Boundedness of the anisotropic maximal and anisotropic singular integral operators in generalized Morrey spaces

In this paper we give the conditions on the pair (ω ₁, ω ₂) which ensures the boundedness of the anisotropic maximal operator and anisotropic singular integral operators from one generalized Morrey space M_p,w1 \mathcal{M}_{p,\omega _1 } to another M_p,w2 \mathcal{M}_{p,\omega _2 }, 1 < p < g8, and from the space M_1,w1 \mathcal{M}_{1,\omega _1 } to the weak space WM_1,w2 W\mathcal{M}_{1,\omega _2 }.     

Strongly singular convolution operators on the weighted Herz-type Hardy spaces

Let 0<p≤1<q<0, andw ₁ ,w ₂ ∈ A ₁ (Muckenhoupt-class). In this paper the authors prove that the strongly singular convolution operators are bounded from the homogeneous weighted Herz-type Hardy spacesH K^{α, p} _q(w₁; w₂) to the homogeneous weighted Herz spacesK ^{α, p} _q (w₁; w₂), provided α=n(1−1/q). Moreover, the boundedness of these operators on the non-homogeneous weighted Herz-type Hardy spacesH K ^{α, p} _q (w ₁;w ₂) is also investigated. Supported by the National Natural Science Foundation of China     

Integrability of the general product Hardy operators on the product Hardy spaces

In this paper,we prove that the general product Hardy operators are bounded from the product Hardy space H1/n ( Rm1 ×…× Rmn ) to L 1 ( RΣni=1 mi).     

Weighted Hardy and singular operators in Morrey spaces

We study the weighted boundedness of the Cauchy singular integral operator SΓ in Morrey spaces Lp,λ(Γ) on curves satisfying the arc-chord condition, for a class of “radial type” almost monotonic weights. The non-weighted boundedness is shown to hold on an arbitrary Carleson curve. We show that the weighted boundedness is reduced to the boundedness of weighted Hardy operators in Morrey spaces Lp,λ(0,?), ?>0. We find conditions for weighted Hardy operators to be bounded in Morrey spaces. To cover the case of curves we also extend the boundedness of the Hardy-Littlewood maximal operator in Morrey spaces, known in the Euclidean setting, to the case of Carleson curves.     

Rough singular integral operators on Hardy-Sobolev spaces

The authors study the singular integral operator TΩ,αf(x)=p.v.∫R^nb(|y|Ω(y′)|y|^-n-αf(x-y)dy, defined on all test functions f, where b is a bounded function, α＞0, Ω(y′) is an integrable function on the unit sphere S^n-1 satisfying certain cancellation conditions. It is proved that, for n/(n α)＜p＜∞,TΩ,α is a bounded operator from the Hardy-Sobolev space H^pα to the Hardy space H^p. The results and its applications improve some theorems in a previous paper of the author and they are extensions of the main theorems in Wheeden‘s paper(1969). The proof is based on a new atomic decomposition of the space H^pα by Han, Paluszynski and Weiss(1995). By using the same proof,the singluar integral operators with variable kernels are also studied.     

Estimates for oscillatory strongly singular integral operators

In this paper we consider oscillatory integral operators with strong singularities on . We obtain sharp decay estimates for L² operator norm. We also obtain H^p estimates for difference of oscillatory strongly singular integral operators and strongly singular integral operators, which gives L^p mapping properties of oscillatory strongly singular integral operators.     

Marcinkiewicz积分交换子在一些Hardy空间的有界性

讨论了由核函数满足具有某类Dini条件的Marcinkiewicz积分μΩ及函数b∈Lipβ(R~n)生成的交换子μ(_Ω,b)~m的性质.证明了Marcinkiewicz积分交换子μ_(_Ω,b)~m在Hardy型空间H_(bm,s)(R~n)上有界,也在Herz型Hardy空间H_(bm)K_p~(a_q)(R~n)上有界.     

New hardy spaces associated with some anisotropic Herz spaces and their applications

In this paper, a class of anisotropic Herz-type Hardy spaces associated with a non-isotropic dilation on ℝ ⁿ are introduced, and the central atomic and molecular decomposition characterizations of those spaces are established. As some applications of the decomposition theory, the authors study the interpolation problem and the boundedness of the central δ-Calderón-Zygmund operators on the anisotropic Herz-type Hardy spaces. The research is supported by NSF of China (Grant Nos. 10571014 and 10571015) and SRFDP of China (Grant No. 20050027025)     

广义分数次积分算子在加权Herz型Hardy空间的有界性

利用加权Herz型Hardy空间的原子分解理论,讨论了广义分数次积分算子Tl从加权Lp空间到加权Lq空间,以及从加权Herz型Hardy空间到加权Herz空间的有界性问题.将已有的分数次积分算子的结论推广到广义分数次积分算子的情形.     

Composition operators on Hardy spaces of a half-plane

We consider composition operators on Hardy spaces of a half-plane. We mainly study boundedness and compactness. We prove that on these spaces there are no compact composition operators.

     

Anisotropic weak Hardy spaces and interpolation theorems

In this paper, the authors establish the anisotropic weak Hardy spaces associated with very general discrete groups of dilations. Moreover, the atomic decomposition theorem of the anisotropic weak Hardy spaces is also given. As some applications of the above results, the authors prove some interpolation theorems and obtain the boundedness of the singular integral operators on these Hardy spaces.     

各向异性Hardy空间的分子特征及其应用

给定一个扩张矩阵A,得到了某些伴随于A的各向异性Hardy空间Hp(Rn)的分子特征刻画.作为其应用,还研究了与A相关的Calderón-Zygmund奇异积分算子和分数次积分算子在各向异性Hardy空间的有界性.     

The common Hardy space and BMO space for singular integral operators associated with isotropic and anisotropic homogeneity

If _T1$T_1$ and _T2$T_2$ are two singular integral operators associated with isotropic and anisotropic homogeneity, respectively, then _T1$T_1$, _T2$T_2$ and _T1°_T2$T_1°T_2$ are bounded on different Hardy spaces and BMO spaces (see ,  and ). In our paper, we show that these operators are actually bounded on a common Hardy space and a common BMO space.     

Hardy and singular operators in weighted generalized Morrey spaces with applications to singular integral equations

We study the weighted boundedness of the multi‐dimensional Hardy‐type and singular operators in the generalized Morrey spaces , defined by an almost increasing function φ(r) and radial type weights. We obtain sufficient conditions, in terms of numerical characteristics, that is, index numbers of the weight functions and the function φ. In relation with the wide usage of singular integral equations in applications, we show how the solvability of such equations in the generalized Morrey spaces depends on the main characteristics of the space, which allows to better control both the singularities and regularity of solutions. Copyright © 2012 John Wiley & Sons, Ltd.     

A note on the boundedness of Calderón-Zygmund operators on Hardy spaces

It was well known that Calderón-Zygmund operators T are bounded on Hp for provided T^∗(1)=0. A new Hardy space , where b is a para-accretive function, was introduced in [Y. Han, M. Lee, C. Lin, Hardy spaces and the Tb-theorem, J. Geom. Anal. 14 (2004) 291-318] and the authors proved that Calderón-Zygmund operators T are bounded from the classical Hardy space Hp to the new Hardy space if T^∗(b)=0. In this note, we give a simple and direct proof of the boundedness of Calderón-Zygmund operators via the vector-valued singular integral operator theory.     

Convolution operators on the weighted Herz-type Hardy spaces

A molecular characterization of the weighted Herz-type Hardy spaces and is given, by which the boundedness of the Hilbert transform and the Riesz transforms are proved on these space for 0<p1. These results are obtained by first deriving that the convolution operator Tf=k*f is bounded on the weighted Herz-type Hardy spaces.     

Norms of multiplication operators on Hardy spaces and weighted composition operators from Hardy spaces to weighted-type spaces on bounded symmetric domains

Let D be a bounded symmetric domain. We calculate operator norm of the multiplication operator on the Hardy space H^p(D), as well as of the weighted composition operator from H^p(D) to a weighted-type space.     

Weighted Hardy operators and commutators on Morrey spaces

The operator norms of weighted Hardy operators on Morrey spaces are worked out. The other purpose of this paper is to establish a sufficient and necessary condition on weight functions which ensures the boundedness of the commutators of weighted Hardy operators (with symbols in BMO(ℝ ⁿ )) on Morrey spaces.     

Boundedness of higher order commutators of generalized fractional integral operators on Hardy spaces

Let Tμ,b,m be the higher order commutator generated by a generalized fractional integral operator Tμ and a BMO function b. In this paper, we will study the boundedness of Tμ,b,m on classical Hardy spaces and Herz-type Hardy spaces.