Lp(u)
kχ S(χ σ)k kχ S∗(χ u)k
Q Q Lp(u) Q Q Lp(σ)
6 τkM(·σ)k +sup +sup .
Lp(σ)→Lp(u) 1 1
Q σ(Q)p Q u(Q)p
Remark 6. Testing conditions of this kind were first proved by E. Sawyer [37] for
positive operators. Later, this result was proved by Nazarov, Treil andVolberg [20, 24]
for all well localized operators (in particular, all Haar shifts) when p = 2. Also see
Theorem 5.1 below. It is not known if Theorem 3.1 is true for all Haar shifts when
p 6= 2 even if S is replaced by S.
♯
We will now use Theorem 3.1 to show that if S is any positive Haar shift operator,
then the right-hand side of (15) is finite if our bump conditions are satisfied. As
before, by Theorem 2.2 the first term is bounded by Cτ. It remains to estimate the
two testing conditions. We will estimate the first; the estimate for S∗ is gotten in
essentially the same fashion. (See Remark 7 below.)
Fix a cube Q ; using the notation from the definition of a Haar shift, we have
0
that
(16) χ S(χ σ) = S (σ)+χ S (χ σ) 6 S (σ)+χ hσi .
Q0 Q0 R Q0 R Q0 R Q0 Q0
RX⊂Q0 R,XQ0⊂R RX⊂Q0
LOGARITHMIC BUMP CONDITIONS 9
The second inequality is straightforward: see, for instance, [9, 11, 12, 14]. As we
noted above, the pair (u,σ) satisfies the two-weight A condition (11). Therefore,
p
the Lp(u) norm of the second term is bounded by
kχ k hσi = hui1/phσi1/p′σ(Q )1/p ≤ Cσ(Q )1/p.
Q0 Lp(u) Q0 Q0 Q0 0 0
To estimate the Lp(u) norm of the first term, we formthe following decomposition
(see [11]):
K = K = {Q ⊂ Q : ℓ(I) = 2i+τn}, n ∈ Z ;
i 0 +
1 1
K = {Q ∈ K: 2a 6 huiphσip′ < 2a+1};
a Q Q
Pa = all maximal cubes in K ;
0 a
Pa = maximal cubes P′ ⊂ P ∈ Pa , such that hσi > 2hσi ;
n n−1 P′ P
n o
Pa = Pa.
n
n>0
[
Hereafter we suppress the index i; this will give us a sum with τ +1 terms. Given
Q ∈ K , let Π(Q) denote the minimal principal cube that contains it, and define
a
K (P) = {Q ∈ K : Π(Q) = P}.
a a
We will estimate the Lp(u) norm of the first sum on the right-hand side of (16)
using the exponential decay distributional inequality originated in [16]. (This in-
equality was subsequently improved inthesense ofestablishing theright dependence
on the complexity of the shift in [14, 11].) Below, S is any positive generalized Haar
shift that is bounded on unweighted L2. In particular, we will take S to be one of
the positive Haar shifts S from above.
L
Theorem 3.2. There exists a constant c, depending only on the dimension and the
unweighted L2 norm of the shift, such that for any P ∈ Pa,
σ(P)
u x ∈ P : |S (σ)| > t . e−ctu(P).
Ka(P) |P|
(cid:18) (cid:19)
It follows from Theorem 3.2 that for some positive constant c,
1
σ(P) p p
(17) k S (σ)k 6 Cτ u(P) .
R Lp(u)
|P|
!
R(Q a P∈Pa (cid:18) (cid:19)
X X X
We sketch the proof of (17) following the beautiful calculations in [11]:
τ
S (σ) = S (σ),
R Ka(P)
R(Q i=0 a P∈Pa
X XX X
and so
k S (σ)k 6 (τ +1) k S (σ)k .
R Lp(u) Ka(P) Lp(u)
R(Q a P∈Pa
X X X
10 DAVIDCRUZ-URIBE,SFO,ALEXANDERREZNIKOV,AND ALEXANDERVOLBERG
Fix a. Using Fubini’s theorem we write
k S (σ)k
Ka(P) Lp(u)
P∈Pa
X
p 1/p
= χ S (σ)(x) u(x)dx
{SKa(P)(σ)∈(j,j+1)v|(PP|)} Ka(P)
(cid:18)Z (cid:18) j P∈Pa (cid:19) (cid:19)
X X
p 1/p
σ(P)
≤ (j +1) χ u(x)dx .
{SKa(P)(σ)∈(j,j+1)σ|(PP|)} |P|
j (cid:18)Z (cid:20)P∈Pa (cid:21) (cid:19)
X X
By the choice of the stopping cubes P ∈ Pa we have that
p p
σ(P) σ(P)
χ . χ .
{SKa(P)(σ)∈(j,j+1)σ|(PP|)} |P| {SKa(P)(σ)∈(j,j+1)σ|(PP|)} |P|
(cid:20)P∈Pa (cid:21) P∈Pa (cid:18) (cid:19)
X X
This follows because the ratios σ(P) in the sum on the left are super-exponential.
|P|
This beautiful observation from [11] lets us write
p 1/p
σ(P) σ(P)
k S (σ)k . (j+1) u(S (σ) ∈ (j,j+1) ) .
Ka(P) Lp(u) Ka(P)
|P| |P|
P∈Pa j (cid:18)P∈Pa(cid:18) (cid:19) (cid:19)
X X X
Then by the distributional inequality from Theorem 3.2:
p 1/p
σ(P)
k S (σ)k . (j +1)e−cj/p u(P) .
Ka(P) Lp(u)
|P|
P∈Pa j (cid:18)P∈Pa(cid:18) (cid:19) (cid:19)
X X X
This gives us (17).
It is at this point in the proof that we can no longer assume that our pair of
weights (u,v) satisfies the general A bump condition and we must instead make
p
the more restrictive assumption that we have log bumps. Before doing so, however,
we want to showhowtheproofgoesandwhere theproblem arisesforgeneral bumps.
We will then give the modification necessary to make this argument work for log
bumps.
Define the sequence
|P|, Q = P ,for some cube P ∈ Pa
µ =
Q
0, otherwise;
(
then the inner sum in (17) becomes
p
u(Q) σ(Q)
µ .
Q
|Q| |Q|
QX⊂Q0 (cid:18) (cid:19)
But by H¨older’s inequality in the scale of Orlicz spaces,
σ(Q)
1 1 1 1 1 1
(18) |Q| = hσpσp′iQ 6 Ckσp′kQ,BkσpkQ,B¯ 6 kσp′kQ,Bxin∈Qf MB¯(σpχQ).
Therefore, by (9),
p
u(Q) σ(Q)
(19) |Q| |Q| µQ 6 Kp µQxin∈Qf MB¯(σp1χQ)p.
QX⊂Q0 (cid:18) (cid:19) QX⊂Q0