Fractional Integral Operators on Anisotropic Hardy Spaces

In this paper, we introduce the fractional integral operator T of degree α of order m with respect to a dilation A for 0 < α < 1 and . First we establish the Hardy-Littlewood-Sobolev inequalities for T on anisotropic Hardy spaces associated with dilation A, which show that T is bounded from H ^p to H ^q , or from H ^p to L ^q , where 0 < p ≤ 1/(1 + α) and 1/q = 1/p − α. Then we give anisotropic Hardy spaces estimates for a class of multilinear operators formed by fractional integrals or Calderón-Zygmund singular integrals. Finally, we apply the above results to give the boundedness of the commutators of T and a BMO function. Research supported by NSF of China (Grant: 10571015) and SRFDP of China (Grant: 20050027025).     

The maximal (C, α, β) operator of two-parameter walsh-fourier series

The two-parameter dyadic martingale Hardy spacesH _p are introduced and it is proved that the maximal operator of the (C, α, β) means of a two-dimensional Walsh-Fourier series is bounded from H_p to L_p (1/(α+1), 1/(β+1)<p<∞) and is of weak type (H ₁ ^# , L₁), where the Hardy space H ₁ ^# is defined by the hybrid maximal function. As a consequence, we obtain that the (C, α, β) means of a function f∈H ₁ ^# converge a.e. to the function in question. Moreover, we prove that the (C, α, β) means are uniformly bounded on H_p whenever 1/(α+1), 1/(β+1)<p<∞. Thus in case f∈H_p, the (C, α, β) means converge to f in H_p norm. The same results are proved for the conjugate (C, α, β) means, too.     

Convolutions, multipliers and maximal regularity on vector-valued Hardy spaces

We extend the recently developed L^p-theory for the maximal regularity of the abstract Cauchy problem and the related Fourier multiplier techniques to the real-variable Hardy space H¹. Some results for H^p, 0 < p < 1, are also proved.     

Norm Estimate of the Cauchy Transform on L P (Ω)

We give two sided estimates of the norm of Cauchy transform on L^p(Ω)(1 ≤ p < ∞) in the case when Ω is the unit disc or the bounded simply connected domain with piecewise C¹ boundary. As a consequence we get the better constant in Poincare inequality. Also, we conjecture the exact value norm of Cauchy transform on L^p(D), where D is the unit disc.     

Wavelet characterization of weighted spaces

We give a characterization of weighted Hardy spaces H ^p (w), valid for a rather large collection of wavelets, 0 <p ≤ 1,and weights w in the Muckenhoupt class A _∞ We improve the previously known results and adopt a systematic point of view based upon the theory of vector-valued Calderón-Zygmund operators. Some consequences of this characterization are also given, like the criterion for a wavelet to give an unconditional basis and a criterion for membership into the space from the size of the wavelet coefficients.     

Cesaro Summability with Respect to Two-Parameter Walsh Systems

 The one- and two-parameter Walsh system will be considered in the Paley as well as in the Kaczmarz rearrangement. We show that in the two-dimensional case the restricted maximal operator of the Walsh–Kaczmarz (C, 1)-means is bounded from the diagonal Hardy space H ^p to L ^p for every . To this end we consider the maximal operator T of a sequence of summations and show that the p-quasi-locality of T implies the same statement for its two-dimensional version T _α. Moreover, we prove that the assumption is essential. Applying known results on interpolation we get the boundedness of T _α as mapping from some Hardy–Lorentz spaces to Lorentz spaces. Furthermore, by standard arguments it will be shown that the usual two-parameter maximal operators of the (C, 1)-means are bounded from L ^p spaces to L ^p if . As a consequence, the a.e. convergence of the (C, 1)-means will be obtained for functions such that their hybrid maximal function is integrable. Of course, our theorems from the two-dimensional case can be extended to higher dimension in a simple way. (Received 20 April 2000; in revised form 25 September 2000)     

On modular inequalities in variable L p spaces

We show that the Hardy-Littlewood maximal operator and a class of Calderón-Zygmund singular integrals satisfy the strong type modular inequality in variable L^p spaces if and only if the variable exponent p(x) ∼ const. Received: 15 September 2004     

Singular integrals with rough kernels along real-analytic submanifolds in R n

L ^p mapping properties will be established in this paper for singular Radon transforms with rough kernels defined by translated of a real-analytic submanifold in R ⁿ .Work in this paper was done during the second author's visit at the Department of Mathematics, University of Pittsburgh.Supported in part by NSF Grant DMS-9622979.     

一类粗糙核参数型Marcinkiewicz积分

In this paper,the L2-boundedness of a class of parametric Marcinkiewicz integral μρΩ,h with kernel function Ω in Bq0.0 (Sn-1) for some q＞ 1,and the radial function h (x)∈ l∞ (Ls) (R+) for 1＜s≤∞ are given. The Lp(Rn) (2≤p＜∞) boundedness of μ*Ω,ph,λ and μρΩ,h,s with Ω in Bq0,0(Sn-1) and h(|x|)∈l∞(Ls)(R+) in application are obtained. Here μ*Ω,p h,λ and μpΩ,h,s are parametric Marcinkiewicz integrals corresponding to the Littlewood-Paley gλ* function and the Lusin area function S,respectively.     

Some remarks on the Hardy-Littlewood maximal function on variable <Emphasis Type="Italic">L</Emphasis><Superscript><Emphasis Type="Italic">p</Emphasis></Superscript> spaces

We show that any pointwise multiplier for BMO(ℝⁿ) generates a function p from the class (ℝⁿ) of those functions for which the Hardy-Littlewood maximal operator is bounded on the variable L^p space. In particular, this gives a positive answer to Diening's conjecture saying that there are discontinuous functions which nevertheless belong to (ℝⁿ).     

Hardy spaces and the<Emphasis Type="Italic">Tb</Emphasis> theorem

It is well-known that Calderón-Zygmund operators T are bounded on H^p for\(\frac{n}{{n + 1}}< p \leqslant 1\) provided T*(1) = 0. In this article, it is shown that if T*(b) = 0, where b is a para-accretive function, T is bounded from the classical Hardy space H^p to a new Hardy space H _b ^p . To develop an H _b ^p theory, a discrete Calderón-type reproducing formula and Plancherel-Pôlya-type inequalities associated to a para-accretive function are established. Moreover, David, Journé, and Semmes’ result [9] about the L^P, 1 < p < ∞, boundedness of the Littlewood-Paley g function associated to a para-accretive function is generalized to the case of p ≤ 1. A new characterization of the classical Hardy spaces by using more general cancellation adapted to para-accretive functions is also given. These results complement the celebrated Calderón-Zygmund operator theory.     

Marcinkiewicz integral on hardy spaces

In this paper we prove that the Marcinkiewicz integral is an operator of type (H ¹,L ¹) and of type (H ^1,,L ^1,). As a corollary of the results above, we obtain again the the weak type (1,1) boundedness of , but the smoothness condition assumed on is weaker than Stein's condition.The research was supported partly by Doctoral Programme Foundation of Institution of Higher Education (Grant No. 98002703) of China.The author was supported partly by NSF of China (Grant No. 19971010).The author was supported partly by NSF of China (Grant No. 19131080).     

Littlewood-paley operators on the generalized Lipschitz spaces

Littlewood-Paley operators defined on a new kind of generalized Lipschitz spaces ₀ ^,p are studied. It is proved that the image of a function under the action of these operators is either equal to infinity almost everywhere or is in ₀ ^,p , where –n<<1 and 1<p<.     

A new proof of certain littlewood-paley inequalities

The aim of this article is to give a new proof of the L^p-inequalities for the Littlewood-Paley g_*-function. Our main tool is a pointwise equality, relating a function f, and the associated functional g_*(f), which has the form f²=h(f)+g _* ² (f), where h(f) is an explicit function. We obtain this equality as a particular case of a more general one, which is reminiscent of a well-known identity in the stochastic calculus setting, namely the Itô formula. Once the above equality is proved, L^p-estimates for g_*(f) are obviously equivalent to L^p/2-estimates for h(f). We obtain these last estimates (more precisely, H^p/2-estimates for h(f) by using a slight extension of the Coifman-Meyer-Stein theorem relating the so-called tent-spaces and the Hardy spaces. We observe that our methods clearly show that the restriction p>2n/n+1 is closely related to cancellation and size properties of the gradient of the Poisson kernel.     

On Marcinkiewicz Integral Operators with Rough Kernels

This paper is devoted to the study on the L^p-mapping properties of Marcinkiewicz integral operators with rough kernels along “polynomial curves” on The boundedness of the Marcinkiewicz integrals for some fixed 1 < p < ∞ are obtained under some size conditions, which essentially improve or extend some well-known results.     

A necessary condition for Calderón-Zygmund singular integral operators

Calderón-Zygmund singular integral operators have been extensively studied for almost half a century. This paper provides a context for and proof of the following result: If a Calderón-Zygmund convolution singular integral operator is bounded on the Hardy space H¹ (Rⁿ), then the homogeneous of degree zero kernel is in the Hardy space H¹(S^n–1) on the sphere.     

Boundedness of generalized marcinkiewicz integral on the <Emphasis Type="Italic">Hp</Emphasis>-sobolev spaces

In this paper, the general Marcinkiewicz integral operator μΩ,α on the Hp Sobolev spaces under the proper condition of kernel Ω(x′) is considered. It is obtained that μΩ,α is bounded from H{ } to Lp for some 0<p≤1. Jiang and Jia were supported in part by Education Department of Zhejiang province.     

Marcinkiewicz Integrals with Non-Doubling Measures

Let μ be a positive Radon measure on which may be non doubling. The only condition that μ must satisfy is μ(B(x, r)) ≤ Cr ⁿ for all , r > 0 and some fixed constants C > 0 and n ∈ (0, d]. In this paper, we introduce the Marcinkiewicz integral related to a such measure with kernel satisfying some H?rmander-type condition, and assume that it is bounded on L ²(μ). We then establish its boundedness, respectively, from the Lebesgue space L ¹(μ) to the weak Lebesgue space L ^1,∞(μ), from the Hardy space H ¹(μ) to L ¹(μ) and from the Lebesgue space L ^∞(μ) to the space RBLO(μ). As a corollary, we obtain the boundedness of the Marcinkiewicz integral in the Lebesgue space L ^p (μ) with p ∈ (1,∞). Moreover, we establish the boundedness of the commutator generated by the RBMO(μ) function and the Marcinkiewicz integral with kernel satisfying certain slightly stronger H?rmander-type condition, respectively, from L ^p (μ) with p ∈ (1,∞) to itself, from the space L log L(μ) to L ^1,∞(μ) and from H ¹(μ) to L ^1,∞(μ). Some of the results are also new even for the classical Marcinkiewicz integral. The third (corresponding) author was supported by National Science Foundation for Distinguished Young Scholars (No. 10425106) and NCET (No. 04-0142) of China.     

SPACELIKE SUBMANIFOLDS IN THE DE SITTER SPACE Sp^（n＋p）（c）WITH CONSTANT SCALAR CURVATURE

Let M^n be a closed spacelike submanifold isometrically immersed in de Sitter space Sp^(n p)(c)， Denote by R,H and S the normalized scalar curvature,the mean curvature and the square of the length of the second fundamental form of M^n ,respectively. Suppose R is constant and R≤c. The pinching problem on S is studied and a rigidity theorem for M^n immersed in Sp^(n p)(c) with parallel normalized mean curvature vector field is proved. When n≥3, the pinching constant is the best. Thus, the mistake of the paper “Space-like hypersurfaces in de Sitter space with constant scalar curvature”(see Manus Math, 1998,95 :499-505) is corrected. Moreover,the reduction of the codimension when M^n is a complete submanifold in Sp^(n p)(c) with parallel normalized mean curvature vector field is investigated.