An Approximation Method for Convolution Calderón-Zygmund Operators

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 space 𝐹̇₁^0;𝑞(2<𝑞<∞). Our proof mainly relies on the 𝑛-dimensional Daubechies wavelet bases and the atomic-molecular approach.

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 𝐿^𝑝(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 Theorem 1).

Now, we introduce a class of Calderón-Zygmund operators. Let 𝒟=𝒟(ℝ^𝑛) 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∈𝐶𝑍𝑂_𝛾 if the following three conditions are satisfied.(I)𝐾(𝑥,𝑦) is continuous on Ω=ℝ^𝑛×ℝ^𝑛⧵{(𝑥,𝑦)∶𝑥=𝑦} and satisfies 𝐾𝑥,𝑦≤𝐶𝑥−𝑦^−𝑛; (1) for all 𝑥,𝑥^′,𝑦 with 2|𝑥−𝑥^′|≤|𝑥−𝑦|, 𝐾𝑥,𝑦−𝐾𝑥^′,𝑦+𝐾𝑦,𝑥−𝐾𝑦,𝑥^′≤𝐶𝑥−𝑥^′𝛾𝑥−𝑦^𝑛+𝛾, (2) where 0<𝛾≤1.(II)Weak boundedness condition: 𝑇𝑓,𝑔≤𝐶𝑅^𝑛𝑓_∞+𝑅∇𝑓_∞𝑔_∞+𝑅∇𝑔_∞,∀𝑅>0,𝑢∈ℝ^𝑛,𝑓,𝑔∈𝐶¹₀(𝐵(𝑢,𝑅)). (3)(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 operator 𝑇, its kernel 𝐾(𝑥,𝑦) can be written as 𝐾(𝑥,𝑦)=𝐾(𝑥−𝑦). In this case, the conditions for the operator 𝑇∈𝐶𝐶𝑍𝑂_𝛾 are reduced to the following: |𝐾(𝑥)|≤𝐶|𝑥|^−𝑛; (4) for 2|𝑥−𝑥^′|≤|𝑥|, 𝐾(𝑥)−𝐾𝑥^′≤𝐶𝑥−𝑥^′𝛾|𝑥|^𝑛+𝛾;𝐾(𝑥)𝜓_𝑡(𝑥)𝑑𝑥≤𝐶𝜓_∞+∇𝜓_∞, (5)(6) where 𝜓∈𝐶¹₀(𝐵(0,1)) and 𝜓_𝑡(𝑥)=𝜓(𝑥/𝑡) for 𝑡>0. For convenience, let 𝐶𝐶𝑍𝑂_𝛾 denote the collection of all convolution operators in 𝐶𝑍𝑂_𝛾.

In what follows, we restrict our attention to the operator 𝑇 in 𝐶𝐶𝑍𝑂_𝛾. 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 𝐵̇_𝑝^0,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 𝐹̇₁^0,𝑞,2<𝑞<∞, whose definitions will be given in Section 2.

We first introduce some notations. Let Φ⁰(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∈ℝ^𝑛 and 𝜀=(𝜀₁,𝜀₂,…,𝜀_𝑛)∈{0,1}^𝑛, denote Φ^𝜀(𝑥)=∏^𝑛_𝑖=1Φ^𝜀𝑖(𝑥_𝑖). For any 𝑗∈ℤ and 𝑘∈ℤ^𝑛, let Φ^𝜀_𝑗,𝑘(𝑥)=2^𝑗𝑛/2Φ^𝜀2^𝑗𝑥−𝑘. (7) We also put 𝐸_𝑛={0,1}^𝑛⧵{0},Λ_𝑛=𝜆=𝜀,𝑗,𝑘,𝜀∈𝐸_𝑛,𝑗∈ℤ,𝑘∈ℤ^𝑛. (8) Then, {Φ^𝜀_𝑗,𝑘(𝑥)}_{𝜆∈Λ𝑛} forms orthonormal bases in 𝐿²(ℝ^𝑛), and it can be used to characterize general functions or distributions.

For any 𝜆∈Λ_𝑛, let 𝑏^𝜀_𝑗,𝑘=⟨𝐾(𝑥),Φ^𝜀_𝑗,𝑘(𝑥)⟩, then we have the representation 𝐾=_{𝜀,𝑗,𝑘∈Λ𝑛}𝑏^𝜀_𝑗,𝑘Φ^𝜀_𝑗,𝑘(𝑥) (9) in the sense of distribution. Now, we present the approximation of 𝑇 by the banded operator 𝑇_𝑢. For any integer 𝑢≥0 and 𝜀∈𝐸_𝑛, we define 𝐾^𝜀_𝑢(𝑥)=∑_𝑗𝑏^𝜀_𝑗,0Φ^𝜀_𝑗,0(𝑥)∑,𝑢=0;_{2𝑢−1≤|𝑘|<2𝑢}∑_𝑗𝑏^𝜀_𝑗,𝑘Φ^𝜀_𝑗,𝑘(𝑥),𝑢≥1. (10) Let 𝑇^𝜀_𝑢 be the annular operator associated to the kernel 𝐾^𝜀_𝑢(𝑥), and let 𝑇_𝑢=_{0≤𝑢′≤𝑢𝜀∈𝐸𝑛}𝑇^𝜀_𝑢′. (11) Then, we can approximate 𝑇 by the banded operator 𝑇_𝑢 and get the uniform approximation accuracy on the endpoint Triebel-Lizorkin space 𝐹̇₁^0,𝑞(1<𝑞<∞). Our result is stated as follows.

Theorem 1. Let 1<𝑞<∞ and 0<𝛾≤1. If 𝑇∈𝐶𝐶𝑍𝑂_𝛾, then 𝑇−𝑇_𝑢̇𝐹1^0,𝑞_→̇𝐹1^0,𝑞≤𝐶𝑢2^−𝑢𝛾,𝑢≥1. (12)

Throughout this paper, the symbol 𝐶 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

Let 𝒮(ℝ^𝑛) be the space of tempered test functions. Let 𝜑∈𝒮(ℝ^𝑛) with supp 𝜑⊂{𝜉∈ℝ^𝑛∶1/2≤|𝜉|≤2} and |𝜑|≥𝑐>0 for {3/5≤|𝜉|≤5/3}; let 𝜑_𝑗(𝑥)≡2^𝑗𝑛𝜑(2^𝑗𝑥). Let 𝛼∈ℝ, 0<𝑝<∞, and 0<𝑞≤∞. Then, the homogeneous Triebel-Lizorkin space 𝐹̇_𝑝^𝛼,𝑞 is defined as the collection of all 𝑓∈𝒮^′/𝑃 (the tempered distributions modulo polynomials) such that 𝑓_̇𝐹𝑝^𝛼,𝑞≡_𝑗2^𝑗𝛼𝜑_𝑗∗𝑓^𝑞1/𝑞_𝑝<∞ (13) 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 {Φ^𝜀_𝑗,𝑘(𝑥)}_{𝜆∈Λ𝑛} is an unconditional bases in the space 𝐹̇_𝑝^0,𝑞. For any 𝑗∈ℤ and 𝑘∈ℤ^𝑛, let 𝑄_𝑗,𝑘=𝑥∈ℝ^𝑛∶2^𝑗𝑥−𝑘∈0,1)^𝑛, (14) and let 𝜒(2^𝑗𝑥−𝑘) be the characteristic function of the dyadic cube 𝑄_𝑗,𝑘. For any 𝑓∈𝒮^′/𝑃, if we can define 𝑓^𝜀_𝑗,𝑘=⟨𝑓,Φ^𝜀_𝑗,𝑘⟩ for all (𝜀,𝑗,𝑘)∈Λ_𝑛, then we have the representation 𝑓=∑_{(𝜀,𝑗,𝑘)∈Λ𝑛}𝑓^𝜀_𝑗,𝑘Φ^𝜀_𝑗,𝑘(𝑥) in the sense of distribution. The space 𝐹̇_𝑝^0,𝑞 is characterized in terms of wavelets in the following way (for more details, see also [6, 7, 14]).

Proposition 2. For 1≤𝑝<∞ and 1≤𝑞≤∞, there exist two positive constants 𝐶_𝑝,𝑞 and 𝐶^′_𝑝,𝑞 such that 𝐶_𝑝,𝑞𝑓_̇𝐹𝑝^0,𝑞≤_{𝜀,𝑗,𝑘}2^{𝑗𝑞𝑛/2}𝑓^𝜀_𝑗,𝑘^𝑞𝜒2^𝑗𝑥−𝑘^1/𝑞_𝐿𝑝≤𝐶^′_𝑝,𝑞𝑓_̇𝐹𝑝^0,𝑞. (15)

Triebel-Lizorkin spaces have been studied by means of the atomic and molecular decompositions. Next, we state the atomic-molecular decomposition for the endpoint space 𝐹̇₁^0,𝑞; see Meyer and Yang [7] for more details. Let 𝔔 denote the collection of all dyadic cubes 𝑄_𝑗,𝑘,𝑗∈ℤ,𝑘∈ℤ^𝑛. Now, we recall the following two definitions which can be found in [6, 7, 15].

Definition 3. Let 1≤𝑞<∞. 𝑎 is said to be a 𝐹̇_𝑞^0,𝑞-atom in 𝐹̇₁^0,𝑞 with the norm 𝐶 if there exists a ball 𝐵 such that(i)Supp 𝑎⊂𝐵; (ii)‖𝑎‖_̇𝐹𝑞^0,𝑞≤𝐶|𝐵|^(1/𝑞)−1;(iii)∫𝑎𝑑𝑥=0.

Definition 4. Let 1≤𝑞<∞. 𝑎 is said to be a 𝐹̇_𝑞^0,𝑞-molecule in 𝐹̇₁^0,𝑞 with the norm 𝐶 if there exists a cube 𝑄_𝑠,𝑝∈𝔔 such that(i)𝑎=∑_{(𝜀,𝑗,𝑘)∈Λ𝑛𝑄𝑗,𝑘⊂𝑄𝑠,𝑝}𝑎^𝜀_𝑗,𝑘Φ^𝜀_𝑗,𝑘;(ii)‖𝑎‖_̇𝐹𝑞^0,𝑞≤𝐶|𝑄_𝑠,𝑝|^(1/𝑞)−1.
For convenience, in the above two definitions, 𝑎 is said to be a 𝐹̇_𝑞^0,𝑞-atom or a 𝐹̇_𝑞^0,𝑞-molecule when 𝐶=1.

Proposition 5. Let 1≤𝑞<∞. The following three conditions are equivalent: (i)𝑓∈𝐹̇₁^0,𝑞. (ii)There exist {𝜇_𝑚}_𝑚∈ℤ∈𝑙¹ and 𝐹̇_𝑞^0,𝑞-atoms 𝑎_𝑚 such that 𝑓=∑_𝑚∈ℤ𝜇_𝑚𝑎_𝑚.(iii)There exist {𝜆_𝑚}_𝑚∈ℤ∈𝑙¹ and 𝐹̇_𝑞^0,𝑞-molecules 𝑎_𝑚 such that 𝑓=∑_𝑚∈ℤ𝜆_𝑚𝑎_𝑚.

In fact, for any 𝑓∈𝐹̇₁^0,𝑞, we can write it as 𝑓=∑_𝑚𝜆_𝑚𝑎_𝑚, where each 𝑎_𝑚 is a 𝐹̇_𝑞^0,𝑞-molecule and {𝜆_𝑚}∈𝑙¹, and it can be verified that 𝑓_̇𝐹1^0,𝑞∼_𝑚𝜆_𝑚, (16) namely, 𝐶₁𝑓_̇𝐹1^0,𝑞≤_𝑚𝜆_𝑚≤𝐶₂𝑓_̇𝐹1^0,𝑞 (17) 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 of 𝑇^𝜀′_𝑢 and Proof of Theorem 1

To prove Theorem 1, we first estimate the annular operator 𝑇^𝜀′_𝑢. For any 𝜀^′∈𝐸_𝑛 and integer 𝑢≥1, let 𝐵^𝜀′_𝑢=sup_{𝑗′2𝑢−1≤𝑘′<2𝑢}2^{−𝑗′𝑛/2}𝑏^𝜀′_{𝑗′,𝑘′}. (18)

Lemma 6. For 1<𝑞<∞, let 𝜏_𝑞=1−(1/𝑞). One has 𝑇^𝜀′_{𝑢̇𝐹10,𝑞→̇𝐹10,𝑞}≤𝐶𝑢^𝜏𝑞𝐵^𝜀′_𝑢. (19)

Proof. In terms of the molecular decomposition for the space 𝐹̇₁^0,𝑞, we only need to prove that for an arbitrary 𝐹̇_𝑞^0,𝑞-molecule 𝑎, 𝑇^𝜀′_𝑢𝑎(𝑥)_{̇𝐹10,𝑞}≤𝐶𝑢^𝜏𝑞𝐵^𝜀′_𝑢, (20) where 𝐶 is independent of 𝑎.
Denote 𝑚=𝑗^′−𝑗, then we have 𝑇^𝜀′_𝑢𝑎(𝑥)=_{𝜀,𝑚2𝑢−1≤|𝑘′|<2𝑢}×_{𝑄𝑗,𝑘⊂𝑄𝑠,𝑝}𝑏^𝜀′_{𝑗+𝑚,𝑘′}𝑎^𝜀_𝑗,𝑘Φ^𝜀_𝑗,𝑘∗Φ^𝜀′_{𝑗+𝑚,𝑘′}(𝑥). (21) Thus, 𝑇^𝜀′_𝑢𝑎(𝑥)_{̇𝐹10,𝑞}≤_{𝜀𝑚≤02𝑢−1≤|𝑘′|<2𝑢}×_{𝑄𝑗,𝑘⊂𝑄𝑠,𝑝}𝑏^𝜀′_{𝑗+𝑚,𝑘′}𝑎^𝜀_𝑗,𝑘Φ^𝜀_𝑗,𝑘∗Φ^𝜀′_{𝑗+𝑚,𝑘′}(𝑥)_{̇𝐹10,𝑞}+_{𝜀𝑚>02𝑢−1≤|𝑘′|<2𝑢}×_{𝑄𝑗,𝑘⊂𝑄𝑠,𝑝}𝑏^𝜀′_{𝑗+𝑚,𝑘′}𝑎^𝜀_𝑗,𝑘Φ^𝜀_𝑗,𝑘∗Φ^𝜀′_{𝑗+𝑚,𝑘′}(𝑥)_{̇𝐹10,𝑞}∶=𝐼₁+𝐼₂. (22) In the following, we consider the estimates of 𝐼₁ and 𝐼₂.
(i) For any 𝑚≤0, put Φ^𝜀,𝜀′_𝑚(𝑥)=2^−𝑚𝑛(Φ^𝜀′_𝑚,0∗Φ^𝜀)(𝑥), then 𝐼₁≤𝐶_𝑚≤02^𝑚𝑛_{2𝑢−1≤|𝑘′|<2𝑢}×_{𝑄𝑗,𝑘⊂𝑄𝑠,𝑝}𝑏^𝜀′_{𝑗+𝑚,𝑘′}𝑎^𝜀_𝑗,𝑘Φ^𝜀,𝜀′_𝑚2^𝑗𝑥−𝑘+2^−𝑚𝑘^′_{̇𝐹10,𝑞}. (23) For fixed 𝑚 and 𝑘^′, let 𝑇^𝑚_𝑘′ be the operator associated with the kernel 𝐾^𝑚_𝑘′𝑥,𝑦=_{𝑄𝑗,𝑘⊂𝑄𝑠,𝑝}2^𝑗𝑛Φ^𝜀,𝜀′_𝑚2^𝑗𝑥−𝑘+2^−𝑚𝑘^′×Φ^𝜀2^𝑗𝑦−𝑘+2^−𝑚𝑘^′. (24) By the properties of Daubechies wavelets, we can get that the operator 𝑇^𝑚_𝑘′ is bounded on 𝐹̇₁^0,𝑞 (see also [16, lemma 3.1]).
Set 𝑔^𝑚_𝑘′=_{𝑄𝑗,𝑘⊂𝑄𝑠,𝑝}𝑏^𝜀′_{𝑗+𝑚,𝑘′}𝑎^𝜀_𝑗,𝑘Φ^𝜀2^𝑗𝑥−𝑘+2^−𝑚𝑘^′. (25) By means of the orthonormality of the wavelet bases, the right side of (23) is equal to 𝐶_𝑚≤02^𝑚𝑛_{2𝑢−1≤|𝑘′|<2𝑢}𝑇^𝑚_𝑘′𝑔^𝑚_{𝑘′̇𝐹10,𝑞}. (26) Moreover, 𝐼₁ is bounded by 𝐶_𝑚≤02^𝑚𝑛_{2𝑢−1≤|𝑘′|<2𝑢}𝑔^𝑚_{𝑘′̇𝐹10,𝑞}. (27) Namely, we have 𝐼₁≤𝐶_𝑚≤02^𝑚𝑛_{2𝑢−1≤|𝑘′|<2𝑢}×_{𝑄𝑗,𝑘⊂𝑄𝑠,𝑝}𝑏^𝜀′_{𝑗+𝑚,𝑘′}𝑎^𝜀_𝑗,𝑘Φ^𝜀2^𝑗𝑥−𝑘+2^−𝑚𝑘^′_{̇𝐹10,𝑞}. (28) 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 𝐼₁≤𝐶_𝑚≤02^𝑚𝑛×_{2𝑢−1≤|𝑘′|<2𝑢𝑄𝑠≤𝑗<𝑠+𝑢𝑗,𝑘⊂𝑄𝑠,𝑝}𝑏^𝜀′_{𝑗+𝑚,𝑘′}𝑎^𝜀_𝑗,𝑘Φ^𝜀×2^𝑗𝑥−𝑘+2^−𝑚𝑘^′_{̇𝐹10,𝑞}+_{𝑄𝑗≥𝑠+𝑢𝑗,𝑘⊂𝑄𝑠,𝑝}𝑏^𝜀′_{𝑗+𝑚,𝑘′}𝑎^𝜀_𝑗,𝑘Φ^𝜀×2^𝑗𝑥−𝑘+2^−𝑚𝑘^′_{̇𝐹10,𝑞}∶=𝐶_𝑚≤02^𝑚𝑛_{2𝑢−1≤|𝑘′|<2𝑢}𝐽^𝑚,𝑢_1,𝑘′+𝐽^𝑚,𝑢_2,𝑘′. (29) Now, we consider 𝐽^𝑚,𝑢_1,𝑘′ and 𝐽^𝑚,𝑢_2,𝑘′. As for 𝐽^𝑚,𝑢_1,𝑘′, we use Proposition 2 to get that 𝐽^𝑚,𝑢_1,𝑘′≤𝐶_{𝑄𝑠≤𝑗<𝑠+𝑢𝑗,𝑘⊂𝑄𝑠,𝑝}𝑏^𝜀′_{𝑗+𝑚,𝑘′}^𝑞×𝑎^𝜀_𝑗,𝑘^𝑞𝜒2^𝑗𝑥−𝑘+2^−𝑚𝑘^′1/𝑞𝑑𝑥≤𝐶2^𝑚𝑛/2sup_𝑗′2^{−𝑗′𝑛/2}𝑏^𝜀′_{𝑗′,𝑘′}×_{𝑄𝑠≤𝑗<𝑠+𝑢𝑗,𝑘⊂𝑄𝑠,𝑝}2^{𝑗𝑞𝑛/2}𝑎^𝜀_𝑗,𝑘^𝑞×𝜒2^𝑗𝑥−𝑘+2^−𝑚𝑘^′1/𝑞𝑑𝑥. (30) Let ⊗^𝑛₁[𝑎_𝑖,𝑏_𝑖] denote the Cartesian product [𝑎₁,𝑏₁]×[𝑎₂,𝑏₂]×⋯×[𝑎_𝑛,𝑏_𝑛]. Let 𝑓^𝑚_𝑗,𝑘′(𝑥)=_{𝑘∶𝑄𝑗,𝑘⊂𝑄𝑠,𝑝}2^{𝑗𝑞𝑛/2}𝑎^𝜀_𝑗,𝑘^𝑞𝜒2^𝑗𝑥−𝑘+2^−𝑚𝑘^′, (31) then the support of 𝑓^𝑚_𝑗,𝑘′(𝑥) is contained in ⊗^𝑛₁[(𝑝_𝑖/2^𝑠)+(𝑘^′_𝑖/2^𝑗+𝑚),((𝑝_𝑖+1)/2^𝑠)+(𝑘^′_𝑖/2^𝑗+𝑚)]. Moreover, we can obtain that supp𝑓^𝑚_𝑗,𝑘′(𝑥)∩supp𝑓^𝑚_{𝑗1,𝑘′}(𝑥)=∅ (32) when |𝑗−𝑗₁|≥𝐶. In other words, the number of the nonzero terms in the sum ∑_{𝑠≤𝑗<𝑠+𝑢}𝑓^𝑚_𝑗,𝑘′(𝑥) is a constant, which is independent of 𝑢. By the inequality (𝛼+𝛽)^1/𝑞≤𝛼^1/𝑞+𝛽^1/𝑞, we have 𝐽^𝑚,𝑢_1,𝑘′≤𝐶2^𝑚𝑛/2sup_𝑗′2^{−𝑗′𝑛/2}𝑏^𝜀′_{𝑗′,𝑘′}×_{𝑄𝑠≤𝑗<𝑠+𝑢𝑗,𝑘⊂𝑄𝑠,𝑝}2^𝑗𝑛/2𝑎^𝜀_𝑗,𝑘𝜒2^𝑗𝑥−𝑘+2^−𝑚𝑘^′𝑑𝑥≤𝐶2^𝑚𝑛/2sup_𝑗′2^{−𝑗′𝑛/2}𝑏^𝜀′_{𝑗′,𝑘′}×_{𝑄𝑠≤𝑗<𝑠+𝑢𝑗,𝑘⊂𝑄𝑠,𝑝}2^𝑗𝑛/2𝑎^𝜀_𝑗,𝑘𝜒2^𝑗𝑥−𝑘𝑑𝑥. (33) From Hölder's inequality, we obtain 𝐽^𝑚,𝑢_1,𝑘′≤𝐶2^𝑚𝑛/2sup_𝑗′2^{−𝑗′𝑛/2}𝑏^𝜀′_{𝑗′,𝑘′}𝑢^𝜏𝑞×_{𝑠≤𝑗<𝑠+𝑢}×_{𝑘∶𝑄𝑗,𝑘⊂𝑄𝑠,𝑝}2^𝑗𝑛/2𝑎^𝜀_𝑗,𝑘𝜒2^𝑗𝑥−𝑘^𝑞1/𝑞𝑑𝑥≤𝐶2^𝑚𝑛/2sup_𝑗′2^{−𝑗′𝑛/2}𝑏^𝜀′_{𝑗′,𝑘′}𝑢^𝜏𝑞×_{𝑄𝑠≤𝑗<𝑠+𝑢𝑗,𝑘⊂𝑄𝑠,𝑝}2^{𝑗𝑞𝑛/2}𝑎^𝜀_𝑗,𝑘^𝑞𝜒2^𝑗𝑥−𝑘^1/𝑞𝑑𝑥. (34) Since ‖𝑎‖_̇𝐹𝑞^0,𝑞≤𝐶|𝐵|^1/𝑞−1, then we get that ‖𝑎‖_̇𝐹1^0,𝑞≤𝐶. Thus 𝐽^𝑚,𝑢_1,𝑘′≤𝐶2^𝑚𝑛/2𝑢^𝜏𝑞sup_𝑗′2^{−𝑗′𝑛/2}𝑏^𝜀′_{𝑗′,𝑘′}. (35) As for 𝐽^𝑚,𝑢_2,𝑘′, we have 𝐽^𝑚,𝑢_2,𝑘′≤𝐶2^𝑚𝑛/2sup_𝑗′2^{−𝑗′𝑛/2}𝑏^𝜀′_{𝑗′,𝑘′}×_{𝑄𝑗≥𝑠+𝑢𝑗,𝑘⊂𝑄𝑠,𝑝}2^𝑗𝑛/2𝑎^𝜀_𝑗,𝑘Φ^𝜀2^𝑗𝑥−𝑘+2^−𝑚𝑘^′_{̇𝐹10,𝑞}∶=𝐶2^𝑚𝑛/2sup_𝑗′2^{−𝑗′𝑛/2}𝑏^𝜀′_{𝑗′,𝑘′}ℎ^𝑚_𝑢,𝑘′(𝑥)_{̇𝐹10,𝑞}. (36) By Definition 3, we can verify that (2^−𝑚+2^𝑀)^{−𝜏𝑞𝑛}ℎ^𝑚_𝑢,𝑘′(𝑥) is a 𝐹̇_𝑞^0,𝑞-atom with the norm 𝐶, see also [16] for more details. Furthermore, we obtain ‖ℎ^𝑚_𝑢,𝑘′(𝑥)‖_{̇𝐹10,𝑞}≤(2^−𝑚+2^𝑀)^𝜏𝑞𝑛. Hence, 𝐽^𝑚,𝑢_2,𝑘′≤𝐶2^𝑚𝑛/2sup_𝑗′2^{−𝑗′𝑛/2}𝑏^𝜀′_{𝑗′,𝑘′}2^−𝑚+2^{𝑀𝜏𝑞𝑛}. (37) By (29) and (35), it follows that 𝐼₁≤𝐶_𝑚≤02^3𝑚𝑛/2𝑢^𝜏𝑞𝐵^𝜀′_𝑢≤𝐶𝑢^𝜏𝑞𝐵^𝜀′_𝑢. (38)
(ii) The estimate for 𝐼₂ can be treated as that for 𝐼₁. For convenience of the reader, we repeat some details as follows. Let Φ^𝜀,𝜀′_𝑚(𝑥)=2^𝑚𝑛(Φ^𝜀′_𝑚,0∗Φ^𝜀)(2^−𝑚𝑥). Following the idea used to get (28), we have 𝐼₂≤𝐶_𝑚>02^−𝑚𝑛_{2𝑢−1≤|𝑘′|<2𝑢}×_{𝑄𝑗,𝑘⊂𝑄𝑠,𝑝}𝑏^𝜀′_{𝑗+𝑚,𝑘′}𝑎^𝜀_𝑗,𝑘Φ^𝜀2^𝑗+𝑚𝑥−𝑘^′+2^𝑚𝑘_{̇𝐹10,𝑞}. (39) Splitting ∑_{𝑗∶𝑄𝑗,𝑘⊂𝑄𝑠,𝑝} into ∑_{𝑄𝑠≤𝑗<𝑠+𝑢−𝑚𝑗,𝑘⊂𝑄𝑠,𝑝} and ∑_{𝑄𝑗≥𝑠+𝑢−𝑚𝑗,𝑘⊂𝑄𝑠,𝑝}, then 𝐼₂≤𝐶_𝑚>02^−𝑚𝑛_{2𝑢−1≤|𝑘′|<2𝑢}×_{𝑄𝑠≤𝑗<𝑠+𝑢−𝑚𝑗,𝑘⊂𝑄𝑠,𝑝}𝑏^𝜀′_{𝑗+𝑚,𝑘′}𝑎^𝜀_𝑗,𝑘Φ^𝜀2^𝑗𝑥−𝑘+2^−𝑚𝑘^′_{̇𝐹10,𝑞}+_{𝑄𝑗≥𝑠+𝑢−𝑚𝑗,𝑘⊂𝑄𝑠,𝑝}𝑏^𝜀′_{𝑗+𝑚,𝑘′}𝑎^𝜀_𝑗,𝑘Φ^𝜀×2^𝑗𝑥−𝑘+2^−𝑚𝑘^′_{̇𝐹10,𝑞}∶=𝐶_𝑚>02^−𝑚𝑛_{2𝑢−1≤|𝑘′|<2𝑢}𝐿^𝑚,𝑢_1,𝑘′+𝐿^𝑚,𝑢_2,𝑘′. (40)
The estimates of 𝐿^𝑚,𝑢_1,𝑘′ and 𝐿^𝑚,𝑢_2,𝑘′ can be obtained as we handle 𝐽^𝑚,𝑢_1,𝑘′ and 𝐽^𝑚,𝑢_2,𝑘′, respectively. In conclusion, it follows that 𝐼₂≤𝐶𝑢^𝜏𝑞𝐵^𝜀′_𝑢. (41) This completes the proof of Lemma 6.

In addition to the estimate of the annular operator 𝑇^𝜀′_𝑢, we need the estimate for wavelet coefficients of 𝐾(𝑥), which can be found in [10].

Proposition 7. Let 𝐾(𝑥) be the distribution kernel associated to 𝑇∈𝐶𝐶𝑍𝑂_𝛾, then one has 𝑏^𝜀′_{𝑗′,𝑘′}≤𝐶2^{𝑗′𝑛/2}1+𝑘^′𝑛+𝛾,𝜀^′,𝑗^′,𝑘^′∈Λ_𝑛. (42)

We are now in a position to prove Theorem 1. By virtue of (42), we obtain that 𝐵^𝜀′_𝑢=sup_{𝑗′2𝑢−1≤|𝑘′|<2𝑢}2^{−𝑗′𝑛/2}𝑏^𝜀′_{𝑗′,𝑘′}≤𝐶2^−𝑢𝛾. (43) Together with Lemma 6, we have 𝑇−𝑇_𝑢̇𝐹1^0,𝑞_→̇𝐹1^0,𝑞≤_{𝑢′>𝑢𝜀′}𝑇^𝜀′_{𝑢′̇𝐹10,𝑞→̇𝐹10,𝑞}≤𝐶^+∞_{𝑢′=𝑢+1}𝑢^′𝜏𝑞2^{−𝑢′𝛾}. (44) On the other hand, since 𝜏_𝑞 belongs to the interval (0,1), then a direct computation leads to that 𝑇−𝑇_𝑢̇𝐹1^0,𝑞_→̇𝐹1^0,𝑞≤𝐶𝑢2^−𝑢𝛾. (45) This completes the proof of Theorem 1.

Remark 8. Notice that the atomic and molecular decompositions for the endpoint space 𝐹̇₁^0,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 𝐹̇_𝑝^𝑠,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 𝐹̇_𝑝^𝑠,𝑞(𝑠∈ℝ,1<𝑝,𝑞<∞).

Acknowledgments

This research is supported by the Fundamental Research Funds for the Central Universities, South-Central University for Nationalities (no. CZY12015; no. CTZ13026), and the Research Fund for the Doctoral Program of Higher Education (no. 20090141120010).