Rough singular integrals associated with surfaces of Van der Corput type

In this paper, the authors establish the LV-mapping properties for a class of singular integrals along surfaces in Rn of the form {Ф（lul）u＇ ： u ε ]t^n} as well as the related maimal operators provided that the function Ф satisfies certain oscillatory integral estimates of Van der Corput type, and the integral kernels are given by the radial function h E ε△γ（R＋） for γ 〉 1 and the sphere function ΩεFβ（S^n-1） for someβ 〉 0 which is distinct from HI（Sn-1）.

Hardy空间上的高阶交换子定理(英文)

Coifman ,Rochberg和Weiss发现Caldero幃ｎ -Zygmund算子T与BMO函数b构成的交换子 [T ,b]的Lp 到Lp 有界性 (1<p <∞ ) ,(AnnofMath .10 3 (1976) ,611-63 5 ) ;Perez发现其Hpb(Hp 的一个子空间 )到Lp 有界性 (p =1) ,(J .Funct .Anal.12 8(1995 ) ,163 -185 ) ;该文考虑其Hpb 到Hp 有界性 ,(0 <p≤ 1) ,建立了C -Z算子与BMO函数b构成的交换子在Hardy空间上的有界性定理 ,包括从Hardy空间Hp ,q ,sb ,m 到Hp,Hp ,q ,sb ,m 到Lp;和从HerztypeHardy空间HKα ,p ,sq ,b ,m 到H· Kα ,pq ,H· Kα ,p ,sq ,b ,m到Kα ,pq .

Orlicz-Hardy spaces associated with operators

Let L be a linear operator in L2(Rn) and generate an analytic semigroup {e-tL}t 0 with kernel satisfying an upper bound of Poisson type, whose decay is measured by θ(L) ∈ (0, ∞). Let ω on (0, ∞) be of upper type 1 and of critical lower type p0(ω) ∈ (n/(n + θ(L)), 1] and ρ(t) = t-1/ω-1(t-1) for t ∈ (0, ∞). We introduce the Orlicz-Hardy space Hω, L(Rn) and the BMO-type space BMOρ, L(Rn) and establish the John-Nirenberg inequality for BMOρ, L(Rn) functions and the duality relation between Hω, L(Rn) and BMOρ, L*(Rn), where L* denotes the adjoint operator of L in L2(Rn). Using this duality relation, we further obtain the ρ-Carleson measure characterization of BMOρ, L*(Rn) and the molecular characterization of Hω, L(Rn); the latter is used to establish the boundedness of the generalized fractional operator Lρ- γfrom Hω, L(Rn) to HL1(Rn) or Lq(Rn) with certain q > 1, where HL1(Rn) is the Hardy space introduced by Auscher, Duong and McIntosh. These results generalize the existing results by taking ω(t) = tp for t ∈ (0, ∞) and p ∈ (n/(n + θ(L)), 1].

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.

An integral estimate of Bessel function and its application

In this paper the authors give a new integral estimate of the Bessel function,which is an extension of Calder(?)n-Zygmund’s result.As an application of this result,we prove that the parameterized Marcinkiewicz integralμ_Ω~p with variable kernels is of type (2,2),where the kernel functionΩdoes not have any smoothness on the unit sphere in R^n.

Quaternion H-type group and differential operatorΔ_λ

We study the relations between the quaternion H-type group and the boundary of the unit ball on the two-dimensional quaternionic space.The orthogonal projection of the space of square integrable functions defined on quaternion H-type group into its subspace of boundary values of q- holomorphic functions is considered.The precise form of Cauchy-Szeg(?)kernel and the orthogonal projection operator is obtained.The fundamental solution for the operatorΔ_λis found.

Maximal function characterizations of Hardy spaces on RD-spaces and their applications

Let X be an RD-space, i.e., a space of homogeneous type in the sense of Coifman and Weiss, which has the reverse doubling property. Assume that X has a "dimension" n. For α∈ (0, ∞) denote by Hαp(X ), Hdp(X ), and H?,p(X ) the corresponding Hardy spaces on X defined by the nontangential maximal function, the dyadic maximal function and the grand maximal function, respectively. Using a new inhomogeneous Calder′on reproducing formula, it is shown that all these Hardy spaces coincide with Lp(X ) when p ∈ (1, ∞] and with each other when p ∈ (n/(n + 1), 1]. An atomic characterization for H*,p(X ) with p ∈ (n/(n + 1), 1] is also established; moreover, in the range p ∈ (n/(n + 1), 1], it is proved that the space H?,p(X ), the Hardy space Hp(X ) defined via the Littlewood-Paley function, and the atomic Hardy space of Coifman and Weiss coincide. Furthermore, it is proved that a sublinear operator T uniquely extends to a bounded sublinear operator from Hp(X ) to some quasi-Banach space B if and only if T maps all (p, q)-atoms when q ∈ (p, ∞)∩[1, ∞) or continuous (p, ∞)-atoms into uniformly bounded elements of B.

Multipliers and Herz type spaces

Hrmander condition for boundedness of multiplier operators will be replaced by a weaker condition described by certain weighted or non-weighted Herz spaces. Some results on boundedness of multiplier operators are then established. As direct corollaries of main theorems in this paper, several celebrated results on boundedness of multiplier operators will be improved or deduced.