Lipschitz estimates for commutators of singular integral operators associated with the sections

R E S E A R C H

Open Access

Lipschitz estimates for commutators of

singular integral operators associated with

the sections

Guangqing Wang and Jiang Zhou

*_{Correspondence:}

[email protected] College of Mathematics and System Sciences, Xinjiang University, Urumqi, 830046, Republic of China

Abstract

LetHbe Monge-Ampère singular integral operator,b∈Lipβ_F, and 1/q= 1/p–

β

_μ

_μ

β

_μ

_β

β

MSC: 42B20; 42B30

Keywords: Hardy spaces; commutator; Lipschitz function; sections

1 Introduction

In , Caffarelli and Gutiérrez [] introduced the new concept of a family of sections in studying real variable theory related to the Monge-Ampère equation. They defined the

Hardy-Littlewood maximal operatorMF andBMOF(Rn) spaces associated to sections,

and the weak (, ) ofMF and the John-Nirenberg inequality forBMOF(Rn_{) were}

ob-tained. Caffarelli and Gutiérrez [] defined the singular integral operatorHrelated to the

Monge-Ampère equation and provedL_{-boundedness of it. Applying the theory of}

ho-mogeneous spaces, Incognito [] obtained a weak type (, ) estimate ofH. In [], Tang

considered the commutator of Coifman-Rochberf-Weiss [b,H] and obtained weighted

es-timates for the operatorHand the commutator [b,H], whereb∈BMOF. And from [], it follows that [b,H] withb∈BMO_Fis bounded onLp_(Rn_{,dμ) for  <p<∞. Inspired by} the above work, we will study the behaviors of commutator [b,H] withb∈Lipβ_Facting on Lebesgue spacesLp(Rn,dμ) and Hardy spacesH_Fp(Rn), where the Lipschitz spacesLipβ_F

and Hardy spacesH_Fp(Rn_{) associated with sections are defined by Lin [].}

As is well known, linear commutators are naturally appearing operators in harmonic analysis that have been extensively studied already. In general, the boundedness results of commutators in harmonic analysis can be used to characterize some important

func-tion spaces such asBMOspaces, Lipschitz spaces, Besove spaces and so on (see [–]).

Coifmanet al.[] applied the boundedness to some non-linear PDEs, which perfectly il-lustrate the intrinsic links between the theory of compensated compactness and the clas-sical tools of harmonic and real analysis. As for some other essential applications to PDEs such as characterizing pseudodifferential operators, studying linear PDEs with measur-able coefficients and the integrability theory of the Jacobians, interested researchers can

refer to [–]. It is perhaps for this important reason that the boundedness of commuta-tors attracted vast attention among researchers in harmonic analysis and PDEs. Thus, it is

meaningful to identify the behaviors of commutator [b,H] associated with the

Monge-Ampère equation. In the sense of Euclidean space Rn_{, the boundedness of}

commuta-tor [b,T] withb∈Lipβ acting on Lebesgue spaces, is easily obtained by the inequality

|[b,T]f(x)| ≤CIβ(|f|)(x), whereTis a Calderón-Zygmund singular integral operator and

Iβis the Riesz potential of orderβ. However, we cannot find a suitable operator to control the commutator [b,H] withb∈Lipβ_F - just as controlling of [b,T] withb∈Lipβ- due to the particularity of the operatorH. Therefore, in this paper we investigate [b,H] directly, and obtain some relatively important properties.

This paper is organized as follows. In Section , we recall some elementary properties of sections. In the first part of Section , we demonstrate the (Lp,Lq) boundedness of [b,H] when  <p< /βand /q= /p–β. It is worth mentioning that boundedness of [b,H] from

L_(Rn_{,dμ) to weakL}/(–β)_(Rn_{,dμ) is also obtained, which indicates the differences} be-tween the commutator [b,H] withb∈Lipβ_Fand that commutator withb∈BMO_F. Based on these differences, in the second part of Section , we further discuss the behavior of the commutator [b,H] acting on Hardy spacesH_Fp(Rn), and we see that the commutator [b,H] is bounded fromH_Fp(Rn_{) toL}q_(Rn_{,dμ) if /( +β) <p≤ and /q= /p–β. For the} extreme casep= /( +β), we cannot get the (Hp_,Lq_{) boundedness of [b,H], but we give} a characterization. Instead of the boundedness in the extreme case, a weak estimate for [b,H] is showed.

Now we recall the definition of sections which play an important role in the study of the Monge-Ampère equation and the linearized Monge-Ampère equation (see [, –]). For

x∈Rn_{andt> , letS(x,t) denote an open and bounded convex subset ofR}n_containingx. The setS(x,t) is called asectionif the familyF={S(x,t)⊂Rn_:x∈Rn_{andt> }is} mono-tone increasing int,i.e.,S(x,t)⊂S(x,t) fort≤twhich satisfies the following criteria:

(A) There exist constantsK,K,Kand,such that given two sectionsS(x,t),S(x,t)

witht≤tsatisfying

S(x,t)∩S(x,t)=∅,

and givenT, an affine transformation that “normalizes”S(x,t), that is,

B(, /n)⊂TS(x,t)⊂B(, ),

there existsz∈B(,K) depending onS(x,t) andS(x,t) such that

B(z,K(t/t)⊂_Ts(x,t)⊂_{Bz,K(t/t)} _(.)

and

T(z)∈Bz, (/)K(t/t)_.

HereB(x,t) denotes the Euclidean ball centered atxwith radiust.

(B) There exists a constantδ>  such that given a sectionS(x,t) andy∈/S(x,t), ifTis an affine transformation that “normalizes”S(x,t), then for any  << 

(C)_t>S(x,t) ={x}and_t>S(x,t) =Rn_.

In addition, we also assume that a Borel measureμwhich is finite on compact sets is

given,μ(Rn) =∞, and that it satisfies thedoubling propertywith respect toF, that is, there exists a constantAsuch that

μS(x, t)≤AμS(x,t) (.)

for any sectionS(x,t)∈F. Throughout the paper, the letterCwill denote a positive con-stant that may vary from line to line but remains independent of the main variables.

We write ABto indicate thatA is majorized byB times a constant independent of

AandB, while the notation A≈Bdenotes both ABandBA. Finally, we denote

Lpμ:=Lp(Rn,dμ) (≤p≤ ∞) simply.

2 Elementary properties of section and notions

According to [], the properties of (A) and (B) imply the following properties:

(D) There exists a constantθ ≥, depending only onδ,K and, such that for any y∈S(x,t),

S(x,t)⊂S(y,θt) and S(y,t)⊂S(x,θt). (.)

(E) There exists a quasi-metricd(x,y) onRn_{with respect toFdefined by}

d(x,y) =inft:x∈S(y,t) andy∈S(x,t),

and its triangular constant is just theθappearing in (D); that is,

d(x,y)≤θd(x,z) +d(z,y) for anyx,y,z∈Rn. (.)

Also,

S(x,t/θ)⊂Bd(x,t)⊂S(x,t) for anyx∈Rnandt> , (.)

whereBd(x,t) is ad-ball defined byBd(x,t) :={y∈Rn_{:d(x,y) <t}. Combining (.) and} (.), one can see that there exists a constantn>  and n_{>Asuch that}

μBd(x, r)≤n_{μBd(x,r). (.)}

Thus, (Rn_{,d,μ) becomes a space of homogeneous type. Based on this, one can use the} standard real analysis tools as the maximal functionMf and the sharp functionM_{f. In} this paper, both of them are defined on (Rn,d,μ), namely

Mf(x) =sup

r>



μ(B(x,r))

B(x,r)

f(y) dμ(y),

M_{f(x) =sup} x∈B



μ(B)

B

f(y) –fB dμ(y)≈sup x∈B

inf

c 

μ(B)

B

Here and below,Bis ad-ball andfBmeans the average offonB. If we writeBMO(Rn) :={f :

fBMO(Rn₎<∞}withf_BMO=Mf_L∞

μ, then theBMOF(R

n_{) space coincides withBMO}

and fBMO_F(Rn₎ ≈f_BMO(Rn₎ (see []). Denote M_δf(x) =M(|f|δ)/δf(x) and M_δf(x) = M_(|f|δ₎/δ_(x).

Macías and Segovia [] have found that the quasi-metricdcan be replaced by another

quasi-metricρsuch that (Rn_{,ρ,μ) is a normal space. Moreover, for the quasi-metricρ} there exist constantsC>  and∈(, ) such that

⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩

ρ(x,y)≈inf{μ(Bd) :Bdared-balls containingxandy};

μ(Bρ(x,r))≈r, ∀x∈Rn,r> , whereBρ(x,y) :={y∈Rn:ρ(x,y) <r}; |ρ(x,y) –ρ(x,y)| ≤Cρ(x,y)_{[ρ(x,y) +ρ(x,y)]}–_{, ∀x,x,y∈R}n_.

(.)

Letρsatisfy (.) above andf be a continuous function onRn_{. Lin [] defined Lipschitz} spacesLipβ_Fassociated with sections as follows.

Definition . Let  <β≤. There exists a positive constantCsuch that

sup

ρ(x,y)≤h

f(x) –f(y) ≤Chβ

for∀h> . The “norm” off inLipβ_Fis defined by the lower bound of the constantsC.

Letbe given in (.) above. Lin [] found that the function spacesβ_q,FandLipβ_F co-incide with equivalent norms for  <β<and ≤q≤ ∞, whereβ_q,Fdenotes the Cam-panato spaces associated to the familyFof the section. Also, he proved thatβ_q,Fare the dual spaces of Hardy spacesH_Fp(Rn_{) (/ <p≤).}

For a locally integral functionb, the commutator of Cofiman-Rochberg-Weiss [b,H] is defined as follows:

[b,H]f(x) =b(x)Hf(x) –H(bf)(x) =

Rn

b(x) –b(y)k(x,y)f(y)dμ(y),

whereHis defined by the formula

Hf(x) =

Rnk

(x,y)f(y)dμ(y),

withk(x,y) =_iki(x,y), and each kernelkisatisfies the following properties:

suppki(·,y)⊂Si(y), ∀y∈Rn; suppki(x,·)⊂Si(x), ∀x∈Rn;

Rnki(x,y)d

μ(y) =

Rnki(x,y)d

μ(x) = , ∀x,y∈Rn;

sup

i

Rn

ki(x,y) dμ(y)≤C, ∀x∈Rn; sup

i

Rn

ki(x,y) dμ(x)≤C, ∀x∈Rn;

whereSi(x) =S(x, i_{) for anyx∈R}n_{,i∈Z. IfTis an affine transformation that normalizes} the sectionSi(y) then eachkisatisfies the Lipschitz condition,

ki(u,y) –ki(v,y) ≤C



and, finally, ifTis an affine transformation that normalizes the sectionSi(x) thenkisatisfies the Lipschitz condition,

ki(x,u) –ki(x,v) ≤C



μ(Si(x))|Tu–Tv|.

Caffarelli and Gutiérrez [] obtained thatHis bounded onL

μ. Subsequently, Incognito [] has givenLpμ( <p<∞) and the weak-type (, ) estimate ofH.

Using the property (D) and defining a functionσonRn_×Rn_{byσ(x,y) =inf{t>  :y∈}

S(x,t)}, Incognito [] obtained the following conclusions: (E)σ(x,y)≤θ σ(y,x) for allx,y∈Rn_.

(F)σ(x,y)≤θ(σ(x,z) +σ(z,y)) for allx,y,z∈Rn_. It is easy to see that

σ(x,y) <d(x,y) <θ σ(x,y) for allx,y∈Rn, (.)

and for a given sectionS(x,t),y∈S(x,t) if and only ifσ<t.

3 Main results

3.1 Boundedness fromLpμtoLqμ

In this subsection, we discuss the property of the commutator acting on Lebesgue spaces.

Theorem . Suppose that b∈Lipβ_F,  <β< .If/q= /p–βwith <p< /β,then[b,H]

is bounded from Lpμto Lqμ.

Theorem . Suppose that b∈Lipβ_F,  <β<min{,/n},whereand nare given in

(.)and(.)respectively.Then the commutator[b,H]is bounded from L

μto weak L/(–μ β).

In order to prove the theorems above, it is necessary to give the following lemmas.

Lemma .([]) Let <p,δ<∞andω∈A∞.There exists a positive C such that

RnMδf(x)

p_{ω(x)dμ(x)≤C}

RnM

δf(x)pω(x)dμ(x)

for any smooth function f for which the left-hand side is finite.

Lemma .([]) Let K(x,y) =_iKi(x,y).Then there exists a constant C> such that

K(x,y) –K(x,y) + K(y,x) –K(y,x) ≤C  –k

μ(S(y, kσ(y,x)))

ifσ(y,x)≥kθσ(y,y)and k≥.

Lemma . Suppose that b∈Lipβ_F,for <β< .Let <δ<  <r< /β.Then there exists a constant C> such that

Mδ

[b,H]f(x)≤Cb_Lipβ

F

for any smooth function f and every x∈Rn_{,and where}

Mβ,r(f)(x) =sup

x∈B



μ(B)–rβ

B

f(y) rdμ(y)

/r .

Proof Observe that for any constantλ

[b,H]f(x) =b(x) –λHf(x) –H(b–λ)f(x).

For any fixed ballB=B(x,r). Decomposef =f+f, wheref=fχB¯ withB¯=B(x, θr). LetλandcBbe constants to be fixed in the proof. We write



μ(B)

B

[b,H]f(y) δ–|cB|δ dμ(y)

/δ

≤



μ(B)

B

b(y) –λHf(y) δdμ(y)

/δ +



μ(B)

B

H(b–λ)f(y) δdμ(y)

/δ

+



μ(B)

B

H(b–λ)f

(y) –cB δ

dμ(y)

/δ

:=L+L+L.

ForL, we fixλ=bB. The Hölder inequality and (.) give us

Lb_Lipβ

Fμ(B)

β



μ(B)

B

Hf(y) rdμ(y)

/r

b_Lipβ

FMβ,r(Hf)(x).

From Kolmogorov’s inequality and (.), it follows that

L μ(B) /δ–/r

μ(B)/δ

B

b(y) –bB

f(y) rdμ(y)

/r

b_Lipβ

FMβ,r(f)(x).

Finally, we takecB= (H((b–bB)f))B. Then for anyx∈B, Lemma ., (.), (.), and (.)

yield

L≤ 

μ(B)

B

_H

(b–λ)f(y) –H(b–bB)f

B dμ(y)

≤ 

μ(B)

B

B

Rn_{\ ¯B}

K(x,w) –K(z,w) b(w) –bB f(w) dμ(w)dμ(z)dμ(y)

× K(x,w) –K(z,w) f(w) dμ(w)dμ(z)dμ(y)

b_Lipβ

F

∞

k=

k_θ_{r<σ(x,w)≤}k+_θ_r

ρβ(x,w)–k

μ(S(x, k_θ_r)) f(w) dμ(w)

b_Lipβ

FMβ,r(f)(x)

∞

k=

–k

b_Lipβ

FMβ,r(f)(x).

Lemma .([]) Let local integral function f ∈L

μandα> .Then there exists a family

of balls{Bi}such that:

() |f(x)| ≤α,forμ-a.e.x∈Rn_\ iBi;

() _μ(_B

i)

Bi|f(t)|dμ(t)≤Cα;

() ∞_i=μ(Bi)≤C_αf_L

μ;

() there exists an integerN≥,independent off andλ,such that_iχBi(x)≤Nfor

μ-a.e.x∈Rn_.

Now, with the lemmas above, we state the proof of our results.

Proof of Theorem. From Lemma . and Lemma . with  <δ<  <r<p, it follows that

[b,H]f_Lq

μ≤Mδ

[b,H]f_Lq

μ

≤M_δ[b,H]f_Lq

μ

b_Lipβ

FMβ,r(Hf)Lqμ+Mβ,r(f)Lqμ

b_Lipβ

FfL p

μ.

Thus, the proof of the theorem is completed.

Proof of Theorem. Forf ∈L

μand anyα> , applying Lemma . withαreplaced by

αq_withq= 

–β, we obtain, with the same notation as in Lemma .,f =g+h=g+

jhj, where

g(x) =f(x)χRn_\ iBi(x) +

j

(fηj)BjχBj(x) and hj(x) =f(x)ηj(x) – (fηj)BjχBj(x)

withηj(x) =χBj(x)

jχBj(x)χ

iBi(x). By Lemma ., it is easy to obtain the following properties:

(i) |f(x)| ≤αq_{, forμ-a.e.x}∈Rn\

jBj;

(ii) _μ(_B

j)

Bj|f(t)|dμ(t)≤Cα

q_;

(iii) ∞_j=μ(Bj)≤αCqfLμ; (iv) g_L

μ≤CfLμ and|g(x)| ≤Cα

q_{, forμ-a.e.x}∈Rn_;

(v) eachhjis supported inBj,

Rn|hj(x)|dμ(x)≤Cαqμ(Bj)and

Rnhj(x)dμ(x) = .

LetB¯j=B(zj, θrj), we write

μx∈Rn: [b,H]f(x) >α

≤μx∈Rn: [b,H]g(x) >α/+μ

x∈

j

¯

Bj: [b,H]h(x) >α/

+μ

x∈Rn\

j

¯

Bj: [b,H]h(x) >α/

Choose  <p< /βand_q

 =



p –β. The boundedness of [b,H] fromL

p

μ toLqμand (iv) give us that

Kα–q[b,H]g

q

Lqμ α

–q_gq

Lpμ α

–q_αq(p–)qp_f q_ p

Lμα

–q_f

q_ p

Lμ.

By (iii), it is concluded that

K≤μ j ¯ Bj j

μ(Bj)α–qfL

μ.

ForK, we have

K≤μ

x∈Rn

j

¯ Bj:

j

b(x) –b(zj) H(hj)(x) >α/

+μ

x∈Rn

j

¯ Bj:

H

j

b–b(zj)hj

(x) >α/

:=K+K.

From (v), Lemma ., (.), and (iii), it follows that

Kα–

j

Rn_\ jB¯j

b(x) –b(zj) H(hj)(x) dμ(x)

α–

j

Rn_{\ ¯B} j

b(x) –b(zj)

Bj

k(x,y) –k(x,zj) hj(y) dμ(y)dμ(x)

b_Lipβ

Fα

q–

j

μ(Bj)

β+∞

k=

(nβ–)k

b_Lipβ

Ff

β+

Lμ α

–q_.

The boundedness of [b,H] fromL

μto weakLμ, (.), (v), and (iii) give us that

Kα–

j

Bj

b(x) –b(zj) hj(x) dμ(x)b_Lipβ

Fα

q–

j

μ(Bj)

β+

b_Lipβ

Ff

β+

Lμ α

–q_.

Combining the estimates forK,KandK, one can finish the proof.

3.2 Boundedness fromHp_F(Rn_{) toL}q

μ

In this subsection, we discuss the boundedness of the commutator [b,H] on Hardy spaces

H_Fp(Rn_{), and obtain the following results in which the symbols}

andnare given in (.)

and (.) respectively. Firstly, we recall the definition of the (p,∞)-atoms and the atomic Hardy spacesH_Fp(Rn_{) with respect to a familyFof sections and a doubling measureμ.} Definition .([]) Let / <p≤. A functiona∈L∞μ is called a (p,∞)-atom if there exists a sectionS(x,t)∈Fsuch that

() _Rna(x)dμ(x) = ;

() aL∞μ ≤μ(S(x,t))

–/p_.

The atomic Hardy spaceH_Fp(Rn_{) is defined byH}p

F(Rn) ={f ∈S(Rn) :f(x)S

=_jλjaj(x), eachajis a (p,∞)-atom and

j|λj|p <∞}, where S(Rn) denotes the space of Schwartz functions andS(Rn_{) is the dual space ofS(R}n_{). We define theH}p

F(Rn) norm off by

fH_Fp(Rn₎=inf

j

|λj|p

/p ,

where the infimum is taken over all decompositions off =_jλjajabove.

Theorem . Suppose that b∈Lipβ_F,  <β<min{,/n}.If_+β<p≤and/q= /p–β,

then[b,H]maps H_Fp(Rn_{)continuously into L}q_μ.

For the extreme casep= /( +β), one gets the following characterization theorem.

Theorem . Let b∈Lipβ_F,  <β<min{,/n}.Then the following statements are equiv-alent.

(a) [b,H]rapsH_F/(+β)continuously intoL

μ;

(b) for any(,∞)-atomasupported in a sectionS=S(x,r/θ)andu∈S,

_Rn

b(y)a(y)dμ(y)

¯ Bc

K(x,u) dμ(x),

whereB¯=B(x, θr).

In general, the (Hp_,Lq_{) boundedness of [b,H] fails forp= /( +β), then we give a weak} estimate instead.

Theorem . Let b∈Lipβ_F,  <β<min{,/n}.Then[b,H]maps H/(+

β)

F (Rn) continu-ously into weak L

μ.

Next, we show the proofs of the theorems above.

Proof of Theorem. Without loss of generality, we assume thatb_Lipβ

F = . By

Defi-nition ., we only need to prove that for any (p,∞)-atoma,[b,H]a_Lq

μ. Given a

(p,∞)-atomawithsuppa⊂S=S(x,r/θ)∈F. LetB=B(x,r),B¯ =B(x, θ_{r). Then} S(x,r/θ)⊂B(x,r). Write

[b,H]a_Lq

μ≤

¯ B

[b,H]a(x) qdμ(x)

/q +

Rn_{\ ¯B}

[b,H]a(x) qdμ(x)

/q =I+II.

Choosing  <p< /βand /q= /p–β, and noting the (p,q) boundedness of [b,H]

and the size condition ofa, one can get

I[b,H]a_Lq_

μμ(B)

/q–/qa

Lpμμ(B)

/q–/qa

L∞μμ(B)

On the other hand, the cancellation condition of the atomayields

II≤

Rn_{\ ¯B}

b(x) –b(x) B

K(x,y) –K(x,x)a(y)dμ(y) q

dμ(x)

/q

+

Rn_{\ ¯B}

_BK(x,y)b(x) –b(y)a(y)dμ(y) q

dμ(x)

/q

:=II+II.

Lemma ., (.), (.), and (.) imply that

II b_Lipβ

F

μ(B)/p

_∞

k=

k_θ_{r≤σ(x,x)<}k+_θ_r

ρβ(x,x)

×

B

K(x,y) –K(x,x) dμ(y) q

dμ(x)

/q

μ(B)/q–/p+β ∞

k=

k(n(β–+/q)–)

.

Finally, noting that

b(x) –ba_Lq

μbLipβ_Fρ β

(x,y)aL∞μμ(B)

/q_μ(B)β_μ

(B)/q–/p,

by the (q,q) boundedness ofH, one obtains

II=

Rn_{\ ¯B}

Hb(x) –b

a(x) qdμ(x)

/q

b(x) –b

a_Lq

μ.

Combining the estimates forIandII, one can finish the proof.

Proof of Theorem. Now that [b,H] is bounded fromH_F/(+β)(Rn_{) toL}

μ is equivalent to the fact that [b,H]a_L

μ  holds for any (/( +β),∞) atom. Thus, we will study

the behavior of [b,H] acting on any (/( +β),∞) atom. Letabe an atom withsupp⊂ S=S(x,r/θ)∈F, letB=B(x,r),B¯ =B(x, θ_{r), thenS(x,r/θ)⊂B(x,r). For any} u∈S(x,r/θ), one writes

[b,H]a(x) =χB(x)[b,H]a(x) +χB¯c(x)

B

K(x,y) –K(x,u)b(y) –b(x)a(y)dμ(y)

+χB¯c(x)

b(x) –b(x)Ha(x) +χB¯c(x)K(x,u)

B

b(y)a(y)dμ(y)

:=M+M–M–M.

Similar to the estimate for I and II in the proof of Theorem ., we can show that

MLμ≤

¯ Bc

b(x) –b(x)

B

K(x,y) –K(x,x) a(y) dμ(y)dμ(x)

∞

k=

k(nβ–)

.

The estimation above yields that [b,H]a_L

μ if and only ifMLμ. Hence the

proof is finished.

Proof of Theorem . Letf ∈H_F/(+β)(Rn) andf(x) =_iλiai(x) with eachai an (/( +

β),∞)-atom and_i|λi|/(+β)<∞. Suppose thatsuppa⊂Si=S(xi,ri). Write

[b,H]f(x) = i

λi

b(x) –b(xi)Hai(x)χB¯(x) +

i

λi

b(x) –b(xi)Hai(x)χ₍B¯)c(x)

–H

i

λi

b–b(xi)ai

(x) :=J+J+J.

By the Hölder inequality and the (, ) boundedness ofH, we have

b–b(xi)(Hai)χB¯L

μ.

Using the same method asMtogether with  <β</n, it is easy to check

b–b(xi)(Hai)χB¯c

L

μ.

So, we have

μx∈Rn:|Jj|>λ/

λ–JjLμλ

–

i

|λi|, j= , . (.)

Finally, noting that

b–b(xi)a_L

μbLipβ_Fρ β

(xi,y)aL∞μμ(B)μ(B) β_μ

(B)–(+β),

and we have the weak (, ) boundedness ofH, one obtains

μx∈Rn:|J|>λ/ i

λi

b–b(xi)ai

Lμ

λ–

i

|λi|. (.)

Equations (.) and (.) imply that the proof is completed.

4 Conclusions

perform this work in the future. Moreover, the smoothing effect and the compactness of the commutator [b,H] can be investigated as well.

Competing interests

The authors declare that they do not have any commercial or associative interests that represent a conflict of interest in connection with the work submitted.

Authors’ contributions

All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.

Acknowledgements

The research was supported by National Natural Science Foundation of China (Grant No. 11661075).

Received: 11 May 2016 Accepted: 15 January 2017 References

1. Caffarelli, LA, Gutiérrez, CE: Real analysis related to the Monge-Ampère equation. Trans. Am. Math. Soc.348, 1075-1092 (1996)

2. Caffarelli, LA, Gutiérrez, CE: Singular integrals related to the Monge-Ampère equation. In: Wavelet Theory and Harmonic Analysis in Applied Sciences (Buenos Aires, 1995). Appl. Numer. Harmon. Anal., pp. 3-13. Birkhauser Boston, Boston (1997)

3. Incognito, A: Weak-type (1, 1) inequality for the Monge-Ampère SIO’s. J. Fourier Anal. Appl.7, 41-48 (2001) 4. Tang, L: Weighted estimates for singular integral operators and commutators associated with sections. J. Math. Anal.

Appl.333, 577-590 (2007)

5. Lin, CC: Boundedness of Monge-Ampère singular integral operators acting on Hardy spaces and their duals. Trans. Am. Math. Soc.368, 3075-3104 (2015)

6. Coifman, R, Rochberg, R, Weiss, G: Factorization theorems for Hardy spaces in several variables. Ann. Math.103, 611-635 (1976)

7. Paluszy ´nski, M: Characterization of the Besov spaces via the commutator operator of Coifman, Rochberg and Weiss. Indiana Univ. Math. J.44, 1-17 (1995)

8. Janson, S: Mean oscillation and commutators of singular integral operators. Ark. Mat.16, 263-270 (1978) 9. Ferguson, SH, Lacey, MT: A characterization of product BMO by commutators. Acta Math.189, 143-160 (2002) 10. Coifman, R, Lions, P, Meyer, Y, Semmes, S: Compensated compactness and Hardy spaces. J. Math. Pures Appl.72,

247-286 (1993)

11. Iwaniec, T: Nonlinear commutators and Jacobians. J. Fourier Anal. Appl.3, 775-796 (1997)

12. Turunen, V: Commutator characterization of periodic pseudodifferential operators. Z. Anal. Anwend.19, 95-108 (2000)

13. Shi, S, Fu, Z, Zhao, F: Estimates for operators on weighted Morrey spaces and their applications to nondivergent elliptic equations. J. Inequal. Appl.2013, 390 (2013)

14. Iwaniec, T, Sbordone, C: Riesz transforms and elliptic PDEs with VMO coefficients. J. Anal. Math.74, 183-212 (1998) 15. Caffarelli, LA: Some regularity properties of solutions of Monge Ampère equation. Commun. Pure Appl. Math.44,

965-969 (1991)

16. Caffarelli, LA: Boundary regularity of maps with convex potentials. Commun. Pure Appl. Math.45, 1141-1151 (1992) 17. Caffarelli, LA, Gutiérrez, CE: Properties of the solutions of the linearized Monge-Ampère equation. Am. J. Math.119,

423-465 (1997)

18. Aimar, H, Forzani, L, Toledano, R: Balls and quasi-metrics: a space of homogeneous type modeling the real analysis related to the Monge-Ampère equation. J. Fourier Anal. Appl.4, 377-381 (1998)

19. Macías, RA, Segovia, C: Lipschitz functions on spaces of homogeneous type. Adv. Math.33, 257-270 (1979) 20. Garcá-Cuerva, J, Rubio de Francia, J-L: Weighted Norm Inequalities and Related Topics. North-Holland, Amsterdam

(1985)