Real Inversion Formulas of the Laplace Transform on Weighted Function Spaces

We investigate compactness of linear operators associated with the real inversion formulas of the Laplace transform, coming with weighted Sobolev reproducing kernel Hilbert spaces on the half line R ⁺. We present concrete reproducing kernels along with several typical examples. Submitted: October 13, 2007. Accepted: November 11, 2007.     

Uncertainty Principles for the Continuous Dunkl Gabor Transform and the Dunkl Continuous Wavelet Transform

In this paper we consider the Dunkl operators T _j , j = 1, . . . , d, on and the harmonic analysis associated with these operators. We define a continuous Dunkl Gabor transform, involving the Dunkl translation operator, by proceeding as mentioned in [20] by C.Wojciech and G. Gigante. We prove a Plancherel formula, an inversion formula and a weak uncertainty principle for it. Then, we show that the portion of the continuous Dunkl Gabor transform lying outside some set of finite measure cannot be arbitrarily too small. Similarly, using the basic theory for the Dunkl continuous wavelet transform introduced by K. Trimèche in [18], an analogous of this result for the Dunkl continuous wavelet transform is given. Finally, an analogous of Heisenberg’s inequality for a continuous Dunkl Gabor transform (resp. Dunkl continuous wavelet transform) is proved.      

The Hardy-Littlewood Maximal Function and Weighted Lorentz Spaces

We study the boundedness of the Hardy–Littlewood maximalfunction in weighted Lorentz spaces, using some new techniqueson distribution and rearrangement functions. Our goal is togive a unified version of some well-known weighted inequalitiesin Rⁿ and R⁺, showing the relation between the theories.     

Characterization of Function Spaces for the Dunkl Type Operator on the Real Line

In this work we consider the Dunkl operator on the real line, defined by $$ {\cal D}_kf(x):=f'(x)+k\dfrac{f(x)-f(-x)}{x},\,\,k\geq0. $$ We define and study Dunkl–Sobolev spaces \(L^p_{n,k}(\mathbb{R})\) , Dunkl–Sobolev spaces \({\cal L}^p_{\alpha,k}(\mathbb{R})\) of positive fractional order and generalized Dunkl–Lipschitz spaces \(\wedge^k_{\alpha,p,q}(\mathbb{R})\) . We provide characterizations of these spaces and we give some connection between them.     

两个解析函数空间之间的加权复合算子

本文刻画了空间F(p,q,s)到空间β~α之间加权复合算子T_((?),ψ),的有界性和紧性条件并得到了一系列相关结果．     

Linear and Bilinear Multiplier Operators for the Dunkl Transform

The main purpose of this article is to study the L _p -boundedness of linear and bilinear multiplier operators for the Dunkl transform in the one dimensional case.     

Weighted Dunkl transform inequalities and application on radial Besov spaces

In Dunkl theory on $\mathbb R ^d$ which generalizes classical Fourier analysis, we prove first weighted inequalities for certain Hardy-type averaging operators. In particular, we deduce for specific choices of the weights the $d$ -dimensional Hardy inequalities whose constants are sharp and independent of $d$ . Second, we use the weight characterization of the Hardy operator to prove weighted Dunkl transform inequalities. As consequence, we obtain Pitt’s inequality which gives an integrability theorem for this transform on radial Besov spaces.     

加权Hardy空间与加权L^p空间之间的算子内插

利用Calderon-Zygmund分解,给出了当w∈A1时加权Hardy空间与加权L^p空间之间的算子内插。     

Weighted Tent Spaces

We generalize the theory of tent spaces introduced in [9] and [10], to consider weighted norms related to some function parameters (see [11]). We study their atomic decomposition, from which we obtain a weighted inequality for a certain fractional maximal operator. We also find the dual spaces, and get a new class of CARLESON measures and we identify the intermediate spaces when using several methods of interpolation.     

Characterization of the Weak-Type Boundedness of the Hilbert Transform on Weighted Lorentz Spaces

We characterize the weak-type boundedness of the Hilbert transform H on weighted Lorentz spaces $\varLambda^{p}_{u}(w)$ , with p>0, in terms of some geometric conditions on the weights u and w and the weak-type boundedness of the Hardy–Littlewood maximal operator on the same spaces. Our results recover simultaneously the theory of the boundedness of H on weighted Lebesgue spaces L ^p (u) and Muckenhoupt weights A _p , and the theory on classical Lorentz spaces Λ ^p (w) and Ariño-Muckenhoupt weights B _p .     

Harmonic Analysis Associated with the One-Dimensional Dunkl Transform

This paper presents a systematic study for harmonic analysis associated with the one-dimensional Dunkl transform, which is based upon the generalized Cauchy–Riemann equations D _x u?? _y v=0,? _y u+D _x v=0, where D _x is the Dunkl operator (D _x f)(x)=f′(x)+(λ/x)(f(x)?f(?x)). Various properties about the λ-subharmonic function, the λ-Poisson integral, the conjugate λ-Poisson integral, and the associated maximal functions are obtained, and the λ-Hilbert transform , a crucial analog to the classical one, is introduced and studied by a stringent method. The theory of the associated Hardy spaces $H_{\lambda}^{p}({\mathbb{R}}^{2}_{+})$ on the half-plane ${\mathbb{R}}^{2}_{+}$ for p≥p ₀=2λ/(2λ+1) with λ>0 extends the results of Muckenhoupt and Stein about the Hankel transform to a general case and contains a number of further results. In particular, the λ-Hilbert transform is shown to be a bounded mapping from $H_{\lambda}^{1}({\mathbb{R}})$ to $L^{1}_{\lambda}({\mathbb{R}})$ ; and associated to the Dunkl transform, an analog of the well-known Hardy inequality is proved for $f\in H^{1}_{\lambda}({\mathbb{R}})$ .     

加权Lebesgue空间的对偶和嵌入

本文建立了加权Lebesgue空间的对偶空间,并且给出了在这些空间之间嵌入的必要条件。     

Weighted Composition Operators on Weighted Dirichlet Spaces

We characterize the boundedness and compactness of weighted composition operators on weighted Dirichlet spaces in terms of Nevanlinna counting functions and Carleson measure.     

Dunkl Translations,Dunkl-Type BMO Space,and Riesz Transforms for the Dunkl Transform on $$L^\infty$$

Functional Analysis and Its Applications - In this paper we will give some results on the support of Dunkl translations on compactly supported functions. Then we will define the Dunkl-type BMO...     

Weighted Coorbit Spaces and Banach Frames on Homogeneous Spaces

This article is concerned with frame constructions on domains and manifolds. The starting point is a unitary group representation which is square integrable modulo a suitable subgroup and therefore gives rise to a generalized continuous wavelet transform. Then generalized coorbit spaces can be defined by collecting all functions for which this wavelet transform is contained in a weighted L_p-space. Moreover, we show that a judicious discretization of the representation leads to an atomic decomposition and to Banach frames for these coorbit spaces.     

伴随BMO函数的交换子在Herz型Hardy空间上的加权有界性

记 [b ,T]为由BMO函数b与广义Calder幃ｎ Zygmund算子T生成的交换子 .研究了 [b ,T]在Herz型Hardy空间上的有界性及其加权有界性 .     

The Cauchy Transform of Continuous Linear Functionals and Projections on the Weighted Spaces of Analytic Functions

Under some general assumptions on weight, we give a complete characterization of the Cauchy transform of continuous linear functionals on the weighted spaces of holomorphic functions in a ball. We construct an integral projection that sends the weighted spaces of measurable and n-harmonic functions in the ball onto the corresponding spaces of holomorphic functions.     

Volterra Type and Weighted Composition Operators on Weighted Fock Spaces

Bounded and compact product of Volterra type integral and composition operators acting between weighted Fock spaces are described. We also estimate the norms of these operators in terms of Berezin type integral transforms on the complex plan ${\mathbb{C}}$ . All our results are valid for weighted composition operators acting on the class of Fock spaces considered under appropriate interpretation of the weights.     

Dual Spaces of Weighted Spaces of Analytic Functions

Siberian Mathematical Journal -