Direct approach to <Emphasis Type="Italic">L</Emphasis><Superscript><Emphasis Type="Italic">p</Emphasis></Superscript> estimates in homogenization theory

We derive interior L ^p -estimates for solutions of linear elliptic systems with oscillatory coefficients. The estimates are independent of ε, the small length scale of the rapid oscillations. So far, such results are based on potential theory and restricted to periodic coefficients. Our approach relies on BMO-estimates and an interpolation argument, gradients are treated with the help of finite differences. This allows to treat coefficients that depend on a fast and a slow variable. The estimates imply an L ^p -corrector result for approximate solutions.      

SOME ENDPOINT ESTIMATES FOR COMMUTATORS OF θ-TYPE CALDERON-ZYGMUND OPERATORS

In this note, the authors prove that the commutator generated by θ-type Calderón-Zygmund operator T and a Lipschitz function is bounded from L ^p (R ⁿ ) into (R ⁿ ) and also maps from (R ⁿ ) into BMO (R ⁿ ). Supported by NSFC(10571014), NSFC(10571156), the Doctor Foundation of Jxnu (2443), the Natural Science Foundation of Jiangxi province(2008GZS0051).     

SOME ENDPOINT ESTIMATES FOR COMMUTATORS OFθ-TYPE CALDERON-ZYGMUND OPERATORS

In this note, the authors prove that the commutator T→b generated by 0-type Calder6n-Zygmund operator T and a Lipschitz function b→ is bounded from LP(Rn) into Lip(β-n/p)(Rn) and also maps from Ln/β(Rn) into BMO(Rn).     

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 Lp (X), 1 ＜ p ＜∞. We give a sufficient condition on the kernel k(x,y) of Tso that when a function b ∈ BMO (X) ,the commutator [b, T] (f) = T (b f) - bT (f) is aounded on spaces Lp for all p, 1 ＜ p ＜∞.     

Toeplitz operators related to strongly singular Calderón-Zygmund operators

In this paper, the boundedness of Toeplitz operator T _b(f) related to strongly singular Calderón-Zygmund operators and Lipschitz function b ε (ℝⁿ) is discussed from L ^p(ℝⁿ) to L ^q(ℝⁿ), , and from L ^p(ℝⁿ) to Triebel-Lizorkin space . We also obtain the boundedness of generalized Toeplitz operator Θ _α0 ^b from L ^p(ℝⁿ) to L ^q(ℝⁿ), . All the above results include the corresponding boundedness of commutators. Moreover, the boundedness of Toeplitz operator T _b(f) related to strongly singular Calderón-Zygmund operators and BMO function b is discussed on L ^p(ℝⁿ), 1 < p < ∞.     

Some new characterizations on spaces of functions with bounded mean oscillation

Let X be a space of homogeneous type. The authors introduce some generalized approximations to the identity (for short, GAI) with optimal decay conditions in the sense that these conditions are the sufficient and necessary conditions for these GAI's to characterize BMO(X), the space of functions with bounded mean oscillation on X. The authors also obtain a new John‐Nirenberg‐type inequality associated with GAI's, which leads some new characterizations of BMO(X) in terms of rearrangement functions, and certain maximal functions related to GAI's. Some variants of these characterizations for BMO‐type spaces on X of Duong and Yan are also established. Moreover, the equivalence between BMO and the space of BMO type on Ahlfors n ‐regular quasimetric measure spaces is obtained, which confirms an assertion of Duong and Yan (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Boundedness of Rough Singular Integral Operators on Homogeneous Herz Spaces with Variable Exponents

We establish the boundedness of rough singular integral operators on homogeneous Herz spaces with variable exponents. As an application, we obtain the boundedness of related commutators with BMO functions on homogeneous Herz spaces with variable exponents.     

Weighted estimates for bilinear square functions with non-smooth kernels and commutators

Under weaker conditions on the kernel functions,we discuss the boundedness of bilinear square functions associated with non-smooth kernels on the product of weighted Lebesgue spaces.Moreover,we investigate the weighted boundedness of the commutators of bilinear square functions(with symbols which are BMO functions and their weighted version,respectively)on the product of Lebesgue spaces.As an application,we deduce the corresponding boundedness of bilinear Marcinkiewicz integrals and bilinear Littlewood-Paley^-functions.     

混合齐次向量值多线性交换子的奇异积分

The boundedness of vector-valued multilinear commutators of singular integral operators whose kernels are variable with mixed homogeneity on Lebesgue spaces is obtained.     

带非光滑核的多线性奇异积分算子的有界性

该文对带非光滑核的多线性奇异积分算子建立了sharp函数估计,作为应用,得到了该多线性奇异积分算子的L~p(1p∞)范数不等式.