On pointwise estimates involving sparse operators
Andrei Lerner
Bar-Ilan University
10th International Conference on Harmonic Analysis and PDEs
El Escorial (Spain)
June 12-17, 2016
Sparse families and operators
Given 0 < η < 1, we say that a family S of cubes from Rn is
η-sparse if for any Q ∈ S there is a subset EQ ⊂ Q such that
1
2
|EQ | ­ η|Q|;
the sets {EQ }Q∈S are pairwise disjoint.
Sparse families and operators
Given 0 < η < 1, we say that a family S of cubes from Rn is
η-sparse if for any Q ∈ S there is a subset EQ ⊂ Q such that
1
2
|EQ | ­ η|Q|;
the sets {EQ }Q∈S are pairwise disjoint.
1 R
Denote fQ =
|Q| Q f ,
and define the dyadic maximal operator:
M D f (x) =
sup
|f |Q ,
Q3x,Q∈D
where D = {2−k ([0, 1)n + j), k ∈ Z, j ∈ Zn }.
Sparse families and operators
Given 0 < η < 1, we say that a family S of cubes from Rn is
η-sparse if for any Q ∈ S there is a subset EQ ⊂ Q such that
1
2
|EQ | ­ η|Q|;
the sets {EQ }Q∈S are pairwise disjoint.
1 R
Denote fQ =
|Q| Q f ,
and define the dyadic maximal operator:
M D f (x) =
sup
|f |Q ,
Q3x,Q∈D
where D = {2−k ([0, 1)n + j), k ∈ Z, j ∈ Zn }.
The standard claim (80’s): for every f ∈ L1 (Rn ), there is a 12 -sparse
family S ⊂ D such that
X
M D f (x) ¬ 2n+1
|f |Q χEQ (x).
Q∈S
Sparse families and operators
Given 0 < η < 1, we say that a family S of cubes from Rn is
η-sparse if for any Q ∈ S there is a subset EQ ⊂ Q such that
1
2
|EQ | ­ η|Q|;
the sets {EQ }Q∈S are pairwise disjoint.
1 R
Denote fQ =
|Q| Q f ,
and define the dyadic maximal operator:
M D f (x) =
sup
|f |Q ,
Q3x,Q∈D
where D = {2−k ([0, 1)n + j), k ∈ Z, j ∈ Zn }.
The standard claim (80’s): for every f ∈ L1 (Rn ), there is a 12 -sparse
family S ⊂ D such that
X
M D f (x) ¬ 2n+1
|f |Q χEQ (x).
Q∈S
M D f (x)
Proof: write Ωk = {x :
> 2(n+1)k } = ∪j Qkj and set
Ejk = Qkj \ Ωk+1 . Then the claim holds with S = {Qkj }.
Sparse families and operators
The standard claim (80’s): for every f ∈ L1 (Rn ), there is a 12 -sparse
family S ⊂ D such that
M D f (x) ¬ 2n+1
X
|f |Q χEQ (x).
Q∈S
A one-third trick: there are 3n dyadic lattices D (j) such that for every
cube Q ⊂ Rn , there is a cube P ∈ D (j) for some j, containing Q and
such that |P | ¬ 6n |Q|.
Sparse families and operators
The standard claim (80’s): for every f ∈ L1 (Rn ), there is a 12 -sparse
family S ⊂ D such that
M D f (x) ¬ 2n+1
X
|f |Q χEQ (x).
Q∈S
A one-third trick: there are 3n dyadic lattices D (j) such that for every
cube Q ⊂ Rn , there is a cube P ∈ D (j) for some j, containing Q and
such that |P | ¬ 6n |Q|.
Hence, the usual maximal operator M is bounded as follows:
n
M f (x) ¬ 6
n
3
X
j=1
M D f (x).
(j)
Sparse families and operators
The standard claim (80’s): for every f ∈ L1 (Rn ), there is a 12 -sparse
family S ⊂ D such that
M D f (x) ¬ 2n+1
X
|f |Q χEQ (x).
Q∈S
A one-third trick: there are 3n dyadic lattices D (j) such that for every
cube Q ⊂ Rn , there is a cube P ∈ D (j) for some j, containing Q and
such that |P | ¬ 6n |Q|.
Hence, the usual maximal operator M is bounded as follows:
n
M f (x) ¬ 6
n
3
X
M D f (x).
(j)
j=1
By the standard claim, for every f ∈ L1 (Rn ), there are 12 -sparse
families Sj ⊂ D (j) , j = 1, . . . , 3n , such that
n
n
M f (x) ¬ 2 · 12
3
X
X
j=1 Q∈Sj
|f |Q χEQ (x).
Sparse families and operators
By the standard claim, for every f ∈ L1 (Rn ), there are 12 -sparse
families Sj ⊂ D (j) , j = 1, . . . , 3n , such that
n
n
M f (x) ¬ 2 · 12
3
X
X
|f |Q χEQ (x).
j=1 Q∈Sj
We will show that almost the same pointwise domination holds for
Calderón-Zygmund operators T :
n
|T f (x)| ¬ C(n, T )
3
X
X
j=1 Q∈Sj
|f |Q χQ (x).
Sparse families and operators
We will show that almost the same pointwise domination holds for
Calderón-Zygmund operators T :
n
|T f (x)| ¬ C(n, T )
3
X
X
|f |Q χQ (x).
j=1 Q∈Sj
We say that T is an ω-Calderón-Zygmund operator if
1
2
T is L2 bounded;
T is represented as
Z
T f (x) =
K(x, y)f (y)dy
for all x 6∈ supp f ;
Rn
K satisfies the size condition |K(x, y)| ¬
4 K satisfies the regularity condition
3
0
CK
|x−y|n , x
0
|K(x, y) − K(x , y)| + |K(y, x) − K(y, x )| ¬ ω
6= y;
|x − x0 |
|x − y|
1
|x − y|n
for |x − y| > 2|x − x0 |, where ω : [0, 1] → [0, ∞) is continuous,
increasing, subadditive and ω(0) = 0.
Sparse families and operators
We will show that almost the same pointwise domination holds for
Calderón-Zygmund operators T :
n
|T f (x)| ¬ C(n, T )
3
X
X
|f |Q χQ (x).
j=1 Q∈Sj
If S is a sparse family, then the operator
AS f (x) =
X
Q∈S
is called the sparse operator.
fQ χQ (x)
A very brief history
The standard methods of 70’s-80’s (good-λ inequalities,
rearrangement inequalities, sharp-function estimates) provide an
indirect relation between T and the maximal operator M .
A very brief history
The standard methods of 70’s-80’s (good-λ inequalities,
rearrangement inequalities, sharp-function estimates) provide an
indirect relation between T and the maximal operator M .
The A2 conjecture (90’s):
kT kL2 (w) ¬ c(n, T )[w]A2 ,
where [w]A2 = supQ
1 R
|Q| Q w
1 R
−1
|Q| Q w
.
A very brief history
The standard methods of 70’s-80’s (good-λ inequalities,
rearrangement inequalities, sharp-function estimates) provide an
indirect relation between T and the maximal operator M .
The A2 conjecture (90’s):
kT kL2 (w) ¬ c(n, T )[w]A2 ,
where [w]A2 = supQ
1 R
|Q| Q w
1 R
−1
|Q| Q w
.
S. Petermichl (2004) (the Hilbert transform), T. Hytönen (2010)
(general Calderón-Zygmund operators) gave a positive answer
(via a representation of T in terms of Haar shift operators).
A very brief history
The standard methods of 70’s-80’s (good-λ inequalities,
rearrangement inequalities, sharp-function estimates) provide an
indirect relation between T and the maximal operator M .
The A2 conjecture (90’s):
kT kL2 (w) ¬ c(n, T )[w]A2 ,
where [w]A2 = supQ
1 R
|Q| Q w
1 R
−1
|Q| Q w
.
S. Petermichl (2004) (the Hilbert transform), T. Hytönen (2010)
(general Calderón-Zygmund operators) gave a positive answer
(via a representation of T in terms of Haar shift operators).
D. Cruz-Uribe, J. Martell, C. Pérez (2010): the linear A2 bound for
AS f =
X
Q∈S
fQ χQ .
A very brief history
The A2 conjecture (90’s):
kT kL2 (w) ¬ c(n, T )[w]A2 ,
where [w]A2 = supQ
1 R
|Q| Q w
1 R
−1
|Q| Q w
.
S. Petermichl (2004) (the Hilbert transform), T. Hytönen (2010)
(general Calderón-Zygmund operators) gave a positive answer
(via a representation of T in terms of Haar shift operators).
D. Cruz-Uribe, J. Martell, C. Pérez (2010): the linear A2 bound for
AS f =
X
fQ χQ .
Q∈S
A.L. (2012): Let T be an ω-Calderón-Zygmund
operator, and assume
R1
that ω satisfies the log-Dini condition 0 ω(t) log 1t dtt < ∞. Then for
every Banach function space X over Rn ,
kT f kX ¬ c(n, T ) sup kAS |f |kX .
D,S
A very brief history
A.L. (2012): Let T be an ω-Calderón-Zygmund
operator, and assume
R
that ω satisfies the log-Dini condition 01 ω(t) log 1t dtt < ∞. Then for
every Banach function space X over Rn ,
kT f kX ¬ c(n, T ) sup kAS |f |kX .
D,S
J. RConde-Alonso and G. Rey, A.L. and F. Nazarov (2014):
if 01 ω(t) log 1t dtt < ∞, then for every f ∈ L1 , there are ηn -sparse
families Sj ⊂ D (j) , j = 1, . . . , 3n , such that for a.e. x,
n
|T f (x)| ¬ c(n, T )
3
X
j=1
ASj |f |(x).
A very brief history
A.L. (2012): Let T be an ω-Calderón-Zygmund
operator, and assume
R
that ω satisfies the log-Dini condition 01 ω(t) log 1t dtt < ∞. Then for
every Banach function space X over Rn ,
kT f kX ¬ c(n, T ) sup kAS |f |kX .
D,S
J. RConde-Alonso and G. Rey, A.L. and F. Nazarov (2014):
if 01 ω(t) log 1t dtt < ∞, then for every f ∈ L1 , there are ηn -sparse
families Sj ⊂ D (j) , j = 1, . . . , 3n , such that for a.e. x,
n
|T f (x)| ¬ c(n, T )
3
X
ASj |f |(x).
j=1
M. Lacey (2015): the same estimate (for compactly
supported f )
R1
holds under the usual Dini condition [ω]Dini = 0 ω(t) dtt < ∞.
A very brief history
J. RConde-Alonso and G. Rey, A.L. and F. Nazarov (2014):
if 01 ω(t) log 1t dtt < ∞, then for every f ∈ L1 , there are ηn -sparse
families Sj ⊂ D (j) , j = 1, . . . , 3n , such that for a.e. x,
n
|T f (x)| ¬ c(n, T )
3
X
ASj |f |(x).
j=1
M. Lacey (2015): the same estimate (for compactly
supported f )
R1
holds under the usual Dini condition [ω]Dini = 0 ω(t) dtt < ∞.
A quantitative form: (T. Hytönen, L. Roncal and O. Tapiola (2015))
denote CT = kT kL2 →L2 + CK + [ω]Dini . Then
n
|T f (x)| ¬ cn CT
3
X
j=1
ASj |f |(x).
A very brief history
J. RConde-Alonso and G. Rey, A.L. and F. Nazarov (2014):
if 01 ω(t) log 1t dtt < ∞, then for every f ∈ L1 , there are ηn -sparse
families Sj ⊂ D (j) , j = 1, . . . , 3n , such that for a.e. x,
n
|T f (x)| ¬ c(n, T )
3
X
ASj |f |(x).
j=1
M. Lacey (2015): the same estimate (for compactly
supported f )
R1
holds under the usual Dini condition [ω]Dini = 0 ω(t) dtt < ∞.
A quantitative form: (T. Hytönen, L. Roncal and O. Tapiola (2015))
denote CT = kT kL2 →L2 + CK + [ω]Dini . Then
n
|T f (x)| ¬ cn CT
3
X
ASj |f |(x).
j=1
A.L. (2015): an alternative proof of this result, avoiding some
technicalities.
A very brief history
M. Lacey (2015): the same estimate (for compactly
supported f )
R1
holds under the usual Dini condition [ω]Dini = 0 ω(t) dtt < ∞.
A quantitative form: (T. Hytönen, L. Roncal and O. Tapiola (2015))
denote CT = kT kL2 →L2 + CK + [ω]Dini . Then
n
|T f (x)| ¬ cn CT
3
X
ASj |f |(x).
j=1
A.L. (2015): an alternative proof of this result, avoiding some
technicalities.
The key idea behind of all approaches is an iteration:
iteration of the distribution function (rearrangement) of f with a
substitution f → T f (70’s-80’s);
iteration of f (pointwise, “a median decomposition”) with a
substitution f → T f ;
iteration of T f (M. Lacey).
Main steps of the proof
The key recursive claim: there exist pairwise disjoint cubes
P
Pj ∈ D(Q0 ) such that j |Pj | ¬ 12 |Q0 | and for a.e. on Q0 ,
|T (f χ3Q0 )(x)|χQ0 ¬ cn CT |f |3Q0 +
X
j
|T (f χ3Pj )|χPj .
Main steps of the proof
The key recursive claim: there exist pairwise disjoint cubes
P
Pj ∈ D(Q0 ) such that j |Pj | ¬ 12 |Q0 | and for a.e. on Q0 ,
|T (f χ3Q0 )(x)|χQ0 ¬ cn CT |f |3Q0 +
X
|T (f χ3Pj )|χPj .
j
After iteration we obtain that there exists a 12 -sparse family
F ⊂ D(Q0 ) such that
|T (f χ3Q0 )(x)|χQ0 ¬ cn CT
X
Q∈F
|f |3Q χQ (x).
Main steps of the proof
The key recursive claim: there exist pairwise disjoint cubes
P
Pj ∈ D(Q0 ) such that j |Pj | ¬ 12 |Q0 | and for a.e. on Q0 ,
|T (f χ3Q0 )(x)|χQ0 ¬ cn CT |f |3Q0 +
X
|T (f χ3Pj )|χPj .
j
For arbitrary pairwise disjoint cubes Pj ∈ D(Q0 ),
|T (f χ3Q0 )|χQ0
¬ |T (f χ3Q0 )|χQ0 \∪j Pj +
X
j
+
X
j
|T (f χ3Pj )|χPj .
|T (f χ3Q0 \3Pj )|χPj
Main steps of the proof
The key recursive claim: there exist pairwise disjoint cubes
P
Pj ∈ D(Q0 ) such that j |Pj | ¬ 12 |Q0 | and for a.e. on Q0 ,
|T (f χ3Q0 )(x)|χQ0 ¬ cn CT |f |3Q0 +
X
|T (f χ3Pj )|χPj .
j
For arbitrary pairwise disjoint cubes Pj ∈ D(Q0 ),
|T (f χ3Q0 )|χQ0
¬ |T (f χ3Q0 )|χQ0 \∪j Pj +
X
|T (f χ3Q0 \3Pj )|χPj
j
+
X
|T (f χ3Pj )|χPj .
j
Hence, it suffices to find a set E ⊂ Q0 and a covering of E by
disjoint
P cubes Pj ∈ D(Q0 ) such that
1
j
|Pj | ¬ 12 |Q0 |;
2
|T (f χ3Q0 )(x)| ¬ cn CT |f |3Q0 for a.e. x ∈ Q0 \ E;
3
|T (f χ3Q0 \3Pj )(x)| ¬ cn CT |f |3Q0 for a.e. x ∈ Pj .
Main steps of the proof
Hence, it suffices to find a set E ⊂ Q0 and a covering of E by
disjoint
P cubes Pj ∈ D(Q0 ) such that
1
j
|Pj | ¬ 12 |Q0 |;
2
|T (f χ3Q0 )(x)| ¬ cn CT |f |3Q0 for a.e. x ∈ Q0 \ E;
3
|T (f χ3Q0 \3Pj )(x)| ¬ cn CT |f |3Q0 for a.e. x ∈ Pj .
The following local “grand maximal truncated” operator
MT,Q0 f (x) =
controls condition 3 .
sup
P 3x,P ⊂Q0
ess sup |T (f χ3Q0 \3P )(ξ)|
ξ∈P
Main steps of the proof
Hence, it suffices to find a set E ⊂ Q0 and a covering of E by
disjoint
P cubes Pj ∈ D(Q0 ) such that
1
j
|Pj | ¬ 12 |Q0 |;
2
|T (f χ3Q0 )(x)| ¬ cn CT |f |3Q0 for a.e. x ∈ Q0 \ E;
3
|T (f χ3Q0 \3Pj )(x)| ¬ cn CT |f |3Q0 for a.e. x ∈ Pj .
The following local “grand maximal truncated” operator
MT,Q0 f (x) =
sup
P 3x,P ⊂Q0
ess sup |T (f χ3Q0 \3P )(ξ)|
ξ∈P
controls condition 3 .
We have kMT,Q0 kL1 →L1,∞ ¬ αn CT and
|T (f χ3Q0 )(x)| ¬ αn kT kL1 →L1,∞ |f (x)| + MT,Q0 f (x).
Main steps of the proof
Hence, it suffices to find a set E ⊂ Q0 and a covering of E by
disjoint
P cubes Pj ∈ D(Q0 ) such that
1
j
|Pj | ¬ 12 |Q0 |;
2
|T (f χ3Q0 )(x)| ¬ cn CT |f |3Q0 for a.e. x ∈ Q0 \ E;
3
|T (f χ3Q0 \3Pj )(x)| ¬ cn CT |f |3Q0 for a.e. x ∈ Pj .
The following local “grand maximal truncated” operator
MT,Q0 f (x) =
sup
P 3x,P ⊂Q0
ess sup |T (f χ3Q0 \3P )(ξ)|
ξ∈P
controls condition 3 .
We have kMT,Q0 kL1 →L1,∞ ¬ αn CT and
|T (f χ3Q0 )(x)| ¬ αn kT kL1 →L1,∞ |f (x)| + MT,Q0 f (x).
Set
E = {x ∈ Q0 : MT,Q0 f (x) > cn CT |f |3Q0 ∨ |f (x)| > cn |f |3Q0 },
where cn is such that |E| ¬
1
2n+2 |Q0 |.
Main steps of the proof
Hence, it suffices to find a set E ⊂ Q0 and a covering of E by
disjoint
P cubes Pj ∈ D(Q0 ) such that
1
j
|Pj | ¬ 12 |Q0 |;
2
|T (f χ3Q0 )(x)| ¬ cn CT |f |3Q0 for a.e. x ∈ Q0 \ E;
3
|T (f χ3Q0 \3Pj )(x)| ¬ cn CT |f |3Q0 for a.e. x ∈ Pj .
The following local “grand maximal truncated” operator
MT,Q0 f (x) =
sup
P 3x,P ⊂Q0
ess sup |T (f χ3Q0 \3P )(ξ)|
ξ∈P
controls condition 3 .
Set
E = {x ∈ Q0 : MT,Q0 f (x) > cn CT |f |3Q0 ∨ |f (x)| > cn |f |3Q0 },
1
where cn is such that |E| ¬ 2n+2
|Q0 |.
Apply the Calderón-Zygmund decomposition to χE with λ =
1
2n+1 .
Main steps of the proof
Hence, it suffices to find a set E ⊂ Q0 and a covering of E by
disjoint
P cubes Pj ∈ D(Q0 ) such that
1
j
|Pj | ¬ 12 |Q0 |;
2
|T (f χ3Q0 )(x)| ¬ cn CT |f |3Q0 for a.e. x ∈ Q0 \ E;
3
|T (f χ3Q0 \3Pj )(x)| ¬ cn CT |f |3Q0 for a.e. x ∈ Pj .
The following local “grand maximal truncated” operator
MT,Q0 f (x) =
sup
P 3x,P ⊂Q0
ess sup |T (f χ3Q0 \3P )(ξ)|
ξ∈P
controls condition 3 .
Set
E = {x ∈ Q0 : MT,Q0 f (x) > cn CT |f |3Q0 ∨ |f (x)| > cn |f |3Q0 },
1
where cn is such that |E| ¬ 2n+2
|Q0 |.
Apply the Calderón-Zygmund decomposition to χE with λ =
We obtain disjoint cubes Pj ∈ D(Q0 ) such that
which easily implies 1 , 2 and 3 .
1
2n+1
<
1
2n+1 .
|Pj ∩E|
1
|Pj | ¬ 2 ,
Related remarks and questions
The proof shows that if T is a sublinear operator of weak type (1, 1)
and
MT f (x) = sup ess sup |T (f χRn \3Q )(ξ)|
Q3x
ξ∈Q
is of weak type (1, 1), then kT kL2 (w) ¬ c(n, T )[w]A2 .
Related remarks and questions
The proof shows that if T is a sublinear operator of weak type (1, 1)
and
MT f (x) = sup ess sup |T (f χRn \3Q )(ξ)|
Q3x
ξ∈Q
is of weak type (1, 1), then kT kL2 (w) ¬ c(n, T )[w]A2 .
T. Hytönen, L. Roncal and O. Tapiola (2015): for a class of rough
homogeneous singular integrals TΩ ,
kTΩ kL2 (w) ¬ c(n, T )[w]2A2 .
Related remarks and questions
The proof shows that if T is a sublinear operator of weak type (1, 1)
and
MT f (x) = sup ess sup |T (f χRn \3Q )(ξ)|
Q3x
ξ∈Q
is of weak type (1, 1), then kT kL2 (w) ¬ c(n, T )[w]A2 .
T. Hytönen, L. Roncal and O. Tapiola (2015): for a class of rough
homogeneous singular integrals TΩ ,
kTΩ kL2 (w) ¬ c(n, T )[w]2A2 .
A. Seeger (1996): TΩ is of weak type (1, 1).
Related remarks and questions
The proof shows that if T is a sublinear operator of weak type (1, 1)
and
MT f (x) = sup ess sup |T (f χRn \3Q )(ξ)|
Q3x
ξ∈Q
is of weak type (1, 1), then kT kL2 (w) ¬ c(n, T )[w]A2 .
T. Hytönen, L. Roncal and O. Tapiola (2015): for a class of rough
homogeneous singular integrals TΩ ,
kTΩ kL2 (w) ¬ c(n, T )[w]2A2 .
A. Seeger (1996): TΩ is of weak type (1, 1).
It is natural to ask whether MTΩ is of weak type (1, 1), too.
Related remarks and questions
The proof shows that if T is a sublinear operator of weak type (1, 1)
and
MT f (x) = sup ess sup |T (f χRn \3Q )(ξ)|
Q3x
ξ∈Q
is of weak type (1, 1), then kT kL2 (w) ¬ c(n, T )[w]A2 .
T. Hytönen, L. Roncal and O. Tapiola (2015): for a class of rough
homogeneous singular integrals TΩ ,
kTΩ kL2 (w) ¬ c(n, T )[w]2A2 .
A. Seeger (1996): TΩ is of weak type (1, 1).
It is natural to ask whether MTΩ is of weak type (1, 1), too.
Observe that the question whether the maximal singular integral
operator TΩ? is of weak type (1, 1) is still open.
Some words about the commutators
Let [b, T ] denote the commutator of a Calderón-Zygmund operator T
with a locally integrable function b:
[b, T ]f (x) = bT f (x) − T (bf )(x).
Some words about the commutators
Let [b, T ] denote the commutator of a Calderón-Zygmund operator T
with a locally integrable function b:
[b, T ]f (x) = bT f (x) − T (bf )(x).
Introduce the sparse operator TS,b defined by
TS,b f (x) =
X
|b(x) − bQ |fQ χQ (x).
Q∈S
? be the adjoint operator to T
Let TS,b
S,b :
?
TS,b
f (x) =
X 1 Z
Q∈S
|Q|
Q
|b − bQ |f χQ (x).
Some words about the commutators
Let [b, T ] denote the commutator of a Calderón-Zygmund operator T
with a locally integrable function b:
[b, T ]f (x) = bT f (x) − T (bf )(x).
Introduce the sparse operator TS,b defined by
TS,b f (x) =
X
|b(x) − bQ |fQ χQ (x).
Q∈S
? be the adjoint operator to T
Let TS,b
S,b :
?
TS,b
f (x) =
X 1 Z
Q∈S
|Q|
|b − bQ |f χQ (x).
Q
A.L., S. Ombrosi, I. Rivera-Rı́os (2016): for every compactly
supported f ∈ L∞ (Rn ), there are 2·91 n -sparse families
Sj ⊂ D (j) , j = 1, . . . , 3n , such that for a.e. x ∈ Rn ,
n
|[b, T ]f (x)| ¬ cn CT
3
X
j=1
TSj ,b |f |(x) + TS?j ,b |f |(x) .
Some words about the commutators
A.L., S. Ombrosi, I. Rivera-Rı́os (2016): for every compactly
supported f ∈ L∞ (Rn ), there are 2·91 n -sparse families
Sj ⊂ D (j) , j = 1, . . . , 3n , such that for a.e. x ∈ Rn ,
n
|[b, T ]f (x)| ¬ cn CT
3
X
TSj ,b |f |(x) + TS?j ,b |f |(x) .
j=1
In particular, we obtain the following result: if µ, λ ∈ Ap , 1 < p < ∞,
ν = (µ/λ)1/p and
kbkBM O(ν) := sup
Q
1
ν(Q)
Z
|b(x) − bQ |dx < ∞,
Q
then
k[b, T ]f kLp (λ) ¬ cn,p CT [µ]Ap [λ]Ap
max 1, 1
p−1
kbkBM Oν kf kLp (µ) .
Some words about the commutators
A.L., S. Ombrosi, I. Rivera-Rı́os (2016): for every compactly
supported f ∈ L∞ (Rn ), there are 2·91 n -sparse families
Sj ⊂ D (j) , j = 1, . . . , 3n , such that for a.e. x ∈ Rn ,
n
|[b, T ]f (x)| ¬ cn CT
3
X
TSj ,b |f |(x) + TS?j ,b |f |(x) .
j=1
In particular, we obtain the following result: if µ, λ ∈ Ap , 1 < p < ∞,
ν = (µ/λ)1/p and
kbkBM O(ν) := sup
Q
1
ν(Q)
Z
|b(x) − bQ |dx < ∞,
Q
then
k[b, T ]f kLp (λ) ¬ cn,p CT [µ]Ap [λ]Ap
max 1, 1
p−1
kbkBM Oν kf kLp (µ) .
This provides a quantitative form of the two-weighted bound due to
S. Bloom (1985) and I. Holmes, M. Lacey and B. Wick (2015).
Some related ”sparse domination” works
F. Bernicot, D. Frey and S. Petermichl (2015):
singular non-integral operators.
A. Culiuc, F. Di Plinio and Y. Ou (2016):
trilinear multiplier forms.
J. Conde-Alonso and J. Parcet (2016):
non-homogeneous Calderón-Zygmund operators.
F.C. de Franca Silva and P. Zorin-Kranich (2016):
uncentered variational operators.
Some related ”sparse domination” works
F. Bernicot, D. Frey and S. Petermichl (2015):
singular non-integral operators.
A. Culiuc, F. Di Plinio and Y. Ou (2016):
trilinear multiplier forms.
J. Conde-Alonso and J. Parcet (2016):
non-homogeneous Calderón-Zygmund operators.
F.C. de Franca Silva and P. Zorin-Kranich (2016):
uncentered variational operators.
Thank you for your attention!