Some Fourier-type operators for functions on unbounded intervals

In order to approximate functions defined on the real line or on the real semiaxis by polynomials, we introduce some new Fourier-type operators, connected to the Fourier sums of generalized Freud or Laguerre orthonormal systems. We prove necessary and sufficient conditions for the boundedness of these operators in suitable weighted L ^p -spaces, with 1 < p < ∞. Moreover, we give error estimates in weighted L ^p and uniform norms.     

Transferring Boundedness from Conjugate Operators Associated with Jacobi, Laguerre, and Fourier–Bessel Expansions to Conjugate Operators in the Hankel Setting

In this paper we establish transference results showing that the boundedness of the conjugate operator associated with Hankel transforms on Lorentz spaces can be deduced from the corresponding boundedness of the conjugate operators defined on Laguerre, Jacobi, and Fourier–Bessel settings. Our result also allows us to characterize the power weights in order that conjugation associated with Laguerre, Jacobi, and Fourier–Bessel expansions define bounded operators between the corresponding weighted L ^p spaces. This paper is partially supported by MTM2004/05878. Third and fourth authors are also partially supported by grant PI042004/067.     

On <Emphasis Type="Italic">g</Emphasis>-functions for Laguerre function expansions of Hermite type

We examine weighted L ^p boundedness of g-functions based on semigroups related to multi-dimensional Laguerre function expansions of Hermite type. A technique of vector-valued Calderón–Zygmund operators is used.     

Area Littlewood–Paley Functions Associated with Hermite and Laguerre Operators

In this paper we study L ^p —boundedness properties for area Littlewood–Paley functions associated with heat semigroups for Hermite and Laguerre operators.     

Toeplitz and Hankel type operators on the upper half-plane

An orthogonal decomposition of admissible wavelets is constructed via the Laguerre polynomials, it turns to give a complete decomposition of the space of square integrable functions on the upper half-plane with the measurey dxdy. The first subspace is just the weighted Bergman (or Dzhrbashyan) space. Three types of Ha-plitz operators are defined, they are the generalization of classical Toeplitz, small and big Hankel operators respectively. Their boundedness, compactness and Schatten-von Neumann properties are studied.Research was supported by the National Natural Science Foundation of China.     

Weak-Type Inequality for Conjugate First Order Riesz-Laguerre Transforms

In this paper we introduce a conjugate class of Riesz transforms in the context of Laguerre polynomials. We prove their weak-type (1,1) and L ^p , 1<p<∞, boundedness with respect to the Laguerre measure. A similar result is known in the Hermite context, see Aimar et al. (Trans. Am. Math. Soc. 359(5), 2137–2154, 2007).     

Boundedness of Convolution Operators and Input–Output Maps between Weighted Spaces

We ask when convolution operators with scalar- or operator-valued kernel functions map between weighted L² spaces of Hilbert space-valued functions. For a certain class of decreasing weights, including negative powers (t + a)^−m for example, we solve the one-weight problem completely by using Laplace transforms and Bergman-type spaces of vector-valued analytic functions. For a much more general class of decreasing weights, we solve the one-weight problem for all positive real kernels (also for L^p(w) with p > 1), by results on Steklov operators which generalise the weighted Hardy inequality. When the kernel function is a strongly continuous semigroup of bounded linear Hilbert space operators, which arises from input–output maps of certain linear systems, then the most obvious sufficient condition for boundedness, obtained by taking norm signs inside the integrals, is also necessary in many cases, but not in general. Submitted: July 15, 2007.,Revised: November 19, 2007.,Accepted: December 14, 2007.     

On a convolution property characterizing the Laguerre functions

Consider the Laguerre functions (with parameterp>0), where theL _n are the Laguerre polynomials with parameter =0.{l _n ^p (t)} _n=0 forms a complete orthonormal system inL ² ([0, )). A well known and often used property of the Laguerre functions is the convolution property: for alli,j0. It is the objectiveof this note that the system of Laguerre functions is the only complete and orthonormal system ofL ² ([0, )) satisfying the convolution property.     

<Emphasis Type="Italic">L</Emphasis><Superscript><Emphasis Type="Italic">p</Emphasis></Superscript>-convergence of Lagrange interpolation on the semiaxis

In order to approximate functions defined on (0, +∞), the authors consider suitable Lagrange polynomials and show their convergence in weighted L ^p -spaces.      

A New Characterization for Carleson Measures and Some Applications

We provide a new characterization for Carleson measures in terms of the L ^p behaviors of certain functions represented as an integration on a non-tangential cone. Applications for characterizing the boundedness and compactness of Volterra type operators from Hardy spaces to some holomorphic spaces are also presented.     

Relativistic Lamé functions: Completeness vs. polynomial asymptotics

In earlier work we introduced and studied two commuting generalized Lamé operators, obtaining in particular joint eigenfunctions for a dense set in the natural parameter space. Here we consider these difference operators and their eigenfunctions in relation to the Hilbert space L²((0, π/r), w(x)dx), with r > 0 and the weight function w(x) a ratio of elliptic gamma functions. In particular, we show that the previously known pairwise orthogonal joint eigenfunctions need only be supplemented by finitely many new ones to obtain an orthogonal base. This completeness property is derived by exploiting recent results on the large-degree Hilbert space asymptotics of a class of orthonormal polynomials. The polynomials p_n(cos(rx)), n ε , that are relevant in the Lamé setting are orthonormal in L²((0, π/r), w_P(x)dx), with w_p(x) closely related to w(x).     

Transplanting maximal inequalities between Laguerre and Hankel multipliers

We supplement our previous paper [9] by adding a theorem that transplantsL ^p -norm maximal inequalities for Laguerre multipliers. As an immediate consequence we obtain negative results concerningL ^p -estimates of partial sum maximal operators for Laguerre expansions.Research supported in part by KBN grant No. 2 PO3A 030 09.     

Sharp continuity results for the short-time Fourier transform and for localization operators

We completely characterize the boundedness on Wiener amalgam spaces of the short-time Fourier transform (STFT), and on both L ^p and Wiener amalgam spaces of a special class of pseudodifferential operators, called localization operators. Precisely, sufficient conditions for the STFT to be bounded on the Wiener amalgam spaces W(L ^p , L ^q ) are given and their sharpness is shown. Localization operators are treated similarly: using different techniques from those employed in the literature, we relax the known sufficient boundedness conditions for these operators to be bounded on L ^p spaces and prove the optimality of our results. Next, we exhibit sufficient and necessary conditions for such operators to be bounded on Wiener amalgam spaces.     

Weighted norm inequalities, off-diagonal estimates and elliptic operators. Part III: Harmonic analysis of elliptic operators

This is the third part of a series of four articles on weighted norm inequalities, off-diagonal estimates and elliptic operators. For L in some class of elliptic operators, we study weighted norm Lp inequalities for singular “non-integral” operators arising from L; those are the operators φ(L) for bounded holomorphic functions φ, the Riesz transforms ∇L−1/2 (or (−Δ)^1/2L−1/2) and its inverse L1/2(−Δ)^−1/2, some quadratic functionals gL and GL of Littlewood-Paley-Stein type and also some vector-valued inequalities such as the ones involved for maximal Lp-regularity. For each, we obtain sharp or nearly sharp ranges of p using the general theory for boundedness of Part I and the off-diagonal estimates of Part II. We also obtain commutator results with BMO functions.     

Wavelets and Bidemocratic Pairs in Weighted Norm Spaces

A complete characterization of weight functions for which the higher-rank Haar wavelets are greedy bases in weighted L^p spaces is given. The proof uses the new concept of a bidemocratic pair for a Banach space and also pairs (Φ, Φ), where Φ is an orthonormal system of bounded functions in the spaces L^p, p≠2.     

A note on commutators of Bochner-Riesz operator

In terms of continuous decomposition and choosing an appropriate BMO function, the authors obtain a sharp necessary condition for L ^p boundedness of the commutators generated by Bochner-Riesz operators below the critical index and BMO functions.      

带变量核的Marcinkiewicz算子交换子在加权Herz空间上的有界性

本文研究了带变量核的Marcinkiewicz算子交换子的有界性问题.利用其在Lp(ω)空间上有界的方法,获得了该交换子在加权Herz空间上有界的结果.     

Translation-Invariant Bilinear Operators with Positive Kernels

We study L ^r (or L ^{r, ∞}) boundedness for bilinear translation-invariant operators with nonnegative kernels acting on functions on \mathbb Rⁿ{\mathbb {R}^n}. We prove that if such operators are bounded on some products of Lebesgue spaces, then their kernels must necessarily be integrable functions on \mathbb R²ⁿ{\mathbb R^{2n}}, while via a counterexample we show that the converse statement is not valid. We provide certain necessary and some sufficient conditions on nonnegative kernels yielding boundedness for the corresponding operators on products of Lebesgue spaces. We also prove that, unlike the linear case where boundedness from L ¹ to L ¹ and from L ¹ to L ^{1, ∞} are equivalent properties, boundedness from L ¹ × L ¹ to L ^1/2 and from L ¹ × L ¹ to L ^{1/2, ∞} may not be equivalent properties for bilinear translation-invariant operators with nonnegative kernels.     

一类Schrödinger型算子在关于非负位势的变指数Morrey空间上的有界性

本文考虑了一类Schrödinger型算子和其交换子的有界性问题.利用其在Lp空有界性间上的,获得了一类Schrödinger型算子和其交换子在变指数Morrey空间上的有界性.