Characterizations of Hankel multipliers

We give characterizations of radial Fourier multipliers as acting on radial L ^p functions, 1 < p < 2d/(d + 1), in terms of Lebesgue space norms for Fourier localized pieces of the convolution kernel. This is a special case of corresponding results for general Hankel multipliers. Besides L ^p  − L ^q bounds we also characterize weak type inequalities and intermediate inequalities involving Lorentz spaces. Applications include results on interpolation of multiplier spaces. G. Garrigós partially supported by grant “MTM2007-60952” and Programa Ramón y Cajal, MCyT (Spain). A. Seeger partially supported by NSF grant DMS 0652890.     

On singular integrals along surfaces related to black spaces

Leth(t) be an arbitrary bounded radial function and let (x) be a real measurable and radial function defined onR ^n–1. Forx, yR ^n–1, we establish that the singular integral along surfacex (x, (x)):

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.     

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.     

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     

Inequalities related toH p smoothness of Sobolev type

By exploiting a class of maximal functions and Littlewood-Paley theory, a list of embedding inequalities onH ^p-Sobolev spaces andH ^p boundedness results for Riesz and Bessel potentials are obtained at one stroke.This work was supported in part by the Chung-Ang University Academic Research Special Grants, 1997.     

On maximal operators along surfaces

In this paper, we establish the boundedness of the following maximal operator

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.     

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.     

Weighted estimates for rough oscillatory singular integrals

Weighted L^p estimates (1<p<∞) are shown for oscillatory singular integral operators with polynomial phase and a rough kernel of the form e^iP(x,y)Ω(x−y)h(|x−y|)|x−y|⁻ⁿ. We assume that Ω∈L logL(Sⁿ⁻¹) is homogeneous of degree zero and ∫_Sn-1Ω=0. The radial factor h has bounded variation. The necessary condition on the weight is similar to the A_p condition but involves rectangles (instead of cubes) arising from a covering of a star-shaped set related to Ω.     

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).     

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)     

Characterization ofL p (R n ) using Gabor frames

We characterize L^p norms of functions onR ⁿ for 1<p<∞ in terms of their Gabor coefficients. Moreover, we use the Carleson-Hunt theorem to show that the Gabor expansions of L^p functions converge to the functions almost everywhere and in L^p for 1<p<∞. In L¹ we prove an analogous result: the Gabor expansions converge to the functions almost everywhere and in L¹ in a certain Cesàro sense. Consequently, we are able to establish that a large class of Gabor families generate Banach frames for L^p (R ⁿ) when 1≤p<∞.     

Sharp weighted norm inequalities for Littlewood–Paley operators and singular integrals

We prove sharp Lp(w) norm inequalities for the intrinsic square function (introduced recently by M. Wilson) in terms of the Ap characteristic of w for all 1<p<∞. This implies the same sharp inequalities for the classical Lusin area integral S(f), the Littlewood–Paley g-function, and their continuous analogs Sψ and gψ. Also, as a corollary, we obtain sharp weighted inequalities for any convolution Calderón–Zygmund operator for all 1<p?3/2 and 3?p<∞, and for its maximal truncations for 3?p<∞.     

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.     

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<.     

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.     

The ρ-variation as an operator between maximal operators and singular integrals

The ρ-variation and the oscillation of the heat and Poisson semigroups of the Laplacian and Hermite operators (i.e. Δ and −Δ + |x|²) are proved to be bounded from into itself (from into weak- in the case p = 1) for 1 ≤ p < ∞ and w being a weight in the Muckenhoupt’s A _p class. In the case p = ∞ it is proved that these operators do not map L ^∞ into itself. Even more, they map L ^∞ into BMO but the range of the image is strictly smaller that the range of a general singular integral operator. R. Crescimbeni was partially supported by Fundación Carolina, Ministerio de Educación de la República Argentina and Universidad Nacional del Comahue. R. A. Macías and B. Viviani were partially supported by Facultad de Ingeniera Química-UNL.     

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).