INTERIOR ESTIMATES IN MORREY SPACES FOR SOLUTIONS OF ELLIPTIC EQUATIONS AND WEIGHTED BOUNDEDNESS FOR COMMUTATORS OF SINGULAR INTEGRAL OPERATORS

It is proved that, for the nondivergence elliptic equations Σi,jn=1aijuxixj=f, if f belongs to the generalized Morrey spaces Lp, (w), then uxixj ∈ Lp, (w), where u is the W2,p-solution of the equations. In order to obtain this, the author first establish the weighted boundedness for the commutators of some singular integral operators on Lp, (w).

Boundary properties for several singular integral operators in real Clifford analysis

On the basis of the Cauchy integral formulas for regular and biregular functions, we define some Cauchy-type singular integral operators. Then we discuss the Holder continuous property of some singular integral operators with one integral variable. Then we divide a singular integral operator with two variables into three parts and prove its Holder continuous property on the boundary.

The Boundedness of Singular Integral Operator along an Open Curve

It is well known that, the singular integral operatorS defined as: ifL is a closed smooth contour in the complex plane C, thenS is a bounded linear operator fromH μ(L) intoH μ(L): ifL is an open smooth curve, thenS is just a linear operator fromH * intoH *. In this paper, we define a Banach space , and prove that is a bounded linear operator, then verify the boundedness of other kinds of singular integral operators.

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

ON THE COMMUTATORS OF SINGULAR INTEGRAL OPERATORS

Lp(Rn) (1<p<∞) boundedness and a weak type endpoint estimate are considered for the commutators of singular integral operators. A condition on the associated kernel is given under which the L2(Rn) boundedness of the singular integral operators implies the LP(Rn) boundedness (1<p<∞) and the weak type (H1(Rn), L1(Rn)) boundedness for the corresponding commutators. A new interpolation theorem is also established.

Toeplitz Type Operator Associated to Singular Integral Operator with Variable Kernel on Weighted Morrey Spaces

Suppose T^k,l and T^k,2 are singular integrals with variable kernels and mixed homogeneity or ±I （the identity operator）. Denote the Toeplitz type operator by T^b=k=1∑^QT^k,1M^bT^k,2 where M^bf= bf. In this paper, the boundedness of Tb on weighted Morrey space are obtained when b belongs to the weighted Lipschitz function space and weighted BMO function space, respectively.

Weighted estimates for commutators of singular integral operators with non-smooth kernels

In this paper, we establish the boundedness of commutators of singular integral operators with non-smooth kernels on weighted Lipschitz spaces Lipβ,ω. The condition on the kernel in this paper is weaker than the usual pointwise HSrmander condition.

SINGULAR INTEGRAL OPERATORS IN L^2 SPACES ASSOCIATED WITH THE GENERALIZED JACOBI WEIGHTS

In this paper, a kind of new definitions of singular integral operators in the weighted L-2 space with the generalized Jacobi weights are given, then the boundedness in the weighted L-2 space, the relationships between these operators and the classic singular integral operators are proved. For the convenience in the later use, similar results in the L-2 space with Jacobi weights are given.

Weighted Estimates for the Maximal Commutator of Singular Integral Operator on Spaces of Homogeneous Type

Weighted estimates with general weights are established for the maximal operator associated with the commutator generated by singular integral operator and BMO function on spaces of homogeneous type, where the associated kernel satisfies the HSlder condition on the first variable and some condition which is fairly weaker than the Holder condition on the second variable.

Weighted estimates for strongly singular integral operators with rough kernels

The Fourier transform and the Littlewood-Paley theory are used to give the weighted boundedness of a strongly singular integral operator defined in this paper. The paper shows that the strongly singular integral operator is bounded from the Sobolev space to the Lebesgue space.

COMMUTATORS OF BMO FUNCTIONS AND SINGULAR INTEGRAL OPERATORS WITH NON-SMOOTH KERNELS ON SPACES OF HOMOGENEOUS TYPE OF FINITE MEASURE

Let X be a space of homogeneous type with finite measure. Let T be a singular integral operator which is bounded on L^p （X）, 1 〈 p 〈 ∞. We give a sufficient condition on the kernel k（x,y） of Tso thatwhen a function b ∈ BMO （X）,the commutator [b,T] （f）=T （b f）- bT （f） is bounded on spaces L^p for all p, 1 〈 p 〈 ∞.

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.

On Fatou Type Convergence of Convolution Type Double Singular Integral Operators

In this paper, some approximation formulae for a class of convolution type double singular integral operators depending on three parameters of the type(T_λf)(x, y) = ∫_a^b ∫_a^b f(t, s)K_λ(t-x,s-y)dsdt, x,y ∈(a,b), λ ∈ Λ [0,∞),(0.1)are given. Here f belongs to the function space L_1( <a,b >~2), where <a,b> is an arbitrary interval in R. In this paper three theorems are proved, one for existence of the operator(T_λf)(x, y) and the others for its Fatou-type pointwise convergence to f(x_0, y_0), as(x,y,λ) tends to(x_0, y_0, λ_0). In contrast to previous works, the kernel functions K_λ(u,v)don't have to be 2π-periodic, positive, even and radial. Our results improve and extend some of the previous results of [1, 6, 8, 10, 11, 13] in three dimensional frame and especially the very recent paper [15].

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.

SINGULAR INTEGRAL OPERATORS IN L^2 SPACEWITH CHEBYSHEV WEIGHTS

This paper defines a class of singular integral operators (I) over tilde(wj) on L-wj(2) space, where wights w(j)(j = 1 - 4) are lour kinds of Chebyshev weights. The authors prove that (I) over tilde(wj) is an unique linear extension of classic singular integral operator I-wj on Holder space, some important properties of (I) over tilde(wj) and some results of singular integral equation in L-wj(2) space.

A NEW BLO ESTIMATE FOR MAXIMAL SINGULAR INTEGRAL OPERATORS

In this paper, we extend Hu and Zhang＇s results in [2] to different case.

L^p Estimates for Hyperbolic Singular Integral Operators

In this paper, we prove L^P-boundedness of hyperbolic singular integral operators for kernels satisfying weakened regularity conditions, where 1 〈 p 〈 ∞. This extends previous results of A.R. Nahmod.

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.

SINGULAR INTEGRAL OPERATORS ANDSINGULAR QUADRATURE OPERATORSASSOCIATED WITH SINGULAR INTEGRAL EQUATIONS

In this paper, the author discusses some singular integral operators, singular quadrature operators and discretization matrices associated with singular integral equations with Cauchy kernels, and obtain some useful properties for them. These results improve both the classical theory of singular integral equation and the classical theory of singular quadrature.