© Hindawi Publishing Corp. BOUNDEDNESS OF MULTILINEAR OPERATORS ON TRIEBEL-LIZORKIN SPACES

The purpose of this paper is to study the boundedness in the 
context of Triebel-Lizorkin spaces for some multilinear operators 
related to certain convolution operators. The operators include 
Littlewood-Paley operator, Marcinkiewicz integral, and Bochner-Riesz operator.


Introduction.
Let T be a Calderon-Zygmund operator.A well-known result of Coifman et al. [6] states that the commutator [b, T ] = T (bf ) − bT f (where b ∈ BMO) is bounded on L p (R n ) for 1 < p < ∞; Chanillo [1] proves a similar result when T is replaced by the fractional integral operator.In [7,9], Janson and Paluszyński extend these results to the Triebel-Lizorkin spaces and the case b ∈ Lipβ (where Lip β is the homogeneous Lipschitz space).The main purpose of this paper is to discuss the boundedness of some multilinear operators related to certain convolution operators in the context of Triebel-Lizorkin spaces.In fact, we will establish the boundedness on the Triebel-Lizorkin spaces for some multilinear operators related to certain convolution operator only under certain conditions on the size of the operators.As applications, we obtain the boundedness of the multilinear operators related to the Marcinkiewicz integral, Littlewood-Paley operator, and Bochner-Riesz operator in the context of Triebel-Lizorkin spaces.

Preliminaries.
Throughout this paper, M(f ) will denote the Hardy-Littlewood maximal function of f , M p f = (M(f p )) 1/p for p > 0, and Q will denote a cube of R n with sides parallel to the axes.For a cube Q, let For β > 0 and p > 1, let Ḟ β,∞ p be the homogeneous Triebel-Lizorkin space.The Lipschitz space ∧β is the space of functions f such that where ∆ k h denotes the kth difference operator (see [9]).The operators considered in this paper are following several sublinear operators.Let m be a positive integer and let A be a function on R n .We denote Definition 2.1.Let ε > 0 and let ψ be a fixed function which satisfies the following properties: ( n+1+ε) when 2|y| < |x|.The multilinear Littlewood-Paley operator is defined by where which is the Littlewood-Paley g function (see [10]).
Let H be the space may be viewed as a mapping from [0, +∞) to H, and it is clear that Definition 2.2.Let 0 < γ ≤ 1 and let Ω be homogeneous of degree zero on R n such that S n−1 Ω(x )dσ (x ) = 0. Assume that Ω ∈ Lip γ (S n−1 ), that is, there exists a constant M > 0 such that for any x, y ∈ S n−1 , |Ω(x) − Ω(y)| ≤ M|x − y| γ .The multilinear Marcinkiewicz integral operator is defined by where which is the Marcinkiewicz integral (see [11]).
Let H be the space where B δ t (z) = t −n B δ (z/t) for t > 0. The maximal multilinear Bochner-Riesz operator is defined by (2.13) Also define which is the Bochner-Riesz operator (see [7,8]).
Let H be the space H = {h : h = sup t>0 |h(t)| < ∞}, then it is clear that More generally, we consider the following multilinear operators related to certain convolution operators.Definition 2.4.Let K(x, t) be defined on R n × [0, +∞).Denote that (2.16) Let H be the normed space H = {h : h < ∞}.For each fixed x ∈ R n , K t f (x) and K A t (f )(x) are viewed as a mapping from [0, +∞) to H.Then, the multilinear operators related to K t is defined by It is clear that Definitions 2.1, 2.2, and 2.3 are the particular examples of Definition 2.4.Note that when m = 0, T A is just the commutator of K t and A. It is well known that multilinear operators are of great interest in harmonic analysis and have been widely studied by many authors (see [2,3,4,5]).The main purpose of this paper is to consider the continuity of the multilinear operators on Triebel-Lizorkin spaces.We will prove the following theorems in Section 3.

Main theorem and proof.
First, we will establish the following theorem.
T , and T A be the same as in Definition 2.4.If T is bounded on L q (R n ) for q ∈ (1, +∞) and T A satisfies the size condition To prove the theorem, we need the following lemmas.Lemma 3.2 (see [9]).For 0 < β < 1 and 1 < p < ∞, Lemma 3.3 (see [9]).For 0 < β < 1 and 1 ≤ p ≤ ∞, Lemma 3.6 (see [4]).Let A be a function on R n and D α A ∈ L q (R n ) for |α| = m and some q > n.Then where Q(x, y) is the cube centered at x and having side length 5 √ n|x − y|. then (3.9) Now, we estimate I, II, and III, respectively.First, for x ∈ Q and y ∈ Q, using Lemmas 3.3 and 3.6, we get (3.10) Thus, by Holder's inequality and the L r boundedness of T for 1 < r < p, we obtain (3.11) Secondly, for 1 < r < q, using the inequality (see [9]) and similar to the proof of I, we obtain (3.13)For III, using the size condition of T A , we have Putting these estimates together, taking the supremum over all Q such that x ∈ Q, and using the L p boundedness of M r for r < p, we obtain This completes the proof of (a).(b) By the same argument as in the proof of (a), we have (3.17) Now, using Lemma 3.4, we obtain This completes the proof of (b) and the theorem.
To prove Theorems 2.5, 2.6, and 2.7, it suffices to verify that g A ψ , µ A Ω , and B A δ, * satisfy the size condition in the Theorem 3.1.
Suppose supp f ⊂ Qc and x ∈ Q = Q(x 0 ,l).Note that |x 0 − y| ≈ |x − y| for y ∈ Qc .For g A ψ , we write By the condition of ψ, we obtain (3.20) For I 2 , by the formula (see [4]) and Lemma 3.6, we get Thus, similar to the proof of I 1 , (3.23) For I 3 , by Lemma 3.5, we get Thus, similar to the proof of I 1 , we obtain (3.25) So,

.26)
For µ A Ω , we write (3.28) Similarly, we have For J 3 , by the inequality (see [11]) we obtain (3.30) For J 4 , similar to the proof of J 1 , J 2 , and J 3 , we obtain (3.31) For B A δ, * , we write (3.32) We consider the following two cases.

Call for Papers
As a multidisciplinary field, financial engineering is becoming increasingly important in today's economic and financial world, especially in areas such as portfolio management, asset valuation and prediction, fraud detection, and credit risk management.For example, in a credit risk context, the recently approved Basel II guidelines advise financial institutions to build comprehensible credit risk models in order to optimize their capital allocation policy.Computational methods are being intensively studied and applied to improve the quality of the financial decisions that need to be made.Until now, computational methods and models are central to the analysis of economic and financial decisions.However, more and more researchers have found that the financial environment is not ruled by mathematical distributions or statistical models.In such situations, some attempts have also been made to develop financial engineering models using intelligent computing approaches.For example, an artificial neural network (ANN) is a nonparametric estimation technique which does not make any distributional assumptions regarding the underlying asset.Instead, ANN approach develops a model using sets of unknown parameters and lets the optimization routine seek the best fitting parameters to obtain the desired results.The main aim of this special issue is not to merely illustrate the superior performance of a new intelligent computational method, but also to demonstrate how it can be used effectively in a financial engineering environment to improve and facilitate financial decision making.In this sense, the submissions should especially address how the results of estimated computational models (e.g., ANN, support vector machines, evolutionary algorithm, and fuzzy models) can be used to develop intelligent, easy-to-use, and/or comprehensible computational systems (e.g., decision support systems, agent-based system, and web-based systems) This special issue will include (but not be limited to) the following topics: • Computational methods: artificial intelligence, neural networks, evolutionary algorithms, fuzzy inference, hybrid learning, ensemble learning, cooperative learning, multiagent learning

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Application fields: asset valuation and prediction, asset allocation and portfolio selection, bankruptcy prediction, fraud detection, credit risk management • Implementation aspects: decision support systems, expert systems, information systems, intelligent agents, web service, monitoring, deployment, implementation