FAST ALGORITHM FOR CALDER´ON-ZYGMUND OPERATORS：CONVERGENCE SPEED AND ROUGH KERNEL∗

Qixiang YANG（杨奇祥）

School of Mathematics and statistics，Wuhan University，430072 Hubei，China

E-mail∶qxyang@whu.edu.cn

Yong DING（丁勇）

School of Mathematical Sciences，Beijing Normal University，Laboratory of Mathematics and

Complex Systems（BNU），Ministry of Education，Beijing 100875，China

E-mail∶dingy@bnu.edu.cn



FAST ALGORITHM FOR CALDER´ON-ZYGMUND OPERATORS：CONVERGENCE SPEED AND ROUGH KERNEL∗

Qixiang YANG（杨奇祥）

School of Mathematics and statistics，Wuhan University，430072 Hubei，China

E-mail∶qxyang@whu.edu.cn

Yong DING（丁勇）

School of Mathematical Sciences，Beijing Normal University，Laboratory of Mathematics and

Complex Systems（BNU），Ministry of Education，Beijing 100875，China

E-mail∶dingy@bnu.edu.cn

In this article，we consider a fast algorithm for first generation Calder´on-Zygmund operators.First，we estimate the convergence speed of the relative approximation algorithm. Then，we establish the continuity on Besov spaces and Triebel-Lizorkin spaces for the operators with rough kernel.

First generation Calder´on-Zygmund operators；wavelets；rough kernel；ring type operators；Besov spaces and Triebel-Lizorkin spaces

2010 MR Subject Classification42B20；42B30

1　Introduction

In this article，we use a new wavelet representation to consider the first generation Calder´on-Zygmund operators.We estimate the convergence speed of such algorithm and consider some applications to rough kernel.

Let T be a linear operator which satisfies the following weak boundedness condition：

Schwartz distribution theory says there exists a kernel-distribution K（x，y）such that And there is an one-to-one correspondence relation between operator T and the kernel-distribution K.Beylkin-Coifman-Rokhlin analyzed the kernel-distribution K（x，y）by using wavelet basisin R2n.They got a matrix representationof theoperator T.See Section 2 for the definitions of notationsThe following equality is true in the sense of distributions

Beylkin-Coifman-Rokhlin applied such idea to study Calder´on-Zygmund operators with smooth kernel.See［2，4］and［30］.For x 6=y，assume that K（x，y）satisfies the size condition

For γ＞0 and m=［γ］，K（x，y）satisfies the following smooth conditions：

T satisfies the following T1 conditions：

Calder´on-Zygmund operators can be defined as follows：

Definition 1.1Given m∈N，m＜γ≤m+1，we say T∈CZOγor K（x，y）∈Aγ，if T satisfies the conditions（1.1），（1.2），（1.3），（1.4），（1.5），and（1.6）.

For Calder´on-Zygmund operators in CZOγ，the coefficientshave very good decreasing property.See［4］and［30］.Furthermore，they approximate T by some pseudodiagonal operator TRsuch that its kernel-distribution KR（x，y）satisfies

where，for all（ǫ，ǫ′，j，k，l）∈Λ2n，R≥1，satisfies

Theorem 1Given γ＞0，if T∈CZOγ，then

and this estimation is sharp.

Beylkin-Coifman-Rokhlin algorithm brings a lot of applications.See［2，9-11，23-25，29，30］and［31］.

For a convolution type operator T，like the Hilbert transform and Riesz transforms，the equation（1.2）becomes

Convolution type operator T satisfies automatically the following T1 conditions

Assume that K satisfies the following：

（i）The weak boundedness condition：

（ii）The size condition：

（iii）The smooth conditions：

Definition 1.2Given m∈N，m＜γ≤m+1，we say T∈CZγor K∈Bγ，if K satisfies（1.10），（1.11），（1.12），and（1.13）.

Note that，for an operator T∈CZγ，its kernel distribution K（x−y）has the same oscillation for x and y.However，the wavelet basishas different oscillation for x and y.If a kernel K（x−y）was analyzed by wavelets in R2n，such algorithm will produce some uncertainty of operator's norm.

In this article，we would present another wavelet algorithm，which is different from Beylkin-Coifman-Rokhlin's algorithm.The first object is to establish a fast approximation algorithm for these operators in Section 3，and estimate the approximation speed in Theorem 3.1.One of the key idea is to use the characterization of the kernel-distribution K（x）under wavelet basisin Rn（see Theorems 3.3）.Our another key idea is to transform the estimation of the boundedness of approximation operator to that of the boundedness of the ring type operators and some relative intermediate operators in equation（3.2）.

Remark 1.3Yang［30√］proved that the estimation in（1.7）is sharp.But，for c√onvolution type operators，the factorR is very strange.Our algorithm killed the factorR.Comparing to Beylkin-Coifman-Rokhlin algorithm，our algorithm needs less computation.But our approximation operator converges more quickly to the original operator.

Beylkin-Coifman-Rokhlin algorithm is also applied to study the continuity of operators with rough kernel.Assume that the smooth conditions（1.4）and（1.5）are weakened to the following Hüormander condition：One would like to establish the continuity of such operators for L2space or even more general Banach spaces.But authors in［10］and［25］proved that Hüormander condition can not ensure the relative continuity for certain Besov spaces and Triebel-Lizorkin spaces.The authors proved the continuity of non convolution Calder´on-Zygmund operators under some conditions stronger than Hüormander condition for Besov spaces and Tribel-Lizorkin spaces in［9］and［25］.For convolution operator，the Hüormander condition becomes the following condition：

Definition 1.4We say T∈CZH or K∈BH，if K satisfies（1.10），the size condition（1.11），and the Hüormander condition（1.15）.

Remark 1.5The L2continuity is equivalent to the conditionKˆ∈L∝.Furthermore，K∈BHcan ensure the continuity on Besov spaces（1≤p，q≤∞）.According to the results in［9-11，23］and［25］，Beylkin-Coifman-Rokhlin algorithm causes loss of a part of regularity when considering convolution type operators with rough kernel.

Another way to consider a rough kernel K（x）is to write K（x）=Ω（x）|x|−n.One assumesΩ（x）satisfies some slightly weaker condition than K（x）∈BH.Such idea has been used heavily more than 60 years.See［1，3，5-8，12-14，16-18］and［29］.In this article，we apply wavelets to consider rough kernel K（x）inand BF.

For s=1，···，n，let esbe the vector whose s−th component is 1 and other components are 0.For u∈N，let Z

Two other kinds of rough kernels are defined as follows.See also［24］and［27］.

Definition 1.6（i）We say T∈CZB，if T satisfies（1.8）and

（ii）We say T∈CZF or K∈BF，if K satisfies（1.10）and

The second object is to establish the continuity on Banach spaces for rough kernel operators in Theorem 5.5.One of the key idea is to apply our special wavelet representation to the study of the continuity of the operators with rough kernel.The second key idea is to use some special method to consider the intermediate operatorsin equation（5.2）.Comparing to the known results，our condition to ensure the continuity on Besov spacesis the most weaker one except for the casewhere T is continuous on

2　Wavelets and Function Spaces

We present some preliminaries on wavelets and function spaces in this section.Let≡，where φ∈S（Rn）with

Besov spaces and Triebel-Lizorkin spaces are defined as follows.See［28］.

Definition 2.1（i）For 1≤p，q≤∞，Besov spacesis defined as the collection of distribution f such that

（ii）For 1≤p＜∞，1≤q≤∞，Triebel-Lizorkin spacesis defined as the collection of distribution f such that

Let Φ0（x）and Φ1（x）be the one-dimensional Daubechies father wavelet and mother wavelet，respectively.For m∈N，Φ0（x），Φ1（x）are assumed to be real valued functions belonging toFor arbitraryWe also put

and

For function f belonging to Besov spacesor Triebel-Lizorkin spaces，we denoteWe have

Furthermore，we can find out the following results in［24，25］and［29］.

Theorem 2.2（i）For 1≤p，q≤∞，f∈if and only if

3　Approximation Operators and Convergence Speed

Beylkin-Coifman-Rokhlin developed their famous algorithm to deal with Calder´on-Zygmund operator.See［4，25］and［29］.According to the conclusion in Remark 1.3，this algorithm is not best adapted to the study of convolution Calder´on-Zygmund operator.In addressing the problem of engineering，we meet often the problem about the calculation of convolution Calder´on-Zygmund operator.For example，Hilbert transformation and Riesz transformation are such special operators.So，we develop a new approximation algorithm for convolution operators in this section.

3.1Approximation operators

According to equation（2.1），the distribution K（x）can be written as

In this section，we develop a new algorithm to calculate convolution type Calder´on-Zygmund operators.For R=1 and∀（ǫ，j，k）∈Λn，we denote=0，∀k 6=0.Fix R≥2 and∀（ǫ，j，k）∈Λn.If 2we denote ；otherwise=0.Let

Let

Then，SRare ring type operators，each TRis an approximation of T，andT˜Ris the relative error operator.

Theorem 3.1Given γ＞0 and T∈CZγ，the following hold：

We will prove this Theorem in Section 4.

Remark 3.2Our algorithm is different from the famous Beylkin-Coifman-Rokhlin algorithm.See［4］and［30］.Comparing to Beylkin-Coifman-Rokhlin algorithm，we need less computation for the approximation operators，but the approximation speed is faster.Hence，our algorithm is adapted better to the study of convolution type Calder´on-Zygmund operators，such as Hilbert operator and Riesz operators.Our algorithm provides a good choice for engineering problems.

3.2Wavelet characterization of kernels in Bγ

In this subsection，we consider the wavelet characterization of the kernel in Bγ.

ProofIt is seen that K∈Bγimplies that K satisfies（3.1）.Now，we prove the inverse conclusion.We divide the proof into three steps.

（i）K（x）satisfies the weak boundedness condition（1.10）.Let

As K（x）=K1（x）+K2（x），we get the desired estimation.（ii）K（x）satisfies the size condition（1.11）.Let

Then，we have K（x）=K1（x）+K2（x），where

and（iii）K（x）satisfies the smooth condition（1.13）.Let

Then，we have K（x）=K1（x）+K2（x）+K3（x）and∀|α|=m，2|x−x′|≤|x|，we have

3.3Ring type operators and some relative intermediate operators

In this subsection，we decompose the ring type operators to a series of intermediate oper-ators.For arbitrary function

For s∈N−and（ǫ，ǫ′，l）∈Gn，s，we have

Hence，

4　Proof of Theorem 3.1

4.1 Besov spaces

Since there are at most C2n|s|number of index（ǫ，ǫ′，l）in Gn，s，Theorem 3.1（i）is a direct corollary of Theorem 3.3 and the following theorem.

Theorem 4.1Given 1≤p，q≤∞，γ＞0 and T∈CZγ，for（ǫ，ǫ′，s，l）∈Gn，we have

For s∈N and（ǫ，ǫ′，l）∈Gn，s，we have

Hence，

For s∈N−and（ǫ，ǫ′，l）∈Gn，s，we have

Hence，

4.2 Tribel-Lizorkin spaces

As there are at most C2n|s|number of index（ǫ，ǫ′，l）in Gn，s，Theorem 3.1（ii）is a direct corollary of Theorem 3.3 and the following theorem.

Theorem 4.2Given 1＜p＜∞，1≤q≤∞，γ＞0，and T∈CZγ，for（ǫ，ǫ′，s，l）∈Gn，

We consider only the cases：R＞|s|+1.

For s∈N and（ǫ，ǫ′，l）∈Gn，s，we have

Hence，

For s∈N−and（ǫ，ǫ′，l）∈Gn，s，we have

Hence，

5　Rough Kernel and General Banach Spaces

To study the continuity of different operators，one introduced different representation algorithm.Seehas different oscillation for x and y.Hence，Beylkin-Coifman-Rokhlin algorithm does not adapt to the structure of first generation Calder´on-Zygmund operators.To overcome this shortage，we will use another wavelet algorithm to consider the rough kernel operators.

5.1Rough kernel

Operators with rough kernel were studied extensively.See［1，6，7，9，17，20，21，23］and［25］.In this subsection，we consider some relations about rough kernel.

To study the operator's continuity on Triebel-Lizorkin spaces，we need often use Hardy-Littlewood maximal operator.To meet this requirement，we introduced rough kernel in WF. Given

By a direct calculation，we know that the condition K（x）∈WFis slightly weaker than the condition K（x）∈BF.In fact，it is easy to see

Theorem 5.2If K（x）∈BF，then K（x）∈WF.

Applying the above theorem，it is seen that the smooth kernel-distributions are all rough kernel-distributions.

Among all the known result concerning operator's continuity，the conditionis the weakest condition except the conditionˆK（ξ）∈L∝.In particularly，we have

5.2Continuity of operators with rough kernel

Boundedness of operators with rough kernel was studied heavily.See［1，2，5，6，9，17，19-22］and［25］.In this section，we will introduce a new method to consider the continuity of the operators with rough kernel.We have

Theorem 5.5（i）If 1≤p，q≤∞and K（x）∈，then‖T‖≤CK.

（ii）If 1＜p＜∞，1≤q≤∞and K（x）∈WF，then

Remark 5.6（i）Comparing to the known results，our condition to ensure the continuity on Besov spaces（1≤p，q≤∞）is the weakest one except for L2=B˙20，2，where the condition T is continuous on L2⇔Kˆ（ξ）∈L∝.

（ii）To consider the operator's continuity on Triebel-Lizorkin spaces，usually，we have to deal with the Hardy-Littlewood maximum operator.So，we have to use a condition slightly stronger than the condition for Besov space.

5.3Intermediate operators

For first generation Calder´on-Zygmund operators，we introduce first a new method to studPy the action of the operators T on the functions f（x）.For arbitrary function f（x）=and j，s∈Z，denote

For s∈Z，let Hn，|s|=｛0，···，2|s|−1｝nand Gn，s=｛（ǫ，ǫ′，l）：ǫ，ǫ′∈En，l∈Hn，|s|｝.Let Gn=｛（ǫ，ǫ′，s，l）：s∈Z，（ǫ，ǫ′，l）∈Gn，s｝.For s∈N and（ǫ，ǫ′，l）∈Gn，s，we have

Z Let Φ∈，∈′，s，l（x）=RΦ∈（x−y）Φ∈′（2sy−l）dy.We have

For s∈N−and（ǫ，ǫ′，l）∈Gn，s，we have

Hence，we can decompose operator T into a series of operators.Formally，

As there are at most C2nsnumber of index in the set Gn，s，Theorem 5.5 is a direct corollary of the following Theorem.

Theorem 5.7For（ǫ，ǫ′，s，l）∈Gn，we have：

6　Proof of Theorem 5.7

6.1Continuity on Besov spaces As Φ∈，∈′，s，lj，k′（x）are just like wavelets，hence we have

For s∈N and（ǫ，ǫ′，l）∈Gn，s，we have

Hence，

For s∈N−and（ǫ，ǫ′，l）∈Gn，s，we have

Hence，

6.2Continuity on Triebel-Lizorkin spaces

For s∈N and（ǫ，ǫ′，l）∈Gn，s，we have

It is seen that

and for u≥1，

We have

It is seen that

and for u≥1，

We have

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May 21，2014；revised October 2，2015.Supported by NNSF of China（11271209，11371057，11571261）and SRFDP（20130003110003）.