ATOMIC DECOATOMIC DECOMPOSITIONS AND DUALS OF WEAK HARDY SPACES OF B-VALUED MARTINGALES

In this article, several weak Hardy spaces of Banach-space-valued martingales are introduced, some atomic decomposition theorems for them are established and their duals are investigated. The results closely depend on the geometrical properties of the Banach space in which the martingales take values.

Atomic Decompositions of Weak Hardy Spaces of B-Valued Martingales

The atomic decompositions of weak Hardy spaces of Banach-space-valued martingales are given. With the help of the atomic decompositions, some inequalities for B-valued martingales are established in the case 0〈r≤1. Here the results are connected closely with the p-uniform smoothness and q-uniform convexity of Banach spaces which the martingales take values in.

Boundedness of Multilinear Oscillatory Singular Integral on Weighted Weak Hardy Spaces

In this paper, by using the atomic decomposition of the weighted weak Hardy space WH;（R;）, the authors discuss a class of multilinear oscillatory singular integrals and obtain their boundedness from the weighted weak Hardy space WH;（R;） to the weighted weak Lebesgue space WL;（R;） for ω∈A;（R;）.

Calderón-Zygmund-Type Operators on Weighted Weak Hardy Spaces over R^n

We introduce certain Calderón-Zygmund-type operators and discuss their boundedness on spaces such as weighted Lebesgue spaces,weighted weak Lebesgue spaces,weighted Hardy spaces and weighted weak Hardy spaces.The sharpness of some results is also investigated.

Estimates of Some Integral Operators with Bounded Variable Kernels on the Hardy and Weak Hardy Spaces over R^n

In this paper,we first introduce Lσ1-（log L）σ2 conditions satisfied by the variable kernelsΩ（x,z） for 0≤σ1≤1 and σ2≥0.Under these new smoothness conditions,we will prove the boundedness properties of singular integral operators TΩ,fractional integrals TΩ,α and parametric Marcinkiewicz integrals μΩρ with variable kernels on the Hardy spaces Hp（Rn） and weak Hardy spaces WHP（Rn）.Moreover,by using the interpolation arguments,we can get some corresponding results for the above integral operators with variable kernels on Hardy-Lorentz spaces Hp,q（Rn） for all p 〈 q 〈 ∞.

Parameterized Littlewood-Paley Operators and Area Integrals on Weak Hardy Spaces

In this paper, we establish the boundedness of parameterized Littlewood-Paley operator μλ^＊,p and parameterized area integralμΩ^σSp with kernel satisfying the logarithmic type Lipschitz condition on the weak Hardy space.

Parametrized Area Integrals on Hardy Spaces and Weak Hardy Spaces

In this paper, the authors prove that if Ω satisfies a class of the integral Dini condition, then the parametrized area integral μΩ,S^ρ is a bounded operator from the Hardy space H1 （R^n） to L1 （R^n） and from the weak Hardy space H^1,∞ （R^n） to L^1,∞ （R^n）, respectively. As corollaries of the above results, it is shown that μΩ,S^ρ is also an operator of weak type These conclusions are substantial improvement and （1, 1） and of type （p,p） for 1 〈 p 〈 2, respectively extension of some known results.

A New Estimate for Bochner–Riesz Operators at the Critical Index on Weighted Hardy Spaces

Let w be a Muckenhoupt weight and Hwp （JRn） be the weighted Hardy space. In this paper, by using the atomic decomposition of Hwp（Rn）, we will show that the Bochner-Riesz operators TRδ are bounded from Hwp（Rn） to the weighted weak Hardy spaces WHwp （Rn） for 0 〈 p 〈 1 and δ = n/p- （n ＋ 1）/2. This result is new even in the unweighted case.

Boundedness of parammetric Marcinkiewicz integrals on weighted Hardy spaces

In this paper, several new results on the boundedness of parammetric Marcinkiewicz integrals on the weighted Hardy spaces and the weak weighted Hardy spaces are established.

Weak Hardy-type spaces associated with ball quasi-Banach function spaces I:Decompositions with applications to boundedness of Calderón-Zygmund operators

Let X be a ball quasi-Banach function space on R^(n).In this article,we introduce the weak Hardytype space WH_(X)(R^(n)),associated with X,via the radial maximal function.Assuming that the powered HardyLittlewood maximal operator satisfies some Fefferman-Stein vector-valued maximal inequality on X as well as it is bounded on both the weak ball quasi-Banach function space WX and the associated space,we then establish several real-variable characterizations of WH_(X)(R^(n)),respectively,in terms of various maximal functions,atoms and molecules.As an application,we obtain the boundedness of Calderón-Zygmund operators from the Hardy space H_(X)(R^(n))to WH_(X)(Rn),which includes the critical case.All these results are of wide applications.Particularly,when X:=M^(q)_(p)(R^(n))(the Morrey space),X:=L^(p)(R^(n))(the mixed-norm Lebesgue space)and X:=(EΦq)t(Rn)(the Orlicz-slice space),which are all ball quasi-Banach function spaces rather than quasiBanach function spaces,all these results are even new.Due to the generality,more applications of these results are predictable.

CONTINUITY FOR MAXIMAL COMMUTATORS OF BOCHNER-RIESZ OPERATORS WITH BMO FUNCTIONS

Let p is an element of (n/(n + 1), 1]. The authors investigate the (H-b(p)(R-n), L-p(R-n))-type and (H-b(p,infinity)(R-n), L-p,L-infinity(R-n))-type continuities for the maximal operators associated with the commutators of Bochner-Riesz operators with BMO(R-n) functions, where H-b(p)(R-n) and H-b(p,infinity)(R-n) are, respectively, the variants of the standard Hardy spaces and the standard weak Hardy spaces.

Intrinsic Square Function Characterizations of Several Hardy-Type Spaces-a Survey

In this article,the authors give a survey about the recent developments of intrinsic square function characterizations and their applications on several Hardy-type spaces,including(weak)Musielak-Orlicz Hardy spaces,variable(weak)Hardy spaces,and Hardy spaces associated with ball quasi-Banach function spaces.The authors also present some open problems.

A New Characterization of p-Smoothable Banach Spaces

In this paper an atomic decomposition theorem for Banach-space-valued weak Hardy regular martingale space wpHα^s（X） is given. As an application, p-smoothable Banach spaces are characterized in terms of bounded sublinear operators defined on Banach-space-valued weak Hardy regular martingale space wpHa^s（X）. Keywords Martingale, weak Hardy space, atomic decomposition, p-smoothable Banach space