An Approximation Method for Convolution Calderón-Zygmund Operators

and Applied Analysis 3 Proposition 2. For 1 ≤ p < ∞ and 1 ≤ q ≤ ∞, there exist two positive constants C p,q and C p,q such that C p,q 󵄩 󵄩 󵄩 󵄩 f 󵄩 󵄩 󵄩 󵄩?̇? 0,q p ≤ 󵄩 󵄩 󵄩 󵄩 󵄩 󵄩 󵄩 󵄩 󵄩 󵄩 󵄩 󵄩 (∑ ε,j,k 2 jqn/2󵄨 󵄨 󵄨 󵄨 󵄨 f ε j,k 󵄨 󵄨 󵄨 󵄨 󵄨 q


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 2dimensional 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][3][4][5][6][7][8][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   (1 <  < ∞).
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  ) 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  : D → D  associated with a kernel (, ) (in the sense that ⟨, ⟩ = ∬ ()(, )()  for test functions  and  with disjoint supports).We write  ∈   if the following three conditions are satisfied.
Convolution Calderón-Zygmund operators, such as Hilbert and Riesz operators, are commonly used in engineering.For a convolution operator , its kernel (, ) can be written as (, ) = ( − ).In this case, the conditions for the operator  ∈   are reduced to the following: where  ∈  1 0 ((0, 1)) and   () = (/) for  > 0. For convenience, let   denote the collection of all convolution operators in   .
In what follows, we restrict our attention to the operator  in   .In general, the operator is analyzed by the 2-dimensional wavelet bases.However, Z. Y. Yang and Q. X. Yang [10], making use of the -dimensional Daubechies wavelet bases, approximated the operator  by the banded operator and gave the approximation accuracy 2 − on the homogeneous Besov spaces Ḃ 0,  (1 ≤ ,  ≤ ∞).In this paper, we focus on an approximation method for the operator  and obtaining the uniform approximation accuracy on the endpoint Triebel-Lizorkin spaces Ḟ 0, 1 , 2 <  < ∞, whose definitions will be given in Section 2.
For any  ∈ Λ  , let   , = ⟨(), Φ  , ()⟩, then we have the representation in the sense of distribution.Now, we present the approximation of  by the banded operator   .For any integer  ≥ 0 and  ∈   , we define Let    be the annular operator associated to the kernel    (), and let Then, we can approximate  by the banded operator   and get the uniform approximation accuracy on the endpoint Triebel-Lizorkin space Ḟ 0, 1 (1 <  < ∞).Our result is stated as follows.
Throughout this paper, the symbol  denotes a constant that is independent of the main parameters involved but whose value may differ from line to line.
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  , ()} ∈Λ  is an unconditional bases in the space Ḟ 0,  .For any  ∈ Z and  ∈ Z  , let and let (2   − ) be the characteristic function of the dyadic cube  , .For any  ∈ S  /, if we can define   , = ⟨, Φ  , ⟩ for all (, , ) ∈ Λ  , then we have the representation  = ∑ (,,)∈Λ    , Φ  , () in the sense of distribution.The space Ḟ 0,  is characterized in terms of wavelets in the following way (for more details, see also [6,7,14]).
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 Ḟ 0, 1 ; see Meyer and Yang [7] for more details.Let Q denote the collection of all dyadic cubes  , ,  ∈ Z,  ∈ Z  .Now, we recall the following two definitions which can be found in [6,7,15].
with the norm  if there exists a ball  such that (i) supp  ⊂ ; For convenience, in the above two definitions,  is said to be a Ḟ 0,  -atom or a Ḟ 0,  -molecule when  = 1.
Proposition 5. Let 1 ≤  < ∞.The following three conditions are equivalent: for some fixed  1 ,  2 > 0 independent of .For more details, we refer the reader to [6,15], This will play a key role in Section 3.

Estimate of 𝑇 𝜀 󸀠 𝑢 and Proof of Theorem 1
To prove Theorem 1, we first estimate the annular operator     .For any   ∈   and integer  ≥ 1, let where  is independent of .
Denote  =   − , then we have In the following, we consider the estimates of  1 and  2 .
(i) For any  ≤ 0, put Φ,   () = 2 − (Φ   ,0 * Φ  )(), then By the properties of Daubechies wavelets, we can get that the operator     is bounded on Ḟ 0, 1 (see also [16, lemma 3.1]).Set By means of the orthonormality of the wavelet bases, the right side of ( 23) is equal to Moreover,  1 is bounded by Namely, we have (ii) The estimate for  2 can be treated as that for  1 .For convenience of the reader, we repeat some details as follows.Let Φ,   () = 2  (Φ   ,0 * Φ  )(2 − ).Following the idea used to get (28), we have The estimates of  , 1,  and  , 2,  can be obtained as we handle  , 1,  and  , 2,  , respectively.In conclusion, it follows that This completes the proof of Lemma 6.
In addition to the estimate of the annular operator     , we need the estimate for wavelet coefficients of (), which can be found in [10].
This completes the proof of Theorem 1.
Remark 8. Notice that the atomic and molecular decompositions for the endpoint space Ḟ 0, 1 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 ,  ( ∈ R, 1 < ,  < ∞) 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 ,  ( ∈ R, 1 < ,  < ∞).