WEAK TYPE（H^1 ,L^1） ESTIMATE FOR COMMUTATOR OF MARCINKIEWICZ INTEGRAL

Let μΩ,b be the commutator generalized by the n-dimensional Marcinkiewicz integral μΩ and a function b∈ BMO（R^n）. It is proved that μΩ,bis bounded from the Hardy space H^1 （R^n） into the weak L^1（R^n） space.     

Dixmier traces as singular symmetric functionals and applications to measurable operators

We unify various constructions and contribute to the theory of singular symmetric functionals on Marcinkiewicz function/operator spaces. This affords a new approach to the non-normal Dixmier and Connes-Dixmier traces (introduced by Dixmier and adapted to non-commutative geometry by Connes) living on a general Marcinkiewicz space associated with an arbitrary semifinite von Neumann algebra. The corollaries to our approach, stated in terms of the operator ideal L^(1,∞) (which is a special example of an operator Marcinkiewicz space), are: (i) a new characterization of the set of all positive measurable operators from L^(1,∞), i.e. those on which an arbitrary Connes-Dixmier trace yields the same value. In the special case, when the operator ideal L^(1,∞) is considered on a type I infinite factor, a bounded operator x belongs to L^(1,∞) if and only if the sequence of singular numbers {s_n(x)}_n?1 (in the descending order and counting the multiplicities) satisfies . In this case, our characterization amounts to saying that a positive element x∈L^(1,∞) is measurable if and only if exists; (ii) the set of Dixmier traces and the set of Connes-Dixmier traces are norming sets (up to equivalence) for the space , where the space is the closure of all finite rank operators in L^(1,∞) in the norm ∥.∥_(1,∞).     

Boundaries for algebras of holomorphic functions on Marcinkiewicz sequence spaces

Let Ab(E) be the Banach algebra of all complex-valued bounded continuous functions on the closed unit ball BE of a complex Banach space E and holomorphic in the interior of BE and let Au(E) be the closed subalgebra of those functions which are uniformly continuous on BE. For the case whose bidual is a Marcinkiewicz sequence space Mw, we describe some sufficient conditions for a set to be a boundary of either Ab(E) or Au(E). Moreover, we consider some analogous problems on to those which were studied on the Gowers space Gp of characteristic p by Grados and Moraes [L.R. Grados, L.A. Moraes, Boundaries for algebras of holomorphic functions, J. Math. Anal. Appl. 281 (2003) 575-586; L.R. Grados, L.A. Moraes, Boundaries for an algebra of bounded holomorphic functions, J. Korean Math. Soc. 41 (1) (2004) 231-242].     

非倍测度下Marcinkiewicz积分交换子在Hardy空间中的有界性

在测度μ仅满足多项式增长性条件的假设下,证明了Marcinkiewicz积分算子M与Lipβ(μ)函数b生成的交换子Mb具有(H1(μ), Ln/(n?β)(μ))有界性,同时得到Mb的(Ln/β(μ), RBMO(μ))有界性.     

变量核的Marcinkiewicz积分的加权有界性

对Marcinkiewicz积分算子的有界性问题进行研究.应用Morrey-Herz空间的分解特征和算子在Lρ上的有界性结果,得到变量核的Marcinkiewicz积分算子在Morrey-Herz空间上的加权有界性.     

Singular integrals with mixed homogeneity in Triebel-Lizorkin spaces

In this paper, we study the singular integral

     

Boundedness of Marcinkiewicz integral related to multilinear commutators with variable kernels on Hardy spaces

Let→b=(b1,b2,…,bm),bi∈∧βi(Rn),1≤I≤m,βi＞0,m∑I=1βi=β,0＜β＜1,μΩ→b(f)(x)=(∫∞0|F→b,t(f)(x)|2dt/t3)1/2,F→b,t(f)(x)=∫|x-y|≤t Ω(x,x-y)/|x-y|n-1 mΠi=1[bi(x)-bi(y)dy.We consider the boundedness of μΩ,→b on Hardy type space Hp→b(Rn).     

变指标 Morrey 空间上的 Marcinkiewicz 积分及交换子的有界性

利用Maxcinkiewicz积分算子μ,Lusin面积积分μs和Littlewood-Paley g_λ~*函数以及相应的交换子在变指标Lebesgue空间上的有界性,得到了它们在变指标Morrey空间上的有界性结果.     

The limiting case of the Marcinkiewicz integral: growth for convex sets

The Marcinkiewicz integral

plays a well-known and prominent role in harmonic analysis. In this paper, we estimate the growth of it in the limiting case . Throughout, we assume that is convex; it is interesting that this condition cannot be dropped.

     

A Note on a Marcinkiewicz Integral Operator

We prove the L_p boundedness of Marcinkiewicz integral operators with rough kernels on ℝⁿ and T ⁿ.