On a problem of Zygmund

It is proved that there exists an integrable function on [0, 1]² whose integral is nondifferentiable in each direction belonging to a set everywhere dense in [0, 2π] but is strongly differentiable. Translated fromMatematicheskie Zametki, Vol. 64, No. 5, pp. 749–762, November, 1998.     

Some Estimates for $θ$-Type Calderόn–Zygmund Operators and Linear Commutators on Certain Weighted Amalgam Spaces

In this paper, we first introduce some new kinds of weighted amalgam spaces. Then we discuss the strong type and weak type estimates for a class of Calderόn–Zygmund type operators $T_θ$ in these new weighted spaces. Furthermore, the strong type estimate and endpoint estimate of linear commutators $[b, T_θ]$ formed by $b$ and $T_θ$ are established. Also we study related problems about two-weight, weak type inequalities for $T_θ$ and $[b, T_θ]$ in the weighted amalgam spaces and give some results.     

Marcinkiewicz–Zygmund Strong Laws for Infinite Variance Time Series

We study rate of convergence in the strong law of large numbers for finite and infinite variance time series in both contexts of weak and strong dependence.     

The Boundedness of Calderón-Zygmund Operators by Wavelet Characterization

This article deals with the boundedness properties of Calderón-Zygmund operators on Hardy spaces Hp(Rn). We use wavelet characterization of Hp(Rn) to show that a Calderón-Zygmund operator T with T*1 = 0 is bounded on Hp(Rn), n/n+ε p ≤ 1, where ε is the regular exponent of kernel of T . This approach can be applied to the boundedness of operators on certain Hardy spaces without atomic decomposition or molecular characterization.     

Regularity of Weak Solutions of Nonlinear Equations with Discontinuous Coefficients

In this paper, we prove that the weak solutions u∈Wloc^1, p （Ω） （1 〈p〈∞） of the following equation with vanishing mean oscillation coefficients A（x）： -div[（A（x）△↓u·△↓u）p-2/2 A（x）△↓u＋│F（x）│^p-2 F（x）]=B（x, u, △↓u）, belong to Wloc^1, q （Ω）（A↓q∈（p, ∞）, provided F ∈ Lloc^q（Ω） and B（x, u, h） satisfies proper growth conditions where Ω ∪→R^N（N≥2） is a bounded open set, A（x）=（A^ij（x）） N×N is a symmetric matrix function.     

(log,θ)-Calderon-Zygmund算子在H~p中的有界性(英文)

本文我们在条件∫１０θｐ（ｔ）／ｔ１＋ｎ（１－ｐ）ｄｔ＜＋∞下，讨论了（ｌｏｇ，θ）－Ｃａｌｄｅｒóｎ－Ｚｙｇｍｕｎｄ算子在Ｈｐ（Ｒｎ），（０＜ｐ≤１）中的有界性．     

Global existence of weak solutions for the 3D chemotaxis-Navier–Stokes equations

In this paper, we consider the Cauchy problem for the three dimensional chemotaxis-Navier–Stokes equations. By exploring the new a priori estimates, we prove the global existence of weak solutions for the 3D chemotaxis-Navier–Stokes equations.     

Composition Cesàro Operator on the Normal Weight Zygmund Space in High Dimensions

Let n>1 and B be the unit ball in n dimensions complex space Cⁿ.Suppose thatφis a holomorphic self-map of B andψ∈H(B)withψ(0)=0.A kind of integral operator,composition Cesàro operator,is defined by T_φψ(f)(z)=∫¹0f[φ(tz)]Rψ(tz)dt/t,f∈(B)z∈B.In this paper,the authors characterize the conditions that the composition Cesàro operator T_φ,ψis bounded or compact on the normal weight Zygmund space Z_μ(B).At the same time,the sufficient and necessary conditions for all cases are given.     

多复变中正规权Zygmund空间上的几个性质

本文讨论了多复变中单位球上正规权Zygmund空间Z_μ(B)的一些性质.首先给出了Z_μ(B)函数的一种积分表示,接着证明了Z_μ(B)是正规权Bergman空间A_v~1(B)的对偶空间,其对偶对按如下形式给出:■,其中v(p)=(1-ρ~2)~(β+1)μ~(-1)(ρ)(0≤ρ<1)并且β>max{0,b-1}.最后作为积分表示和对偶的一个应用,作者给出了Z_μ(B)中每个函数的一个原子分解.     

Interior gradient estimate for steady flows of electrorheological fluids

In this paper, we consider the equations of stationary motion of electrorheological fluids in any dimension $n\ge 2$. We show that the first gradient of local solutions to the system has optimal $L^q$-regularity with some $q>1$ with respect to the one of the external force. This is achieved by comparison principle and a good $\lambda$-estimate.