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.

L^p(R^n) Boundedness for the Maximal Operator of Multilinear Singular Integrals with Calderón-Zygmund Kernels

In this paper, the authors consider a class of maximal multilinear singular integral operators and maximal multilinear oscillatory singular integral operators with standard Calderón-Zygmund kernels, and obtain their boundedness on L^p（R^n） for 1 〈 p 〈 ∞.

Boundedness of Rough Operators on Grand Variable Herz Spaces

In this paper, we prove the boundedness of Calderón-Zygmund singular integral operators TΩ on grand Herz spaces with variable exponent under some conditions.

Boundedness of Singular Integral Operators on Herz-Morrey Spaces with Variable Exponent

Let Ω ∈ Ls(S^n-1)(s>1) be a homogeneous function of degree zero and b be a BMO function or Lipschitz function. In this paper, the authors obtain some boundedness of the Calderón-Zygmund singular integral operator TΩ and its commutator [b, TΩ] on Herz-Morrey spaces with variable exponent.

Weighted estimates for the multilinear singular integral operators with non-smooth kernels

In this paper, by sharp function estimates and certain weak type endpoint estimates, the authors establish some weighted norm inequalities with Ap weights for the multilinear singular integral operators with non-smooth kernels.

Strongly Singular Integral Operators on Weighted Hardy Space

In this paper, we obtain that a strongly singular integral operator is bounded on Lω^p. space for 1 〈 p 〈 ∞. We also obtain that a strongly singular integral operator is a bounded operator from Hω^p, to Lω^p for some weight w and 0 〈 p ≤ 1. And by an atomic decomposition, we obtain that a strongly singular integral operator is a bounded operator on Hω^p for some w and 0 〈 p ≤ 1.

Variation operators for singular integrals and their commutators on weighted Morrey spaces and Sobolev spaces

In this paper,we investigate the boundedness and compactness for variation operators of CalderónZygmund singular integrals and their commutators on weighted Morrey spaces and Sobolev spaces.To be precise,letρ>2 and K be a standard Calderón-Zygmund kernel.Denote by V_(ρ)(T_(K))and V_(ρ)(T_(K,b)^(m))(m≥1)theρ-variation operators of Calderón-Zygmund singular integrals and their m-th iterated commutators,respectively.By assuming that V_(ρ)(T_(K))satisfies an a priori estimate,i.e.,the map V_(ρ)(T_(K)):L^(p0)(R^(n))→L^(p0)(R^(n))is bounded for some p0∈(1,∞),the bounds for V_(ρ)(T_(K))and V_(ρ)(T_(K,b)^(m))on weighted Morrey spaces and Sobolev spaces are established.Meanwhile,the compactness properties of V_(ρ)(T_(K,b)^(m))on weighted Lebesgue and Morrey spaces are also discussed.As applications,the corresponding results for the Hilbert transform,the Hermite Riesz transform,Riesz transforms and rough singular integrals as well as their commutators on the above function spaces are presented.

Boundedness of Multilinear Caldern–Zygmund Singular Operators on Morrey–Herz Spaces with Variable Exponents

In this paper, we introduce Morrey-Herz spaces MKq.p（·）α（·）,λ with variable exponents α（·） and p（·）, and prove the boundedness of multilinear Calderdn-Zygmund singular operators on the product of these spaces.

The Cauchy Integral Operator on Weighted Hardy Space

The authors show that the Cauchy integral operator is bounded from Hωp(R1) to hωp(R1) (the weighted local Hardy space). To prove the results, a kind of generalized atoms is introduced and a variant of weighted "Tb theorem" is considered.

Hardy type estimates for commutators of singular integrals

In this paper we study the Hardy type estimates for commutators Tb of standard Calder(o)n-Zygmund singular integral operators T with a Lipschitz function b. The corresponding results are also obtained on the commutators Sb generated by b with singular integral operators S with variable kernels.

A Class of Oscillatory Singular Integrals

LP mapping properties are considered for a class of oscillatory signular integral operators.Ketwords:Calderon-Zygmund kernel. oscillatory singular integral operator. polynomial growth estimate.

Boundedness of Some Operators and Commutators in Morrey-Herz Spaces on Non-Homogeneous Spaces

The authors introduce the homogeneous Morrey-Herz spaces and the weak homo- geneous Morrey-Herz spaces on non-homogeneous spaces and establish the boundedness in ho- mogeneous Morrey-Herz spaces for a class of sublinear operators including Hardy-Littlewood maximal operators,Calderón-Zygmund operators and fractional integral operators.Further- more,some weak estimate of these operators in weak homogeneous Morrey-Herz spaces are also obtained.Moreover,the authors discuss the boundedness in homogeneous Morrey-Herz spaces of the maximal commutators associated with Hardy-Littlewood maximal operators and multilinear commutators generated by Calderón-Zygmund operators or fractional integral operators with RBMO（μ）functions.

Boundedness of Multilinear Operators in Herz-type Hardy Space

Let k ∈ N.We prove that the lnultilinear operators of finite sums of products of singular integrals on Rn are bounded from H Kqlαl·pl（Rn）×…×HKqkαk,pk（Rn）into HKqα,p（Rn）if they have vanishing moments up to a certain order dictated by the target spaces.These conditions on vanishing moments satisfied by the multilinear operators are also necessary when αj（?）0 and the singular integrals considered here include the Calderón-Zygmund singular integrals and the fractional integrals of any orders.

Hölder,Sobolev,weak-type and BMO estimates in mixed-norm with weights for parabolic equations

We prove weighted mixed-norm Lqt(W2,px)and Lqt(C2,αx)estimates for 1<p,q<∞and 0<α<1,weighted mixed weak-type estimates for q=1,L∞t(Lpx)−BMOt(W2,px)and L∞t(Cαx)−BMOt(C2,xx),and a.e.pointwise formulas for derivatives,for the solutions u=u(t,x)to parabolic equations of the form∂tu−aij(t)∂iju+u=f,t∈,x∈n and for the Cauchy problem{∂tv−aij(t)∂ijv+v=fv(0,x)forfort>0,x∈Rn.x∈Rn,The coefficients a(t)=(aij(t))are just bounded,measurable,symmetric and uniformly elliptic.Furthermore,we show strong,weak type and BMO-Sobolev estimates with parabolic Muckenhoupt weights.It is quite remarkable that most of our results are new even for the classical heat equation∂tu−Δu+u=f.