ON A MULTILINEAR SINGULAR INTEGRAL OPERATOR

In this paper, we prove that the maximal operatorsatisfiesis homogeneous of degree 0, has vanishing moment up to order M and satisfies Lq-Dini condition for some

ESTIMATES FOR THE MAXIMAL MULTILINEAR SINGULAR INTEGRAL OPERATORS

In this paper, some mapping properties are considered for the maximal multilinear singular integral operator whose kernel satisfies certain minimum regularity condition. It is proved that certain uniform local estimate for doubly truncated operators implies the L^P（R^n） （1 〈 p 〈 ∞） boundedness and a weak type LlogL estimate for the corresponding maximal operator.

A NOTE ON THE MULTILINEAR SINGULAR INTEGRAL OPERATORS

LP(Rn) boundedness is considered for the multilinear singular integral operator defined by where Ω is homogeneous of degree zero, integrable on the unit sphere and has vanishing moment of order one. A has derivatives of order one in BMO(Rn). We give a smoothness condition which is fairly weaker than that Ω∈Lipαt(Sn-1) (0 <α≤1) and implies the LP(Rn) (1 < p <∞) boundedness for the operator TA.Some endpoint estimates are also established.

Iterated Commutators for Multilinear Singular Integral Operators on Morrey Space with Non-Doubling Measures

Let μ be a non-negative Radon measure on Rd which only satisfies the following growth condition that there exists a positive constant C such that μ(B(x,r)) ≤ Crn?for all x∈ Rd, r > 0 and some fixed n ∈ (0,d]. This paper is interested in the properties of the iterated commutators of multilinear singular integral operators on Morrey spaces?.Precisely speaking, we show that the iterated commutators generated by multilinear singular integrals operators are bounded from to where (Regular Bounded Mean Oscillation space) and 1 qj ≤ pj ∞ with 1/p = 1/p1 + ... + 1/pm and 1/q = 1/q1+ ... + 1/qm.

A BMO ESTIMATE FOR THE MULTILINEAR SINGULAR INTEGRAL OPERATOR

The behavior on the space L^∞（R^n） for the multilinear singular integral operator defined by TAf（x）=∫RnΩ（x-y）/｜x-y｜^n＋1（A（x）-A（y）△A（y）（x-y））f（y）dy is considered, where 12 is homogeneous of degree zero, integrable on the unit sphere and has vanishing is considered, where Ω is homogeneous of degree zero, integrable on the unit sphere and has vanishingmoment of order one, A has derivatives of order one in BMO（R^n）. It is proved that if Ω satisfies some minimum size condition and an L1-Dini type regularity condition, then for f ∈ L^∞（R^n）, TAf is either infinite almost everywhere or finite almost everywhere, and in the latter case, TAf ∈ BMO（R^n）.

AN ENDPOINT ESTIMATE FOR MAXIMAL MULTILINEAR SINGULAR INTEGRAL OPERATORS

A weak type endpoint estimate for the maximal multilinear singular integral operator TAf（x）=supε〉0｜∫｜x-y｜〉εΩ（x-y）/｜x-y｜^n＋1（A（x）-A（y）-VA（y）（x-y））f（y）dy｜ is established, where Ω is homogeneous of degree zero, integrable on the unit sphere and has vanishing moment of order one, and A has derivatives of order one in BMO（R^n）. A regularity condition on Ω which implies an LlogL type estimate of TA is given.

Multilinear singular integrals and commutators in variable exponent Lebesgue spaces

Boundedness of multilinear singular integrals and their commutators from products of variable exponent Lebesgue spaces to variable exponent Lebesgue spaces are obtained. The vector-valued case is also considered.

A LIPSCHITZ ESTIMATE FOR MULTILINEAR OSCILLATORY SINGULAR INTEGRALS WITH ROUGH KERNELS

In this paper, for the multilinear oscillatory singular integral operators TA1,A2,...Ar defined by TA1,A2,...,Arf（x） = p.v.∫R^n ^e^iP（x,y）Ω（x - y）/｜x - y｜^n＋M r∏s=1 Rms＋1（As;x,y）f（y）dy, n≥2 where P（x,y） is a nontrivial and real-valued polynomial defined on R^n×R^n,Ω（x） is homogeneous of degree zero on R^n, As（x） has derivatives of order ms in ∧βs （0〈βs〈 1）, Rms＋1 （As;x, y） denotes the （ms＋1）-st remainder of the Taylor series of As at x expended about y （s = 1, 2, ..., r）, M = ∑s^r =1 ms, the author proves that if 0 〈=β1=∑s^r=1 βs〈1,and Ω∈L^q（S^n-1） for some q 〉 1/（1 -β）, then for any p∈（1, ∞）, and some appropriate 0 〈β〈 1, TA1,A2,...,Ar, is bounded on L^P（R^n）.

Characterization of Lipschitz Functions via Commutators of Multilinear Singular Integral Operators in Variable Lebesgue Spaces

Let 6=(bi,b2,...,bm)be a collection of locally integrable functions and T,the com-mutator of multilinear singular integral operator T.Denote by L(δ)and L(δ(·))the Lipschitz spaces and the variable Lipschitz spaces,respectively.The main purpose of this paper is to establish some new characterizations of the(variable)Lipschitz spaces in terms of the boundedness of multilinear commutator T∑b in the context of the variable exponent Lebesgue spaces,that is,the authors give the necessary and sufficient conditions for bj(j=1,2,...,m)to be L(δ)or L(δ(·))via the boundedness of multilinear commutator from products of variable exponent Lebesgue spaces to variable exponent Lebesgue spaces.The authors do so by applying the Fourier series technique and some pointwise esti-mate for the commutators.The key tool in obtaining such pointwise estimate is a certain generalization of the classical sharp maximal operator.

Multiple weighted estimates for maximal vector-valued commutator of multilinear Calderon-Zygmund singular integrals

This paper concerns with multiple weighted norm inequalities for maximal vector-valued multilinear singular operator and maximal commutators. The Cotlar-type inequality of maximal vector-valued multilinear singular integrals operator is obtained. On the other hand, pointwise estimates for sharp maximal function of two kinds of maximal vector-valued multilinear singular integrals and maximal vector-valued commutators are also established. By the weighted estimates of a class of new variant maximal operator, Cotlar＇s inequality and the sharp maximal flmction estimates, multiple weighted strong estimates and weak estimates for maximal vector-valued singular integrals of multilinear operators and those for maximal vector-valued commutator of multilinear singular integrals are obtained.

Pointwise estimates for the maximal multilinear singular integral operators

Two pointwise estimates relating the maximal multilinear singular integral operators and some classical maximal operators are established. These pointwise estimates imply the rearrangement estimate and the BLO(Rn) estimate for the maximal multilinear singular integral operators.

Boundedness on Triebel-Lizorkin and Lebesgue Spaces of Multilinear Singular Integral Operators Satisfying a Variant of Hormander’s Condition

The boundedness on Triebel-Lizorkin and Lebesgue spaces of the multilinear operators associated to some singular integral operators satisfying a variant of Hörmander’s condition are obtained.

Weighted variation inequalities for differential operators and singular integrals in higher dimensions

We prove weighted q-variation inequalities with 2<q<∞for sharp truncations of singular integral operators in higher dimensions.The vector-valued extensions of these inequalities are also given.Parallel results are proven for differential operators.

Estimates for the Maximal Bilinear Singular Integral Operators

In this paper, the behavior on the product of Lebesgue spaces is considered for the maximal operators associated with the bilinear singular integral operators whose kernels satisfy certain minimal regularity conditions.

The Boundedness of Maximal Operators and Singular Integrals via Fourier Transform Estimates

In this paper, the author studies the mapping properties for some general maximal op- erators and singular integrals on certain function spaces via Fourier transform estimates. Also, some concrete maximal operators and singular integrals are studied as applications.

向量值多线性奇异积分算子的加权Sharp不等式

利用不等式、原子分解性质研究了向量值多线性算子,证明了某些向量值多线性奇异积分算子的一个加权Sharp不等式,并利用此不等式得到了该向量值多线性算子的加权L^(p)型不等式和L log L型不等式。

Multilinear Singular Integrals with Rough Kernel

For a class of multilinear singular integral operators TA,$$T_A f\left( x \right) = \int {_{\Ropf^n} } {{\Omega \left( {x - y} \right)} \over {\left| {x - y} \right|^{n + m - 1} }}R_m \left( {A;x,y} \right)f\left( y \right)dy,$$where Rm (A; x, y) denotes the m-th Taylor series remainder of A at x expanded about y, A has derivatives of order m m 1 in $\dot \Lambda_\beta $(0 < # < 1), OHgr;(x) ] L^s(S^nm1)($s \ge {n \over {n - \beta }}$) is homogeneous of degree zero, the authors prove that TA is bounded from L^p(A^n) to L^q) (A^n) (${1 \over p} - {1 \over q} = {\beta \over n},\,1 < p < {n \over \beta }$) and from L^1 (A^n) to L^n/(nm#), ^X (A^n) with the bound $C\sum\nolimits_{\left| \gamma \right| = m - 1} {} \left\|\left\| {D^\gamma A} \right\|\right\|_{\dot \Lambda_\beta} $. And if Q has vanishing moments of order m m 1 and satisfies some kinds of Dini regularity otherwise, then TA is also bounded from L^p (A^n) to ${\dot F}^{\beta,\infty}_p$ (A^n)(1 < s' < p < X) with the bound $C\sum\nolimits_{\left| \gamma \right| = m - 1} {} \left\| \left\|{D^\gamma A} \right\|\right\|_{\dot \Lambda _\beta } $.

Multilinear Singular and Fractional Integrals

In this paper, we treat a class of non-standard commutators with higher order remainders in the Lipschitz spaces and give （L^v, L^q）, （H^p, L^q） boundedness and the boundedness in the Triebel- Lizorkin spaces. Our results give simplified proofs of the recent works by Chen, and extend his result.

Hardy spaces estimates for a class of multilinear homogeneous operators

It is proved that a class of multilinear singular integral operators with homogeneous kernels are bounded operators from the product spaces Lp1×Lp2×…× Lpk（Rn） to the Hardy spaces HT （Rn） and the weak Hardy space HT,∞（Rn）. As an application of this result, the LP（Rn） boundedness of a class of commutator for the singular integral with homogeneous kernels is obtained.

Boundedness of Multilinear Operators on Herz Type Spaces: Extreme Cases

In this paper, two classes of closely related multilinear singular andfractional integrals, which include the commutators as special cases, are studied and theirboundedness on Herz type spaces is discussed. In fact, it is proved that these operators areactually not bounded in certain extreme cases.