# An Approximation Method for Convolution Calderón-Zygmund Operators

Authors: Yunxia Yu; Zhanying Yang
Journal: Abstract and Applied Analysis
 (2013)

---

## Abstract

We present an approximation method for convolution Calderón-Zygmund operators. We give a uniform approximation accuracy of the operators on the endpoint Triebel-Lizorkin spaceF˙10;q(2<q<∞). Our proof mainly relies on the n-dimensional Daubechies wavelet bases and the atomic-molecular approach.


---

## Body

## 1. Introduction and Main Result

For rapid application of dense matrices (or integral operators) to vectors, the celebrated work of Beylkin et al. [1] introduced a class of numerical algorithms which are based on the 2n-dimensional wavelet bases with compact supports. These algorithms are also applicable to all Calderón-Zygmund operators and pseudodifferential operators. Since then, their algorithms are widely used in compression of matrices, operator approximation, and establishing boundedness of operators see [2–9]; In particular, Beylkin et al. [1] approximated a class of Calderón-Zygmund operators by banded operators and gave the approximation accuracy. It is intriguing to know whether we can get some similar approximation methods on some more general spaces. Notice that Yang [9] approximated the operators by compact operators and gave the approximation accuracy on Lp(1<p<∞).In this paper, we are interested in considering the approximation method for a class of Calderón-Zygmund operators on Triebel-Lizorkin spaces. However, due to technical reasons, we can only get an approximation method for convolution Calderón-Zygmund operators on some endpoint Triebel-Lizorkin spaces (see Theorem1).Now, we introduce a class of Calderón-Zygmund operators. Let𝒟=𝒟(ℝn) denote the space of Schwartz test functions and 𝒟' the space of Schwartz distributions (the dual of 𝒟). Suppose that we have a linear continuous mapping T:𝒟→𝒟' associated with a kernel K(x,y) (in the sense that 〈Tf,g〉=∬g(x)K(x,y)f(y)dxdy for test functions f and g with disjoint supports). We write T∈CZOγ if the following three conditions are satisfied.(I)
K(x,y) is continuous on Ω=ℝn×ℝn∖{(x,y):x=y} and satisfies
(1)|K(x,y)|≤C|x-y|-n;
for all x,x′,y with 2|x-x′|≤|x-y|,
(2)|K(x,y)-K(x′,y)|+|K(y,x)-K(y,x′)|≤C|x-x′|γ|x-y|n+γ,
where 0<γ≤1.(II)
Weak boundedness condition:(3)|〈Tf,g〉|≤CRn(∥f∥∞+R∥∇f∥∞)(∥g∥∞+R∥∇g∥∞),∀R>0,u∈ℝn,f,g∈C01(B(u,R)).(III)
T1 condition: T1∈BMO,T*1∈BMO.Convolution Calderón-Zygmund operators, such as Hilbert and Riesz operators, are commonly used in engineering. For a convolution operatorT, its kernel K(x,y) can be written as K(x,y)=K(x-y). In this case, the conditions for the operator T∈CCZOγ are reduced to the following:
(4)|K(x)|≤C|x|-n;
for 2|x-x′|≤|x|,
(5)|K(x)-K(x′)|≤C|x-x′|γ|x|n+γ;(6)|∫‍K(x)ψt(x)dx|≤C(∥ψ∥∞+∥∇ψ∥∞),
where ψ∈C01(B(0,1)) and ψt(x)=ψ(x/t) for t>0. For convenience, let CCZOγ denote the collection of all convolution operators in CZOγ.In what follows, we restrict our attention to the operatorT in CCZOγ. In general, the operator is analyzed by the 2n-dimensional wavelet bases. However, Z. Y. Yang and Q. X. Yang [10], making use of the n-dimensional Daubechies wavelet bases, approximated the operator T by the banded operator and gave the approximation accuracy C2-uγ on the homogeneous Besov spaces B˙p0,q(1≤p,q≤∞). In this paper, we focus on an approximation method for the operator T and obtaining the uniform approximation accuracy on the endpoint Triebel-Lizorkin spaces F˙10,q,2<q<∞, whose definitions will be given in Section 2.We first introduce some notations. LetΦ0(x) and Φ1(x) be the one-dimensional Daubechies father and mother wavelets, respectively. Assume that they are the real-valued and sufficiently regular functions. For x∈ℝn and ε=(ε1,ε2,…,εn)∈{0,1}n, denote Φε(x)=∏i=1nΦεi(xi). For any j∈ℤ and k∈ℤn, let
(7)Φj,kε(x)=2jn/2Φε(2jx-k).
We also put
(8)En={0,1}n∖{0},Λn={λ=(ε,j,k),ε∈En,j∈ℤ,k∈ℤn}.
Then, {Φj,kε(x)}λ∈Λn forms orthonormal bases in L2(ℝn), and it can be used to characterize general functions or distributions.For anyλ∈Λn, let bj,kε=〈K(x),Φj,kε(x)〉, then we have the representation
(9)K=∑(ε,j,k)∈Λn‍bj,kεΦj,kε(x)
in the sense of distribution. Now, we present the approximation of T by the banded operator Tu. For any integer u≥0 and ε∈En, we define
(10)Kuε(x)={∑j‍bj,0εΦj,0ε(x),u=0;∑2u-1≤|k|<2u∑jbj,kεΦj,kε(x),u≥1.
Let Tuε be the annular operator associated to the kernel Kuε(x), and let
(11)Tu=∑0≤u′≤u‍‍∑ε∈En‍Tu′ε.
Then, we can approximate T by the banded operator Tu and get the uniform approximation accuracy on the endpoint Triebel-Lizorkin space F˙10,q(1<q<∞). Our result is stated as follows.Theorem 1.
Let1<q<∞ and 0<γ≤1. If T∈CCZOγ, then
(12)∥T-Tu∥F˙10,q→F˙10,q≤Cu2-uγ,u≥1.Throughout this paper, the symbolC denotes a constant that is independent of the main parameters involved but whose value may differ from line to line.


## 2. Endpoint Triebel-Lizorkin Spaces

Let𝒮(ℝn) be the space of tempered test functions. Let φ∈𝒮(ℝn) with supp φ^⊂{ξ∈ℝn:1/2≤|ξ|≤2} and |φ^|≥c>0 for {3/5≤|ξ|≤5/3}; let φj(x)≡2jnφ(2jx). Let α∈ℝ, 0<p<∞, and 0<q≤∞. Then, the homogeneous Triebel-Lizorkin space F˙pα,q is defined as the collection of all f∈𝒮′/P (the tempered distributions modulo polynomials) such that
(13)∥f∥F˙pα,q≡∥(∑j‍(2jα|φj*f|)q)1/q∥p<∞
with the usual modification made when q=∞; see also Triebel [11]. And it is well known that the homogeneous Triebel-Lizorkin space and Besov space are the same when p=q.For the homogeneous Triebel-Lizorkin space, Koskela et al. gave the characterization via grand Littlewood-Paley functions in [12] and gave the pointwise characterization in [13]. Now, we recall its characterization based on wavelets. In fact, the Daubechies wavelet bases {Φj,kε(x)}λ∈Λn is an unconditional bases in the space F˙p0,q. For any j∈ℤ and k∈ℤn, let
(14)Qj,k={x∈ℝn:2jx-k∈[0,1)n},
and let χ(2jx-k) be the characteristic function of the dyadic cube Qj,k. For any f∈𝒮′/P, if we can define fj,kε=〈f,Φj,kε〉 for all (ε,j,k)∈Λn, then we have the representation f=∑(ε,j,k)∈Λn‍fj,kεΦj,kε(x) in the sense of distribution. The space F˙p0,q is characterized in terms of wavelets in the following way (for more details, see also [6, 7, 14]).Proposition 2.
For1≤p<∞ and 1≤q≤∞, there exist two positive constants Cp,q and Cp,q′ such that
(15)Cp,q∥f∥F˙p0,q≤∥(∑ε,j,k‍2jqn/2|fj,kε|qχ(2jx-k))1/q∥Lp≤Cp,q′∥f∥F˙p0,q.Triebel-Lizorkin spaces have been studied by means of the atomic and molecular decompositions. Next, we state the atomic-molecular decomposition for the endpoint spaceF˙10,q; see Meyer and Yang [7] for more details. Let 𝔔 denote the collection of all dyadic cubes Qj,k,j∈ℤ,k∈ℤn. Now, we recall the following two definitions which can be found in [6, 7, 15].Definition 3.
Let1≤q<∞. a is said to be a F˙q0,q-atom in F˙10,q with the norm C if there exists a ball B such that(i)
Suppa⊂B;(ii)
∥a∥F˙q0,q≤C|B|(1/q)-1;(iii)
∫‍adx=0.Definition 4.
Let1≤q<∞. a is said to be a F˙q0,q-molecule in F˙10,q with the norm C if there exists a cube Qs,p∈𝔔 such that(i)
a=∑(ε,j,k)∈ΛnQj,k⊂Qs,paj,kεΦj,kε;(ii)
∥a∥F˙q0,q≤C|Qs,p|(1/q)-1.
For convenience, in the above two definitions,a is said to be a F˙q0,q-atom or a F˙q0,q-molecule when C=1.Proposition 5.
Let1≤q<∞. The following three conditions are equivalent:  (i)
f∈F˙10,q.(ii)
There exist{μm}m∈ℤ∈l1andF˙q0,q-atomsamsuch thatf=∑m∈ℤμmam.(iii)
There exist{λm}m∈ℤ∈l1andF˙q0,q-moleculesamsuch thatf=∑m∈ℤλmam.In fact, for anyf∈F˙10,q, we can write it as f=∑m‍λmam, where each am is a F˙q0,q-molecule and {λm}∈l1, and it can be verified that
(16)∥f∥F˙10,q~∑m‍|λm|,
namely,
(17)C1∥f∥F˙10,q≤∑m‍|λm|≤C2∥f∥F˙10,q
for some fixed C1,C2>0 independent of f. For more details, we refer the reader to [6, 15], This will play a key role in Section 3.


## 3. Estimate ofTuε′ and Proof of Theorem 1

To prove Theorem1, we first estimate the annular operator Tuε′. For any ε′∈En and integer u≥1, let
(18)Buε′=supj′∑2u-1≤|k′|<2u‍2-j′n/2|bj′,k′ε′|.Lemma 6.
For1<q<∞, let τq=1-(1/q). One has
(19)∥Tuε′∥F˙10,q→F˙10,q≤CuτqBuε′.Proof.
In terms of the molecular decomposition for the spaceF˙10,q, we only need to prove that for an arbitrary F˙q0,q-molecule a,
(20)∥Tuε′a(x)∥F˙10,q≤CuτqBuε′,
where C is independent of a.
Denotem=j′-j, then we have
(21)Tuε′a(x)=∑ε,m‍∑2u-1≤|k′|<2u‍×∑Qj,k⊂Qs,p‍bj+m,k′ε′aj,kε(Φj,kε*Φj+m,k′ε′)(x).
Thus,
(22)∥Tuε′a(x)∥F˙10,q≤∑ε‍∥∑m≤0‍∑2u-1≤|k′|<2u‍×∑Qj,k⊂Qs,p‍bj+m,k′ε′aj,kε(Φj,kε*Φj+m,k′ε′)(x)∥F˙10,q+∑ε‍∥∑m>0‍∑2u-1≤|k′|<2u‍×∑Qj,k⊂Qs,p‍bj+m,k′ε′aj,kε(Φj,kε*Φj+m,k′ε′)(x)∥F˙10,q:=I1+I2.
In the following, we consider the estimates of I1 and I2.
(i) For anym≤0, put Φ~mε,ε′(x)=2-mn(Φm,0ε′*Φε)(x), then
(23)I1≤C∑m≤0‍2mn∑2u-1≤|k′|<2u‍×∥∑Qj,k⊂Qs,p‍bj+m,k′ε′aj,kεΦ~mε,ε′(2jx-(k+2-mk′))∥F˙10,q.
For fixed m and k′, let Tk′m be the operator associated with the kernel
(24)Kk′m(x,y)=∑Qj,k⊂Qs,p‍2jnΦ~mε,ε′(2jx-(k+2-mk′))×Φε(2jy-(k+2-mk′)).
By the properties of Daubechies wavelets, we can get that the operator Tk′m is bounded on F˙10,q (see also [16, lemma 3.1]).
Set(25)gk′m=∑Qj,k⊂Qs,p‍bj+m,k′ε′aj,kεΦε(2jx-(k+2-mk′)).
By means of the orthonormality of the wavelet bases, the right side of (23) is equal to
(26)C∑m≤0‍2mn∑2u-1≤|k′|<2u‍∥Tk′mgk′m∥F˙10,q.
Moreover, I1 is bounded by
(27)C∑m≤0‍2mn∑2u-1≤|k′|<2u‍∥gk′m∥F˙10,q.
Namely, we have
(28)I1≤C∑m≤0‍2mn∑2u-1≤|k′|<2u‍×∥∑Qj,k⊂Qs,p‍bj+m,k′ε′aj,kεΦε(2jx-(k+2-mk′))∥F˙10,q.
Notice that the sum ∑j:Qj,k⊂Qs,p is adding among all j with s≤j, we split ∑j:Qj,k⊂Qs,p into ∑j:Qj,k⊂Qs,p=∑s≤j<s+u+∑j≥s+u. Hence, we obtain
(29)I1≤C∑m≤0‍2mn×∑2u-1≤|k′|<2u‍(∥∑s≤j<s+uQj,k⊂Qs,pbj+m,k′ε′aj,kεΦε∑s≤j<s+uQj,k⊂Qs,pbj+m,k′ε′aj,kεΦε×(2jx-(k+2-mk′))∑s≤j<s+uQj,k⊂Qs,pbj+m,k′ε′aj,kεΦε∥F˙10,q+∥∑j≥s+uQj,k⊂Qs,pbj+m,k′ε′aj,kεΦε×(2jx-(k+2-mk′))∑j≥s+uQj,k⊂Qs,pbj+m,k′ε′aj,kεΦε∥F˙10,q):=C∑m≤0‍2mn∑2u-1≤|k′|<2u‍(J1,k′m,u+J2,k′m,u).
Now, we consider J1,k′m,u and J2,k′m,u. As for J1,k′m,u, we use Proposition 2 to get that
(30)J1,k′m,u≤C∫‍(∑s≤j<s+uQj,k⊂Qs,p|bj+m,k′ε′|q×|aj,kε|qχ(2jx-(k+2-mk′))∑s≤j<s+uQj,k⊂Qs,p|bj+m,k′ε′|q)1/qdx≤C2mn/2supj′2-j′n/2|bj′,k′ε′|×∫‍(∑s≤j<s+uQj,k⊂Qs,p2jqn/2|aj,kε|q×χ(2jx-(k+2-mk′))∑s≤j<s+uQj,k⊂Qs,p|bj+m,k′ε′|q)1/qdx.
Let ⊗1n[ai,bi] denote the Cartesian product [a1,b1]×[a2,b2]×⋯×[an,bn]. Let
(31)fj,k′m(x)=∑k:Qj,k⊂Qs,p‍2jqn/2|aj,kε|qχ(2jx-(k+2-mk′)),
then the support of fj,k′m(x) is contained in ⊗1n[(pi/2s)+(ki′/2j+m),((pi+1)/2s)+(ki′/2j+m)]. Moreover, we can obtain that
(32)suppfj,k′m(x)∩suppfj1,k′m(x)=∅
when |j-j1|≥C. In other words, the number of the nonzero terms in the sum ∑s≤j<s+u‍fj,k′m(x) is a constant, which is independent of u. By the inequality (α+β)1/q≤α1/q+β1/q, we have
(33)J1,k′m,u≤C2mn/2supj′2-j′n/2|bj′,k′ε′|×∫‍∑s≤j<s+uQj,k⊂Qs,p2jn/2|aj,kε|χ(2jx-(k+2-mk′))dx≤C2mn/2supj′2-j′n/2|bj′,k′ε′|×∫‍∑s≤j<s+uQj,k⊂Qs,p2jn/2|aj,kε|χ(2jx-k)dx.
From Hölder's inequality, we obtain
(34)J1,k′m,u≤C2mn/2supj′2-j′n/2|bj′,k′ε′|uτq×∫‍(∑s≤j<s+u‍×(∑k:Qj,k⊂Qs,p‍2jn/2|aj,kε|χ(2jx-k))q)1/qdx≤C2mn/2supj′2-j′n/2|bj′,k′ε′|uτq×∫‍(∑s≤j<s+uQj,k⊂Qs,p2jqn/2|aj,kε|qχ(2jx-k))1/qdx.
Since ∥a∥F˙q0,q≤C|B|1/q-1, then we get that ∥a∥F˙10,q≤C. Thus
(35)J1,k′m,u≤C2mn/2uτqsupj′2-j′n/2|bj′,k′ε′|.
As for J2,k′m,u, we have
(36)J2,k′m,u≤C2mn/2supj′2-j′n/2|bj′,k′ε′|×∥∑j≥s+uQj,k⊂Qs,p2jn/2aj,kεΦε(2jx-(k+2-mk′))∥F˙10,q:=C2mn/2supj′2-j′n/2|bj′,k′ε′|∥hu,k′m(x)∥F˙10,q.
By Definition 3, we can verify that (2-m+2M)-τqnhu,k′m(x) is a F˙q0,q-atom with the norm C, see also [16] for more details. Furthermore, we obtain ∥hu,k′m(x)∥F˙10,q≤(2-m+2M)τqn. Hence,
(37)J2,k′m,u≤C2mn/2supj′2-j′n/2|bj′,k′ε′|(2-m+2M)τqn.
By (29) and (35), it follows that
(38)I1≤C∑m≤0‍23mn/2uτqBuε′≤CuτqBuε′.
(ii) The estimate forI2 can be treated as that for I1. For convenience of the reader, we repeat some details as follows. Let Φ~mε,ε′(x)=2mn(Φm,0ε′*Φε)(2-mx). Following the idea used to get (28), we have
(39)I2≤C∑m>0‍2-mn∑2u-1≤|k′|<2u‍×∥∑Qj,k⊂Qs,p‍bj+m,k′ε′aj,kεΦε(2j+mx-(k′+2mk))∥F˙10,q.
Splitting ∑j:Qj,k⊂Qs,p into ∑s≤j<s+u-mQj,k⊂Qs,p and ∑j≥s+u-mQj,k⊂Qs,p, then
(40)I2≤C∑m>0‍2-mn∑2u-1≤|k′|<2u‍×(∥∑s≤j<s+u-mQj,k⊂Qs,pbj+m,k′ε′aj,kεΦε(2jx-(k+2-mk′))∥F˙10,q+∥∑j≥s+u-mQj,k⊂Qs,pbj+m,k′ε′aj,kεΦε×(2jx-(k+2-mk′))∑j≥s+u-mQj,k⊂Qs,pbj+m,k′ε′aj,kεΦε∥F˙10,q):=C∑m>0‍2-mn∑2u-1≤|k′|<2u‍(L1,k′m,u+L2,k′m,u).
The estimates ofL1,k′m,u and L2,k′m,u can be obtained as we handle J1,k′m,u and J2,k′m,u, respectively. In conclusion, it follows that
(41)I2≤CuτqBuε′.
This completes the proof of Lemma 6.In addition to the estimate of the annular operatorTuε′, we need the estimate for wavelet coefficients of K(x), which can be found in [10].Proposition 7.
LetK(x) be the distribution kernel associated to T∈CCZOγ, then one has
(42)|bj′,k′ε′|≤C2j′n/2(1+|k′|)n+γ,(ε′,j′,k′)∈Λn.We are now in a position to prove Theorem1. By virtue of (42), we obtain that
(43)Buε′=supj′∑2u-1≤|k′|<2u‍2-j′n/2|bj′,k′ε′|≤C2-uγ.
Together with Lemma 6, we have
(44)∥T-Tu∥F˙10,q→F˙10,q≤∑u′>u‍∑ε′‍∥Tu′ε′∥F˙10,q→F˙10,q≤C∑u′=u+1+∞‍(u′)τq2-u′γ.
On the other hand, since τq belongs to the interval (0,1), then a direct computation leads to that
(45)∥T-Tu∥F˙10,q→F˙10,q≤Cu2-uγ.
This completes the proof of Theorem 1.Remark 8.
Notice that the atomic and molecular decompositions for the endpoint spaceF˙10,q play an important role in our proof. However, up to the best knowledge of the authors, it is unknown whether the Triebel-Lizorkin space F˙ps,q(s∈ℝ,1<p,q<∞) has similar atomic and molecular decompositions. It would be interesting to know whether our method can be adjusted to get the approximation accuracy for the more general Triebel-Lizorkin space F˙ps,q(s∈ℝ,1<p,q<∞).



---

*Source: 572967-2013-06-15.xml*