© Hindawi Publishing Corp. BOUNDEDNESS FOR MULTILINEAR MARCINKIEWICZ OPERATORS ON CERTAIN HARDY SPACES

The boundedness for the multilinear Marcinkiewicz operators on 
certain Hardy and Herz-Hardy spaces are obtained.


Introduction and definitions.
Suppose that S n−1 is the unit sphere of R n (n ≥ 2) equipped with normalized Lebesgue measure dσ = dσ (x ).Let Ω be homogeneous of degree zero and satisfy the following two conditions: (i) Ω(x) is continuous on S n−1 and satisfies the Li p γ condition on S n−1 (0 ≤ γ ≤ 1), that is, (ii) S n−1 S(x )dx = 0. Let m be a positive integer and A be a function on R n .The multilinear Marcinkiewicz integral operator is defined by where (1.3) We denote that F t (f )(x) = f |x−y|≤t (Ω(x − y)/|x − y| n−1 )f (y)dy.We also denote that which is the Marcinkiewicz integral operator (see [5,6,12]).
Note that when m = 0, µ A Ω is just the commutator of Marcinkiewicz operator (see [5,12]).It is well known that multilinear operators are of great interest in harmonic analysis and have been widely studied by many authors (see [1,2,3,4,5]).The main purpose of this paper is to consider the continuity of the multilinear Marcinkiewicz operators on certain Hardy and Herz-Hardy spaces.We first introduce some definitions (see [7,8,9,10,11]).Definition 1.1.Let A be a function on R n , m a positive integer, and 0 in the Schwartz distributional sense, it can be written as where a j 's are (p, D m A)-atoms, λ j ∈ C, and Definition 1.2.Let 0 < p, q < ∞, and α ∈ R.
(1) The homogeneous Herz space is defined by where (2) The nonhomogeneous Herz space is defined by where where Let m be a positive integer and A a function on R n , α ∈ R, and 1 < q ≤ ∞.A function a(x) on R n is called a central (α,q,D m A)-atom (or a central (α,q,D m A)-atom of restrict type), if (1) supp a ⊂ B(0,r ) for some r > 0 (or for some r ≥ 1), ( 2) where a j is a central (α,q,D m A)-atom (or a central (α,q,D m A)-atom of restrict type) supported on B(0, 2 j ) and

Theorems and proofs.
We begin with some preliminary lemmas.Lemma 2.1 (see [2]).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 is the cube centered at x and having side length 5 Proof.By Minkowski inequality and the condition of Ω, we have Thus, the lemma follows from [3,4].
Proof.It suffices to show that there exists a constant c > 0 such that, for every (p, D m A)-atom a, Let a be a (p, D m A)-atom supported on a ball B = B(x 0 ,r ).We write (2.5) For I, taking q > 1 and by Hölder's inequality and the L q -boundedness of µ A Ω (see Lemma 2.2), we see that To obtain the estimate of II, we need to estimate By the vanishing moment of a, we write x, y a(y)dy (2.10) Thus, (2.11) On the other hand, by the following formula (see [2]): and Lemma 2.1, we get (2.14) For II 3 , and by the vanishing moment of a, we write, (2.15) Similar to the estimate of II 1 , we obtain (2.16) Recalling that p > n/(n+ γ), therefore, which, together with the estimate for I, yields the desired result.This finishes the proof of Theorem 2.3.
j=−∞ λ j a j (x) be the atomic decomposition for f as in Definition 1.3.We write (2.18) For II, and by the boundedness of µ A Ω on L q (R n ) (see Lemma 2.2), we have ( For I, and similar to the proof of Theorem 2.3, we have, for (2.21) To estimate I 1 and I 2 , we consider two cases.

Call for Papers
Thinking about nonlinearity in engineering areas, up to the 70s, was focused on intentionally built nonlinear parts in order to improve the operational characteristics of a device or system.Keying, saturation, hysteretic phenomena, and dead zones were added to existing devices increasing their behavior diversity and precision.In this context, an intrinsic nonlinearity was treated just as a linear approximation, around equilibrium points.Inspired on the rediscovering of the richness of nonlinear and chaotic phenomena, engineers started using analytical tools from "Qualitative Theory of Differential Equations," allowing more precise analysis and synthesis, in order to produce new vital products and services.Bifurcation theory, dynamical systems and chaos started to be part of the mandatory set of tools for design engineers.
This proposed special edition of the Mathematical Problems in Engineering aims to provide a picture of the importance of the bifurcation theory, relating it with nonlinear and chaotic dynamics for natural and engineered systems.Ideas of how this dynamics can be captured through precisely tailored real and numerical experiments and understanding by the combination of specific tools that associate dynamical system theory and geometric tools in a very clever, sophisticated, and at the same time simple and unique analytical environment are the subject of this issue, allowing new methods to design high-precision devices and equipment.
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