Hung Viet Le

APUInternationalSchool，286LanhBinhThang，Ward 11，District11，HoChiMinh City，Viet Nam

?

Commutators of Lipschitz Functions and Singular IntegralswithNon-Smooth KernelsonEuclideanSpaces

Hung Viet Le?

APUInternationalSchool，286LanhBinhThang，Ward 11，District11，HoChiMinh City，Viet Nam

.In this article，we obtain the Lp-boundedness of commutators of Lipschitz functions and singular integrals with non-smooth kernels on Euclidean spaces.

Commutators，singularintegrals，maximalfunctions，sharpmaximalfunctions，muckenhoupt weights，Lipschitz spaces.

AMS Subject Classifications:42B20，42B25，42B35

1　Introduction

Consider the singular integral operator T defined by

where f is a continuous function with compact support，x/∈suppf；and the kernel K（x，y）is a measurable functiondefinedon（Rn×Rn）?with?=｛（x，x）:x∈Rn｝.If b∈BMO（Rn），then the commutator［b，T］of a BMO function b and the singular integral operator T is defined by

The Lp-boundedness（1＜p＜∞）of T and Tbare well known in the Euclidean setting，provided that the kernel K（x，y）of the operator T satisfies H¨ormander's conditions（see［1，15-17］among many other good references）.In 1999，Duong and McIntosh［3］obtained the Lp-boundedness of T，under the assumption that the kernel K（x，y）satisfies some conditions which are weaker than H¨ormander's integral conditions.The boundedness of the operator T with non-smooth kernel on Lp（w）（w∈Ap（Rn），1＜p＜∞）was proved by Martell［12］.Moreover，Duong and Yan［4］obtained the Lp-boundedness ofthe commutator Tbunder some conditions which are weaker than H¨ormander's pointwise conditions.Lin and Jiang［11］also obtained the Lp-boundedness of Tb，but with b∈Lipα，w（Rn）.See also［8，9，13，18］for additional results on these topics.

The purpose of this paper is to extend the results in［11］.That is，we would like to obtain the Lp-boundedness（1＜p＜∞）of the operator T→b，where

2　Background

2.1Apweights

For a ball B in Rn，let|B|denote the measure of the ball B.A weight w is said to belong to the Muckenhoupt class Ap（Rn），1＜p＜∞，if there exists a positive constant C such that

for all balls B in Rn.The smallest constant C for which the above inequality holds is the Apbound of w.The class A1（Rn）consists of non-negative functions w such that

for all balls B in Rn.It is well-known that（see［7，17］for instance）if w∈Ap（Rn）for some p∈［1，∞），then for any measurable subset E?B，there exist positive constants γ and C such that

Inequality（2.1）indeed holds with γ∈（0，1）.This will be used in the estimate of（3.3）below.Furthermore，if w∈Ap（Rn）（1≤p≤∞），thenit satisfies the reverse H¨olderinequality. That is，there exist s′＞1 and c＞0（both depending on w）so that

A weight w is said to belong to the class Ap，q（Rn），1＜p，q＜∞，if there exists a positive constant C such that

for all balls B?Rn.Observe that

When p=1 and q＞1，we say that w∈A1，q（Rn）if there exists a positive constant C such that for all balls B?Rn，

It follows from H¨older's inequality that for 1＜q1＜q2＜∞，

and

The following lemma is necessary for the proof of our theorem.

2.2Approximation of the identity

We assume that there exists a class of operators At（t＞0）which can be represented by the kernels at（x，y）in the sense that

Moreover，the kernels at（x，y）satisfy the following conditions

where

Here s is a positive，bounded，decreasing function satisfying

where k appears in（1.2）.

Remark 2.1.The functions htabove satisfy the following properties（see［4，5］）: i）There exist positive constants C1and C2such that

ii）There exists a positive constant C such that

where M is the Hardy-Littlewood maximal operator.

The class of operators Atplays the role of approximation to the identity.The existence of such a class of operators Atwas verified in［3］.

Now consider the operators T and T→bgiven in（1.1）and（1.2）respectively.Let Atand Bt（t＞0）be two classes of operators which satisfy（2.3），（2.4）and（2.5）.Denote by K（x，y）-Kt（x，y）the kernels ofthe operators（T-TBt），and K（x，y）-Kt（x，y）as the kernels of（T-AtT）.We state below some assumptions which are necessary for our theorem.

（a）T is a bounded linear operator from Lr（Rn）to Lr（Rn）for some r∈（1，∞）；

（b）There exist positive constants c1and CAsuch that

（c）There exist positive constants c2，c3and β＞nk（k appears in（1.2））such that

In the sequel，the letter C will denotea constant，which may vary at different occurrences. However，it is independent of any essential variable.

3　Main theorem

Let τ=max｛τ1，τ2，τ3｝.Assume that w∈A1，τ（Rn），and bi∈Lipαi，w（Rn）for 1≤i≤k.

Let T，given by（1.1），satisfy assumptions（a），（b）and（c）.Then there exists a constant C＞0，independent of f，such that

where

Remark 3.1.Observe that for the case k=1，w is only required to be in A1，τ1（Rn）.

Proof.First，we show that there exist r1，r2，r3＞1 such that 1＜rs＜q0，where r:=r1r2r3.For the case k=1，since 1＜s＜q0，there exists an r＞1 such that 1＜rs＜q0.We then choose some numbers r1，r2，r3＞1 such that r=r1r2r3.Now suppose k＞1.Since s＜√q0，there exists an r3such that 1＜s＜r3＜√q0.Then sr3＜q0.Pick a number t1∈（sr3，q0），and let t=t1/sr3＞1.We choose a number r2∈（1，t）and let r1=t/r2，r:=r1r2r3.Then we have r1，r2，r3＞1，and 1＜rs＜q0.For the rest of the proof，we denote t=r1r2and r=r1r2r3=tr3. Let→λ=（λ1，···，λk）=（（b1）B，···，（bk）B），where

Then，the above equation implies that

Note that Eqs.（3.1）and（3.2b）imply that for 1≤j≤k-1，

We have the following lemmas.

Proof.Observe that

By H¨older's inequality，we have

Thus，

Let

where the last inequality follows from（3.3）.Combining（3.5）and（3.6）yields

So，we complete the proof of the lemma.

Proof.Take a ball B which contains x.We have

Thus，by H¨older's inequality and Lemma 3.1，we see that

On the other hand，y∈B and z∈2j+1B2jB imply that|y-z|≥2j-1rB.So，

Hence，an application of Lemma 3.1 yields

provided that

for some ?＞0.Combining the estimates of J3（x）and J4（x）and taking the supremum over all balls B containing x yields the conclusion.

Lemma 3.3.It holds

where ck，iare constants depending on k and i.

Proof.For an arbitrary fixed x∈Rn，choose a ball B which contains x.Following［16］，we split f=fχ2B+fχ（2B）c≡f1+f2，and write

By H¨older's inequality，Lemma 3.1 and Lemma 3.2 respectively，we have

and

where

Note that

Therefore，by Theorem 5.3［12］and Lemma 3.1，

Observe that

The third and last inequalities are due to Theorem 5.3［12］and Lemma 3.1 respectively.

It remains to estimate K7（x）.By hypothesis and Lemma 3.1，we have

provided that β＞nk.Finally，the result follows from combining all of the estimates above and taking the supremum over all balls B containing x.

Lemma 3.4.If w∈A1，s′（Rn）for some s′＞1，then there exists a constant C＞0 such that

Proof.Let B be a ball which contains x.By H¨older's inequality，

Taking the supremum over all balls B which contain x yields the desired result.

By Eqs.（3.2a）and（3.2b），we have that，for 1≤j≤k，

Thus for 1≤j≤k，

where

Therefore，we may apply Lemma 2.1，Lemma 3.4，Theorem 5.3［12］，together with equation（3.2a）to conclude that

and

Since 1＜rs＜q0，it follows that for 1≤j≤k-1，

Hence

which implies that

or equivalently，

Again，by Lemma 2.1，Lemma 3.4 and Eq.（3.1），we infer that for 1≤i≤k-1，

provided that 1＜rs＜q0＜qk-i＜n/νifor 1≤i≤k-1.Consequently，by Theorems 4.2，5.3［12］，Lemma 3.3，inequalities（3.7）-（3.9），and induction argument，we conclude that

Thus，we complete the proof.

Remark 3.2.We could apply the reverse H¨older inequality to prove Lemma 3.4，thereby eliminating the assumption that w∈A1，s′（Rn）.However，the exponent s′appearing in the reverse H¨older inequality（see（2.2））depends on w and may be very close to 1，which means that its conjugate exponent s could be very large（see［7，17］）.This limits the value of q0for which the theorem holds，since q0is necessarily greater than s.

Remark 3.3.Let ? be a non-decreasing positive function on R+.Denote by ?（f，B），the mean oscillation of a function f on a ball B?Rn，as|B|-1RB|f（x）-fB|dx.Define BMO?as the space of all functions f satisfying ?（f，B）≤C?（r），whenever B is a ball with radius r（see［10］）.Note that when ?≡1，then BMO?=BMO，the space of all functions of bounded mean oscillation.Let Λα，0＜α≤1，be the space of Lipschitz continuous functions，Λα=｛f:|f（x）-f（y）|≤C|x-y|α，?x，y∈Rn｝.A function ψ:［0，∞）→［0，∞）is said to be a Young function if it is continuous，convex，increasing and satisfying ψ（0）=0 and limt→∞ψ（t）=∞.The Orlicz space Lψis defined as the space of all functions f such thatRψ（λ|f|）＜∞，for some λ＞0.

Now consider the singular integral Tf and the commutator Tbf（as defined in the introduction），but with the convolution kernel

With this type of kernel，Janson［10］proved that b belongs to BMO?if and only if Tbmaps Lp（1＜p＜∞）boundedly into Lψ，where ? and ψ are related by the equation ?（r）= rn/qψ-1（r-n），or equivalently，ψ-1（t）=t1/p?（t-1/n）.When ?（t）=tα（0＜α＜1），ψ（t）=tqwith 1/q=1/p-α/n，then it is evident that Lipα（Rn）=BMOtα（Rn）and Lψ（Rn）=Lq（Rn）. In this particular case，Janson's Theorem says that b belongs to Lipα（Rn）if and only if Tbmaps Lp（Rn）（1＜p＜∞）boundedly into Lq（Rn），where 1/q=1/p-α/n.It is interesting to note that the above necessary condition is the same as in Theorem 3.1，when k=1 and w≡1，but with different kernel K.

Acknowledgments

The author would like to express his gratitude toward the referees for their thorough review and precious suggestions.

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.Email address:hvle2008@yahoo.com（H.V.Le）

25 October 2015；Accepted（in revised version）11 April 2016