An endpoint estimate for maximal multilinear singular integral operators

A weak type endpoint estimate for the maximal multilinear singular integral operator T*Af(x)=supε＞0|(f)(x-y)＞ε (Ω(x-y)/(|x-y|(n 1)))(A(x)-A(y)-▽A(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(Rn). A regularity condition on Ω which implies an LlogL type estimate of T*A is given.     

A note on the multilinear singular integral operators

Lp(Rn) boundedness is considered for the multilinear singular integral operator defined by TAf(x) = ∫Rn Ω(x - y)/|x - y|n 1 (A(x) - A(y) - (△)A(y)(x - y))f(y)dy,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∈ Lipα(Sn-1) (0 ＜α≤ 1) and implies the Lp(Rn) (1 ＜ p ＜ oo) boundedness for the operator TA. Some endpoint estimates are also established.     

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)=∫Rn Ω(x-y)/|x-y|n 1(A(x)-A(y)-(△)A(y)(x-y))f(y)dy is considered, 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((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).     

一类多线性奇异积分算子的加权估计

In this paper, weighted estimates with general weights are established for the multilinear singular integral operator defined by TAf(x) = p. v.RnΩ(x- y)|x- y|n+1 A(x)- A(y)- A(y)(x- y) f(y)dy,where Ω is homogeneous of degree zero, has vanishing moment of order one, and belongs to Lipγ(Sn-1) with γ∈(0, 1], A has derivatives of order one in BMO(Rn).     

The Estimates for Sharp Maximal Functions of Multilinear Strongly Singular Integral Operators

In this paper, the author obtains that the multilinear operators of strongly singular integral operators and their dual operators are bounded from some L^p（R^n） to L^p（R^n） when the m-th order derivatives of A belong to L^p（R^n） for r large enough. By this result, the author gets the estimates for the Sharp maximal functions of the multilinear operators with the m-th order derivatives of A being Lipschitz functions. It follows that the multilinear operators are （L^p, L^p）-type operators for 1 〈 p 〈 ∞.     

Weighted estimate for the multilinear oscillatory singular integral operators

In this paper, the weighted L^p(R^n) boundedness for the multilinear oscillatory singular integral operators with polynomial phases is studied.     

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 (x) where P(x,y) is a nontrivial and real-valued polynomial defined on Rn × Rn, Ω(x) is homogeneous of degree zero on Rn, 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=∑rs=1ms, the author proves that if 0<β=∑rs=1βs<1, and Ω∈ Lq(Sn-1) for some q ＞ 1/(1 - β), then for any p ∈ (1, ∞), and some appropriate 0 ＜ β ＜ 1, TA1,A2,…,Ar is bounded on Lp(Rn).     

Optimal integrability of some system of integral equations

We obtain the optimal integrability for positive solutions of the Euler-Lagrange system of the weighted Hardy-Littlewood-Sobolev inequality in R^n ：{u（x）=1/｜x｜^α｜∫R^n v（y）^q｜y｜^β｜x-y｜^λdy,v（x）=1/｜x｜^β∫R^n u（y）^p｜y｜^α｜x-y｜^λdy.C. Jin and C. Li [Calc. Var. Partial Differential Equations, 2006, 26： 447-457] developed some very interesting method for regularity lifting and obtained the optimal integrability for p, q 〉 1. Here, based on some new observations, we overcome the difficulty there, and derive the optimal integrability for the case of p, q ≥1 and pq ≠1. This integrability plays a key role in estimating the asymptotic behavior of positive solutions when ｜x｜ →0 and when ｜x｜→∞.     

Estimate for a class of multilinear fractional integral operator with rough kernel

In this paper, we prove the boundedness from H1(Rn) to L n/n-α ,∞(Rn) for the multilinear fractional integral operator with rough kernel related to the mi-th remainder of Taylor series of Ai at x about y for i = 1,2, ,l.     

Carleson-Type Maximal Operators with Variable Kernels

The author considers the L^p boundedness for two kinds of Carleson-type maximal operators with variable kernels Ω（x,y＇）/｜y｜^n,where Ω（x,y＇）∈L^∞（R^n）×W2^s（S^n-1）for some s〉0.     

On maximal operators along surfaces

In this paper, we establish the boundedness of the following maximal operator

On the riemann zeta-function and the divisor problem II

Let Δ(x) denote the error term in the Dirichlet divisor problem, and E(T) the error term in the asymptotic formula for the mean square of . If E ^*(t)=E(t)-2πΔ^*(t/2π) with , then we obtain

Multivariate Refinement Equations and Convergence of Cascade Algorithms in Lp（0〈p〈1）Spaces

We consider the solutions of refinement equations written in the form

On the riemann zeta-function and the divisor problem

Let Δ(x) denote the error term in the Dirichlet divisor problem, and E(T) the error term in the asymptotic formula for the mean square of . If with , then we obtain

A note on regularity for a class of quasilinear elliptic equations

The following regularity of weak solutions of a class of elliptic equations of the form are investigated.     

L p Boundedness for the Multilinear Singular Integral Operator

L^p (\mathbbRⁿ )L^{p} (\mathbb{R}^{n} ) boundedness is considered for the maximal multilinear singular integral operator which is defined by

Multilinear Singular Integrals with Rough Kernel

For a class of multilinear singular integral operators T _A ,

A note on <Emphasis Type="Italic">q</Emphasis>-multiplicative functions

We prove the following statement. Let , and let . Suppose that, for all and , the sequence satisfies the relation