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There are several treatments of weighted theory in book form: [GCF85] [Duo01] [CUMP11]
[LN15]
...
Some parts of these notes are stubs, in these cases references or
sufficiently common names of results are provided to make the full statements locatable
...
1 (Hardy–Littlewood maximal operator)
...
2 (Marcinkiewicz interpolation)
...

Definition 1
...

Theorem 1
...


ˆ
sup λw{M f > λ} ®
λ>0

2

| f |M w
...
1
...

We fix a measure space (X , µ) and a dyadic grid D
...
2
...


(2
...
We begin with the
weak type estimates since in this case there is an explicit formula for arbitrary pairs of weights
...
4
...

In order to avoid measurability and summability issues we will assume throughout that the dyadic
grid D is finite
...
For notational simplicity we also assume
that all functions are positive
...
5 ([Muc72])
...
Then
0

kM ( f v)k L p,∞ (w) ≤ [v, w](1/p ,1/p) k f k L p (v) ,
where

1/p0

0

[v, w](1/p ,1/p) = sup (v)Q (w)Q

1/p

Q∈D

is a two-weight characteristic
...

The above version of the square bracket notation for weight characteristics has been recently
introduced in [LN15] and is not universally adopted yet
...

1

Proof
...
Then
M ( f v) ≥ (1Q v)Q = (v)Q on Q,
so

1/p0

1/p

kM ( f v)k L p,∞ (w) ≥ (v)Q w({M ( f v) ≥ (v)Q })1/p ≥ (v)Q w(Q)1/p = (v)Q (w)Q k f k L p (v)
...

Now we prove the estimate
...
Notice that the members of Q are pairwise disjoint
...

X

It looks peculiar to estimate M ( f v) and not just M ( f )
...
Hence the above theorem
can be restated in the equivalent form
0

kM ( f )k L p,∞ (w) ≤ [v, w](1/p ,1/p) k f k L p (v 1−p )
...

p

Q∈D

The latter quantity is called the A p characteristic of the weight w
...

Remark (Nestedness of A p classes)
...

We turn to strong type estimates
...
In particular, we may use this with the measure µ replaced by vdµ, where v is a
weight
...

(v)Q
x∈Q∈D
Q
2

For these maximal functions we have the estimate
1 < p < ∞,

kM v f k L p (v) ® p k f k L p (v) ,

where the implicit constant does not depend on v
...

Thee one-weight situation is substantially simpler, so we restrict ourselves to this setting
...
6 ([Buc93, Theorem 2
...
Let 1 < p < ∞ and let w, v be weights with v 1/p w 1/p ≡ 1
...

p

Proof
...
Initialize
S T OC K := D,
S := ;
...

This process terminates after finitely many steps because at each step we remove at least the maximal
elements from S T OC K
...
Since the children Q0 have been chosen after Q, we have ( f v)Q0 > 2( f v)Q
...

2 0
2
2
Q0
Q0
Q
0
0
Q ∈chS (Q)

Q ∈chS (Q)

Q ∈chS (Q)

Therefore the sets E(Q) := Q \ ∪ chS (Q) satisfy |E(Q)| ≥ 12 |Q|, and they are pairwise disjoint
...

Q∈S

Write

ˆ
kM ( f v)k L p (w) ≤ 2

X

( f v)Q 1 E(Q)


p

dw

1/p

Q∈S

ˆ
X
1/p
=2
( f v)Q 1 E(Q) dw
Q∈S

X
1/p
=2
( f )Q,v (v)Q w(E(Q))
Q∈S

X
1/p

1/p

...

In the second term we use the hypothesis on the weights v, w and Hölder’s inequality in the form
ˆ
0
0
|Q| ≤ 2|E(Q)| =
v 1/p w 1/p ≤ v(E(Q))1/p w(E(Q))1/p
...


Theorems 2
...
6 can be extended to the Hardy–Littlewood maximal operator using adjacent
dydic grids
...


2
...

Lemma 2
...
Let w : Rn → (0, ∞), w(x) = |x|α
...


In the above statement notice that p − 1 = p/p0 for Hölder conjugate exponents
...
The A p characteristic is a supremum over all cubes Q ⊂ Rn
...

If dist(Q, 0) ≤ diam(Q), then Q ⊂ B = B(0, R), where R = 2 diam(Q), and |B| ® |Q|
...

A computation shows that the right-hand side does not depend on R, so the supremum over Q is finite
0
if and only if both w and w −p /p are locally integrable, which is equivalent to the claimed condition
on α
...
The following example shows that the dependence on the A p characteristic in
Theorem 2
...

Fix 1 < p < ∞ and consider the power weights w(x) = |x|(p−1)(1−δ) on R1 for small δ
...
Let f (x) = |x|−(1−δ) χ[0,1] (x)
...
Moreover, it is easy to see that M f ≥ δ−1 f , so
p0 /p

kM f k L p (w) ≥ δ−1 k f k L p (w) ¦ [w]A

3

p

k f k L p (w)
...

Definition 3
...
The A1 characterisitc of a weight w is
[w]A1 := sup

Mw
w
4

= sup
x∈Q∈D

(w)Q
w(x)


...


,

Q∈D

Lemma 3
...
If 1 < p < p0 < ∞ then for any weights w, u we have
p −p

[wu p−p0 ]Ap ≤ [w]Ap [u]A0

(3
...
Let Q ∈ D
...


Multiplying these inequalities and taking a supremum over Q we obtain the claim
...
4
...


(3
...
Let Q ∈ D
...


Substituting these inequalities into the definition of the A p0 characteristic and taking the supremum
over Q we obtain the claim
...
6 (Rubio de Francia construction, [RdF84])
...
Then the operator
R f :=

∞ 
X
k=0

M

k

2kM k L p (w)

f

has the following properties
...
7)

kR f k L p (w) ≤ 2k f k L p (w)

(3
...


(3
...
10
...
Then for every 1 < p < ∞ and w ∈ A p we
have
ˆ
ˆ

1/p
p
1/p
g w
≤ K(w)
f pw
,
5

where

(
K(w) =

2N ([w]Ap (2kM k L p (w) ) p0 −p ),
2

p0 (1/p0 −1/p)

(p −1)/(p−1)
N ([w]A 0
(2kM k L p0 (w 1−p0 ) )(p−p0 )/(p−1) ),
p

max(1,(p0 −1)/(p−1))

In particular, K(w) ≤ C N (C[w]A
p
function
...


) if M is the dyadic or the Hardy–Littlewood maximal

Corollary 3
...
Let 1 < p0 < ∞
...


n

P
P
Proof
...
10 with functions ( n | f n | p0 )1/p0 and ( n |Tn f n | p0 )1/p0
...

Applying the corollary twice we obtain
Corollary 3
...
If Tn are operators (not necessary linear) that are bounded on all L p0 (w) with w ∈ A p0
with constant depending only on the weight characteristic (but not on n), then
X
X
k( |Tn f n |q )1/q k L p (w) ≤ K([w]Ap )k( | f n |q )1/q k L p (w) , 1 < p, q < ∞
...
Theorem 3
...

Consider for instance the Hardy–Littlewood maximal function that is bounded on L 2 (w) with norm
® [w]A2
...
10 yields that for p > 2 the maximal function
is bounded on L 2 (w) with norm ® [w]Ap , which is worse than the conclusion of Theorem 2
...
On the other hand, Theorem 4
...
10
...
Hölder’s inequality for strictly positive functions f i > 0 can be formulated as follows:
ˆ Y
Yˆ Y a
a
i, j
fi i ≤
(
fi )b j
i

j

j

P
P
if 0 ≤ b j , j b j = 1, ai , ai, j ∈ R, and j ai, j b j = 1
...

Proof
...

ˆ
ˆ
 p/p0  ˆ
1−p/p0
p
p0
p−p0
g w≤
g w(R f )
w(R f ) p
ˆ
 p/p0  ˆ
1−p/p0
p−p0
p
p0
p−p0
≤ N ([w(R f )
]A p )
f w(R f )
w(R f ) p
0
ˆ
p0 −p p
≤ N ([w]Ap [R f ]A )
w(R f ) p
1
ˆ
p
p0 −p p
≤ 2 N ([w]Ap (2kM k L p (w) )
)
f pw
In the case p > p0 let H = w g p−1 so that kHk

ˆ

ˆ
p

g w=

p0

p

0

L p (w 1/(1−p) )

g p0 w (p0 −1)/(p−1) H (p−p0 )/(p−1)
6

= kgk L p (w) and write

by Hölder
by hypothesis
by (3
...
7)
by (3
...
8)
...


ˆ

]A p )

p0

0

f pw

 p0 /p

ˆ

(2kM k L p0 (w 1/(1−p) ) )

f p0 w (p0 −1)/(p−1) (RH)(p−p0 )/(p−1)
w 1/(1−p) (RH) p/(p−1)

(p−p0 )
(p−1)

)

p0

ˆ

p

f w

by hypothes

1−p0 /p

 p0 /p  ˆ

w 1/(1−p) H p

by (3
...
9) and (3
...


4

Sparse operators

So far we only had one example of an operator that is bounded on A p weighted spaces, namely the
maximal operator
...

Definition 4
...
Let D be a dyadic grid and 0 < η ≤ 1
...

Usually we will not specify the parameter η and just talk about “sparse collections”
...

In the previous lecture we have seen one example of a sparse collection that was constructed
using a stopping time
...
2
...

Q∈S

This operator is strictly larger than the operator that we have encountered in the estimate for the
maximal operator
...

Over the last few years people have found out that sparse operators control many different
interesting oeprators, beginning with Calderón–Zygmund operators
...
3
...

For the time being we concentrate on weighted estimates for sparse operators
...
4
...


The implicit constant does not depend on the collection S or the weight w
...

0

Proof
...
To this end write the left-hand side as
X
X
0
0
( f v)Q (g w)Q |Q| =
( f )Q,v v(E(Q))1/p (g)Q,w w(E(Q))1/p |Q|(v)Q (w)Q v(E(Q))−1/p w(E(Q))−1/p
...


Q∈S

Q

Q∈S

The first thow terms are bounded by kM v f k L p (v) and kMw gk L p0 (w) , respectively, and we can use the
martingale maximal inequality in both
...
Fix Q
...

There are now two cases, p ≤ p0 and p ≥ p0 (equivalently, p ≤ 2 and p ≥ 2)
...

Time permitting: show that S is η-sparse iff 1S is 1/η-Carleson (reference [LN15, Lemma 6
...


5

Calderón–Zygmund (CZ) theory

In this lecture we cover some standard material which can be found e
...
in [Gra14] or [Ste93]
...
1
...
The Dini norm of a modulus of continuity is

ˆ
kωkDini =

1

ω(t)
0

dt
t


...

Definition 5
...
Let ω be a modulus of continuity
...
|K(x, y)| ≤ CK |x − y|−d for some CK < ∞ and all x, y ∈ Rn with x 6= y,
2
...

An ω-CZ operator is a linear operator, initially defined on bounded compactly supported measurable
functions on Rn with values in L 1 (Rn ) + L ∞ (Rn ) that has an associated ω-CZ kernel K such that for
all functions f and points x 6∈ supp( f ) we have
ˆ
T f (x) = K(x, y) f ( y)d y
...
The use of the constant CK is traditional; it can be replaced by a qualitative off-diagonal
decay
...


8


...
1

CZ decomposition

TheoremP5
...
Let f ∈ L 1 (Rn ) and λ > 0
...
kgk1 ≤ k f k1 ,
2
...
Q is a collection of pairwise disjoint dyadic cubes,
P
−1
4
...
kbQ k1 ≤ 2n+1 λ|Q| for every Q ∈ Q,
´
6
...

Sketch
of proof
...
Let bQ = 1Q ( f −
ffl
f
)
...

It follows that kgk∞ ® λ (proved using the fact that | f | ≤ M f pointwise a
...
that is obtained by a
density argument/Lebesgue differentiation theorem), and kgk1 ≤ k f k1
...
4
...
Then
kT k L 1 →L 1,∞ ®d kT k L 2 →L 2 + kωkDini
...
The claim is non-void only if T is bounded on L 2
...
By homogeneity if suffices to show
|{T f > 1}| ® (1 + kωkDini )k f k1
for bounded compactly supported functions f
...
This expansion
in fact converges (unconditionally) in L 2 by the qualitative assumptions on
P
f , so T f = T g + Q∈Q T bQ with unconditional convergence in L 2
...


9

ˆ

Q

Q

|K(x, y) − K(x, x Q )||bQ ( y)|d ydx

ˆ

Q

|K(x, y) − K(x, x Q )||bQ ( y)|d ydx

Hence with Ω = ∪Q∈Q 10Q we have
X
X
k
T bQ k L 1 (Rd \Ω) ® kωkDini
|Q| ® kωkDini k f k1
Q

Q∈Q

Therefore
|{T f > 1}| ≤ |Ω| + |{

X

T bQ > 1/2} ∩ Ωc | + |{T g > 1/2}|

Q

® k f k1 + k

X

T bQ k L 1 (Rd \Ω) + kT gk22

Q

® (1 + kωkDini )k f k1 ,
where we have used kgk22 ≤ kgk1 kgk∞ ® k f k1
...
2

Cotlar’s inequality

Define the maximally truncated operator
T] f (x) :=

ˆ
sup

">0,|x−x 0 |≤"/2

B(x 0 ,")c

K(x 0 , y) f ( y)d y
...
e
...

Lemma 5
...

T] f ®d,δ (kT k L 2 →L 2 + kωkDini )M f + Mδ T f
...
6)

Here Mδ f = (M ( f δ ))1/δ , 0 < δ < 1, where M is the usual Hardy–Littlewood maximal function
...

Proof
...


The first term is estimated using the kernel bound by CK M f (x)
...

k
k+1
k
k+1
(2k ")d
k>0 2 "≤|x− y|<2 "
k>0 2 "≤|x− y|<2 "
The middle term equals
T ( f 1B(x,2")c )(x 00 ) = T ( f )(x 00 ) − T ( f 1B(x,2") )(x 00 ),
where we have used that T is associated to K and linearity of T
...
The contribution of the former term is then clearly bounded by Mδ T f (x)
...
4
...
7
...
6) by M1/2 T f , where
Mλ f (x) = sup( f 1Q )∗ (λ|Q|)
x∈Q

∗

and f denotes the non-increasing rearrangement of f
...
3

Marcinkiewicz interpolation theorem, L p,∞ version

We need the following version of the Marcinkiewicz interpolation theorem in which the conclusion is
a bound on a weak L p space
...
8
...
Let 0 < θ < 1 and 1/pθ = (1 − θ )/p0 + θ /p1
...

Proof
...

Then
{|T f | > η} ≤ {|T f0,λ | > η/(2C)} + {|T f1,λ | > η/(2C)}
® η−p0 kT f0,λ k pp0 ,∞ + η−p1 kT f1,λ k pp1 ,∞
0

1

® η−p0 k f0,λ k pp0 + η−p1 k f1,λ k pp1
0
ˆ
ˆ 1
≤ η−p0
| f | p0 + η−p1
| f | p1
| f |>λ
| f |≤λ
Xˆ
Xˆ
p0
−p0
−p1
≤η
|f | + η
k≥0 | f

|∼2k λ

k≤0 | f

|∼2k λ

| f | p1

X
X
≤ η−p0
(2k λ) p0 −pθ k f k pθ ,∞ + η−p1
(2k λ) p1 −pθ k f k pθ ,∞
...
Hence
{|T f | > η} ® η−p0 (λ) p0 −pθ k f k pθ ,∞ + η−p1 (λ) p1 −pθ k f k pθ ,∞
...


Corollary 5
...
The maximal operator Mδ is bounded on L 1,∞ for 0 < δ < 1
...
By Theorem 5
...
Hence
kMδ f k1,∞ = kM ( f δ )k1/δ,∞ ® k f δ k1/δ,∞ = k f k1,∞
...
Many
simplifications have been made since then The two key simplifications were the introduction of sparse
domination by Lerner [Ler13] and a simple algorithm for constructing sparse collections by Lacey
[Lac15], a streamlined version of which appears in [HRT15]
...

The main example that I am aware of where sharp weighted estimates are useful is the regularity
theory for solutions of the Beltrami equation in [AIS01]
...
The associated A∞
characteristic is defined by
ˆ
[w]A∞ := sup w(Q)−1
Q∈D

11

Q

M (w1Q )
...
1
...

0

Proof
...
Fix Q 0 ∈ D and construct the (minimal)
stopping collection S by the rules
1
...
If Q ∈ S , then the maximal cubes Q0 ⊂ Q with (w)Q0 ≥ 2(w)Q are in S
...
The pairwise disjoint major subsets E(Q) ⊂ Q ∈ S satisfy
ˆ
ˆ
0
0
|Q| ∼ |E(Q)| =
1=
w 1/p v 1/p ≤ w(E(Q))1/p v(E(Q))1/p
E(Q)

by Hölder’s inequality
...
2)

E(Q)

ˆ
M (w1Q 0 ) ®
=

X

Q 0 Q∈S

X

1 E(Q) (w)Q

|E(Q)|(w)Q

Q∈S

=

X

w(E(Q))

|E(Q)|(w)Q
w(E(Q))

Q∈S

≤ sup

|E(Q)|(w)Q X
w(E(Q))

Q∈S

≤ sup

|E(Q)|(w)Q
w(E(Q))

Q∈S

Hence it suffices to show

|E(Q)|(w)Q
w(E(Q))

w(E(Q))

Q∈S

w(Q)
...
To this end multiply the left-hand side by (7
...


Lemma 7
...
Let w be a weight, λ > 0, and Q ∈ D maximal with (w)Q > λ
...
Moreover,
ˆ
Q

M w ≤ 2d [w]A∞ |Q|λ

and
w(Q) ≤ 2d |Q|λ
...
The first conclusion is clear because cubes that strictly contain Q have a smaller contribution
ˆ be the dyadic parent of Q, then
to the maximal function than Q
...

Q
Q

Q

ˆ
Q

Q

The third conclusion is even easier:
ˆ = (w)Qˆ |Q|
ˆ ≤ 2d λ|Q|
...
4 ([HPR12, Lemmas 2
...
3])
...

2d+1 [w]A∞ −1

Then for every Q 0 ∈ D

M (w1Q 0 ) ≤ 2[w]A∞ (w)Q1+" ,

M (w1Q 0 )" w ≤ 2(w)Q1+"
...
5)
(7
...
For notational convenience suppose w = w1Q 0 and Q 0 is the unique maximal element of D
...

(7
...

Q0

Q0

0

0

We split this integral at λ = (w)Q 0
...

0

For λ > (w)Q 0 the superlevel set {M w > λ} is the union of the collection Qλ of maximal dyadic cubes
Q ⊂ Q 0 with
(w)Q > λ
...
5)
...
3 that
M w{M w > λ} ≤ 2d [w]A∞ |{M w > λ}|λ,
so using (7
...

Q0

Notice that the fraction on the right-hand side equals 1/2 by the hypothesis
...
5) follows
...
6)
...
3 we obtain
w{M w > λ} ≤ 2d λ|{M w > λ}|
...
7) and (7
...
The conclusion follows
...
8 (Open property)
...
Then [w]A˜p ® [w]Ap , where ˜p =
p−

p−1
2d+1 [v]A∞

0

< p and v is the dual weight: w 1/p v 1/p ≡ 1
...
The exponent ˜p is chosen in such a way that 1 + " = (p/p0 )(˜p0 /˜p), where " is as in Lemma 7
...
Consider the dual weight v˜ = w −˜p /˜p
...
6) applied to the weight v we
have
(˜
v )Q = (v 1+" )Q ≤ 2(v)Q1+"
...
1

(1+")˜p/˜p0

® (w)Q (v)Q

p/p0

= (w)Q (v)Q

≤ [w]Ap
...

It is not hard to show that A p weights are doubling if p < ∞
...

Exercise 7
...
Find a weight that is A∞ with respect to the standard dyadic filtration but not A∞ (Rd )
...
10
...

To combat these difficulties we define the A∞ (Rd ) by
ˆ
−1
[w]A∞ (Rd ) = sup w(Q)
M (w1Q ),
Q

Q

where the supremum is taken over all non-empty axis-parallel cubes in Rd
...
11
...


The converse is not true: there exist doubling weights that are not A∞ , see [FM74] (a different
version of the A∞ condition was used there)
...
Let k > C[w]A∞ (Rd ) be an integer, where C is a large constant to be chosen later
...
The claim then follows by iterating this estimate log2 k times
...
Also, it suffices to estimate w(P),
˜ since Q
˜ is the union
where P is a parallelepiped of dimensions 1 × · · · × 1 × 2−k at the boundary of Q
of finitely many such parallelepipeds and the cube Q
...
Estimating the maximal
function on this strip by the averages of scale 2−l we obtain
ˆ
ˆ
ˆ
M (w1Q˜ ) ≥
M (w1 P ) ¦ w1 P
...

˜
Q

P

Summing up these estimates for finitely many P’s we obtain
ˆ
ˆ
M (w1Q˜ ) ¦ k
w
...

˜
Q\Q

˜ \ Q) ≤ w(Q)/2
˜
If k was chosen sufficiently large in terms of [w]A∞ this implies w(Q
and consequently
˜ ≤ 2w(Q)
...

where Q
Recall that A∞ weights satisfy the reverse Hölder inequality
(w 1+" )Q ≤ 2(w)Q1+" ,

1/" ∼ [w]A∞
...
12
...
Then w ∈ A r provided r ≥ C41/" log(2Cd b )
...
This quantitative dependence
has been noted in [HP14, Theorem 1
...
e
...
We follow the proof in [Ste93, p
...
5
...

Proof
...
Fix Q 0 ∈ D and
0
let f = w −1 1Q 0
...
We may normalize |Q 0 | = 1 by scaling and
0
w(Q 0 ) = 0 by multiplying w by an absolute constant
...

Q

15

Notice that for Q ∈ Qk we have
ˆ
ˆ
ˆ
−1
−1
−1
−1
ˆ
ˆ
w(Q)
f w ≤ w(Q)
f w = w(Q) w(Q)w(Q)
f w ≤ Cd b N k
...
Then
X

ˆ

X

w(Q0 ) ≤ N −k

Q0

Q0 ∈Qk :Q0 ⊂Q

Q0 ∈Qk :Q0 ⊂Q

ˆ
Q

ˆ
f w ≤ N −k
Q

f w ≤ N −k Cd b N k−1 w(Q) = N −1 Cd b w(Q)
...
Then w(Ek ∩ Q) ≤ w(Q)/2
...

Summing over Q ∈ Qk−1 we obtain
|Ek |/|Ek−1 | ≤ 1 − 2−2/"−1
...


Corollary 7
...
A∞ (Rd ) = ∪ p<∞ A p (Rd )
...
Now we refine
these estimates following [HP13]
...
1
...
Then
0

1/p

kM ( f v)k L p (w) ® p [v, w]1/p ,1/p [v]A k f k L p (v)
...
Moreover, the A∞ characteristic
p
above can be substantially smaller than [v]Ap0 (example: power weights)
...
We start with the stopping collection S as in Lecture 2 which is sparse with respect to the
reference measure µ and such that
X
M ( f v) ®
( f v)Q 1 E(Q)
...
Denote
−1
( f )Q,v = v(Q)
Q f v = ( f v)Q /(v)Q
...
Then for each Q ∈ F we add all maximal Q0 ⊂ Q with Q0 ∈ D and
( f )Q0 ,v > 2( f )Q,v to F
...
Notice that
ˆ
ˆ
X
X
p
p
p
˜
( f ) F,v v( E (F )) ® (M v f ) v ® ( f ) p v
( f ) F,v v(F ) ®
F ∈F

F ∈F

p

by the L estimate for the weighted maximal function M v
...

The next objective is a similar estimate for sparse operators
...
Moreover, the estimate for sparse operators should be symmetric in the weights v and
w (by duality)
...

Proposition 8
...
Let 1 < p < ∞ and v, w be weights
...
Then
0

1/p0

1/p

kAS ( f v)k L p (w) ® [v, w]1/p ,1/p ([v]A + [w]A )k f k L p (v)
...
By duality kAS ( f v)k L p (w) = sup g:kgk p0 =1 AS ( f v)g w, so it suffices to show
L

X

(w)

0

1/p

1/p0

∞

∞

( f v)Q (g w)Q |Q| ® [v, w]1/p ,1/p ([v]A + [w]A

Q∈S

)k f k L p (v) kgk L p0 (w)
...

Then the left-hand side above is bounded by
X
X
X
( f ) F,v
(g)G,w
(v)Q (w)Q |Q|
...
Since both parts are symmetric (under
interchanging f with g, v with w, and p with p0 ) we consider only the second
...


G:πF (G)=F Q∈S :πF (Q)=F,πG (Q)=G

F

The first bracket is bounded by kMw gk L p0 (w) ® kgk L p0 (w)
...


Q∈S :πF (Q)=F

F

Multiplying and dividing each summand by v(F ) and observing that
ˆ
X
p
( f ) F,v v(F ) ® (M v f ) p v ® k f k L p (v)
F

ˆ

it remains to show

X

v(F )−1

(v)Q 1Q


p

w ® [v, w] p−1,1 [v]A∞
...
To get some feeling for what is going on let us first consider the case
p = 2
...

For general p we use the numerical inequality
X
X
X
(
ai ) p ®
ai1 · · · aibpc (
ai ){p} 7
(8
...



{p}

w
...
Then we estimate this by
X
X

{p}
=
(v)Q 1 w(Q 1 ) w(Q 1 )−1
(v)Q (w)Q |Q|
F ⊇Q 1

Q⊆Q 1

X

0

≤ [v, w]{p}(p/p ,1)

(v)Q 1 w(Q 1 ) w(Q 1 )−1

F ⊇Q 1

X

1−p/p0

(v)Q


{p}
|Q|

Q⊆Q 1

Using Lemma 8
...


Consider now the case p ≥ 2, so that 1/p ≤ 1/p0
...
4
1−p0 /p
{p}

X

0

® [v, w]{p}(1,p /p)

(v)Q 1 · · · (v)Q bpc w(Q bpc ) w(Q bpc )−1 |Q bpc |(w)Q

F ⊇Q 1 ⊇···⊇Q bpc
−{p}p0 /p

X

0

= [v, w]{p}(1,p /p)

(v)Q 1 · · · (v)Q bpc w(Q bpc )(w)Q

F ⊇Q 1 ⊇···⊇Q bpc
0

bpc

1−{p}p0 /p−p0 /p

X

0

≤ [v, w]{p}(1,p /p)+(1,p /p)

bpc

(v)Q 1 · · · (v)Q bpc −1 |Q bpc |(w)Q

F ⊇Q 1 ⊇···⊇Q bpc

bpc

Using Lemma 8
...

bpc −1

(v)Q 1 · · · (v)Q bpc −1 |Q bpc−1 |(w)Q

F ⊇Q 1 ⊇···⊇Q bpc−1

Continuing in this manner we obtain inductively
X
0
0
1−{p}p0 /p−mp0 /p
® [v, w]{p}(1,p /p)+m(1,p /p)
(v)Q 1 · · · (v)Q bpc −m |Q bpc−m |(w)Q −m

...


In the above proof we have used repeatedly the following fact
...
4
...
Then for every non-negative function v we have
X
β
β
|Q|(v)Q ® |F |(v) F
...

Proof
...

Proof of (8
...
The claim (8
...

(8
...
For real a, b ≥ 0 we
have
ˆ a+b
ˆ a+b
p
p
p−1
p
p−1
(a + b) = a +
pt
dt ≤ a + p(a + b)
dt = a p + p(a + b) p−1 b
...
5) follows by induction on m
...
1

Weighted weak type (1, 1) for sparse operators
Orlicz spaces

Definition 9
...
A Young function is a convex increasing function ϕ : [0, ∞) → [0, ∞) such that
ϕ(0) = 0 and lim t→∞ ϕ(t) = ∞
...
2
...
Then
ψ(s) = sup(st − ϕ(t))
t>0

is also a Young function, called the complementary Young function of ϕ
...
All properties are easy to verify with the possible exception of convexity
...
Then
ψ((1 − λ)s0 + λs1 ) = sup(((1 − λ)s0 + λs1 )t − ϕ(t))
t>0

= sup((1 − λ)(s0 t − ϕ(t)) + λ(s1 t − ϕ(t)))
t>0

≥ sup(1 − λ)(s0 t − ϕ(t)) + sup λ(s1 t − ϕ(t))
t>0

t>0

= (1 − λ)ψ(s0 ) + λψ(s1 )
...
If ϕ(t) = t p , then ψ(s) = s p (exercise)
...
3
...
The Orlicz space ϕ(L)(X , µ)
is defined by
ˆ
k f kϕ = inf{Λ > 0 :

ϕ(| f |/Λ) ≤ 1}
...

Lemma 9
...
Let ϕ be a continuous Young function and ψ its complementary Young functions
...

g:kgkψ ≤1

X

Proof
...
Notice that ψ(s)+ϕ(t) ≥ ts for all t, s > 0
...

Then
ˆ
ˆ
| f g| ≤
ϕ(| f |) + ψ(|g|) ≤ 2
...
Suppose
now that k f kϕ > 1 and without loss of
´
generality f ≥ 0
...

On the other hand,

ˆ

ˆ
f gΛ =

ˆ
Λϕ( f /Λ) →

as Λ → 1
...
5 ([DSLR16, Theorem 1
...
Let ϕ be a Young function and ψ its complementary function
...
Then
sup λw{AS f > λ} ®
λ>0

ˆ

∞
X

1

k=1

ψ−1 (22 )

| f |Mϕ w,

k

where ψ−1 denotes the inverse function of ψ and
Mϕ w(x) = sup (w)Q,ϕ ,

(w)Q,ϕ = inf{ν > 0 :

x∈Q∈D

ϕ(w/ν) ≤ 1}
...
6
...
5 applies if ϕ(t) = t L(t) with 0 ≤ s L 0 (s) ≤ C and
particular e
...
if L(t) = ln ln t(ln ln ln t)1+"
...

ψ(L(t)) = sup(L(t)τ − ϕ(τ)) = sup τ(L(t) − L(τ)) ≤ sup τ(L(t) − L(τ))
τ>0

0<τ≤t

0<τ≤t

ˆ

® sup τ
0<τ≤t

t

τ

s−1 ds = sup τ ln(t/τ) ® t
...

It is known that Theorem 9
...
We will prove an earlier result
that it fails if ϕ(t) = t [RT12]
...

Lemma 9
...
[CUP00])
...
Assume that
kT 0 f k L 1,∞ (w) ® k f k L 1 (M w)
...


Proof
...
Note
M (w1Ω )(x) ®

w(I) 

sup

x∈I∈∪α D α :w(I)6=0

I

ˆ
−1

w(I)

I

1Ω w) ≤ sup MD α (w)(x)MD α ,w (1Ω )(x)
...
Then

ˆ
w(Ω) ®
W

ˆ
Xˆ
2
2 −1 1/2
| f |M (w1Ω ) ≤
( | f | MD α (w) w ) (
MD α ,w (1Ω )w)1/2
α

W

W

ˆ
® ( | f |2 M (w)2 w −1 )1/2 w(Ω)1/2 ,
W

where we have used the L 2 estimate for the weighted dyadic maximal function
...

By duality for functions g supported in W we have

22

ˆ
kT gk L 2 (W,(M w)−2 w) = k(M w)−1 w 1/2 T gk L 2 (W ) =

ˆ
=

k f k L 2 (W ) =1

hT g| =

|

sup

khk L 2 (W,(M w)2 w −1 ) =1

|

sup

f (M w)−1 w 1/2 T g|

ˆ
|

sup

khk L 2 (W,(M w)2 w −1 ) =1

(T 0 h)g|

ˆ
≤

sup

|

f g|
...


Construct collections of intervals in R as follows
...
Let J1 = {[0, 1]}
...

For each l and J ∈ Jl let P(J) be an interval of length 3−k |J| situated either to the left or to
the right from 13 J (we will decide later on which side each P(J) is situated)
...
Let Ωl = ∪J∈Jl P(J), Ω0l = ∪J∈Jl 13 P(J), and consider the weight
w=

∞ 
X
l=1

3k
3k−1 + 1

l

1Ω l
...
8
...

Proof
...
On Ωl 0 with l 0 ≤ l we have w ≤ w(x), so it suffices to consider

contributions of Ωl 0 with l 0 > l
...

By construction w(J 0 ) = w(P(J)) for each J and J 0 ∈ ch(J), and it follows that on each interval of
length |I| the mass of w does not exceed w(I)
...
9
...

Proof
...
Split
ˆ
ˆ
ˆ
w( y)
w( y)
w( y)
dy =
dy +
dy
1
y−x
y−x
I y−x
J
3
ˆ
ˆ 
w( y)
w( y)
w( y) 
+
dy +
−
d y
...
The third summand only depends on the
choices of P(J) for J ∈ Jl 0 with l 0 < l
...
Choose P(J) so
that the sign of this term matches the sign of the third term
...
7, we obtain
ˆ
ˆ
ˆ
ˆ
2
2
2
−2
k
w®k
w ® |H w| (M w) w ® w
∪l Ω0l

and this is a contradiction for large k
...
Astala, T
...
Saksman
...
In: Duke Math
...
107
...
27–56
...
M
...
“Estimates for operator norms on weighted spaces and reverse Jensen inequalities”
...

Amer
...
Soc
...
1 (1993), pp
...


[CLO17]

M
...
K
...
Ombrosi
...
In: Proc
...
Math
...
145
...
3005–3012
...
V
...
M
...
Pérez
...
Vol
...
Operator Theory: Advances and Applications
...
xiv+280
...
Cruz-Uribe and C
...
“Two weight extrapolation via the maximal operator”
...
Funct
...
174
...
1–17
...
Domingo-Salazar, M
...
Rey
...
In: Bull
...
Math
...
48
...
63–73
...
01804 [math
...


[Duo01]

J
...
“Fourier analysis”
...
29
...
Translated and revised
from the 1995 Spanish original by David Cruz-Uribe
...
xviii+222
...
Duoandikoetxea
...
In: J
...
Anal
...
6 (2011), pp
...


[FM74]

C
...
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...
In: Proc
...

Math
...
45 (1974), pp
...


[FS71]

C
...
M
...
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...
J
...
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...


[GCF85]

J
...
L
...
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...
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...
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...
, Amsterdam, 1985, pp
...


[Gra14]

L
...
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...
Vol
...
Graduate Texts in Mathematics
...
xviii+638
...
P
...
Li
...

Preprint
...
arXiv:1509
...
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...
Hytönen and C
...
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...
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...
777–818
...
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...


[HP14]

P
...
Hagelstein and I
...
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embedding of A∞ into A p ”
...
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...
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...


[HPR12]

T
...
Pérez, and E
...
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type”
...
Funct
...
263
...
3883–3899
...
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...


[HRT15]

T
...
Hytönen, L
...
Tapiola
...
In: Israel J
...
(2015)
...
arXiv:1510
...
CA]
...
P
...
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...
of Math
...
3 (2012), pp
...
arXiv:1007
...
CA]
...
John and L
...
“On functions of bounded mean oscillation”
...
Pure Appl
...
14 (1961),
pp
...


[Lac15]

M
...
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...
In: Israel J
...
(2015)
...
arXiv:1501
...
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K
...
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...
Math
...
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...
3159–3170
...
2824 [math
...


[Ler16]

A
...
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...
In: New York J
...
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...
arXiv:1512
...
CA]
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K
...
Nazarov
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Preprint
...
arXiv:1508
...
CA]
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Moen
...
In: Arch
...
(Basel) 99
...
457–466
...
4207 [math
...


24

[Muc72]

B
...
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...
Amer
...
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...
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...
Nazarov, A
...
Vasyunin, and A
...
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conjecture: the blow-up of the weak norm estimates of weighted singular operators”
...
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[NRVV16]

F
...
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...
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...

Preprint
...
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...
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[RdF84]

J
...
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...
In: Amer
...
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...
3 (1984), pp
...


[RT12]

M
...
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...
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...
Res
...

19
...
1–7
...
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...


[Ste93]

E
...
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...
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...

Princeton Mathematical Series
...
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III
...
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...
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...
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---