Bilinear oscillatory integrals and boundedness for new bilinear multipliers

We consider bilinear oscillatory integrals, i.e. pseudo-product operators whose symbol involves an oscillating factor. Lebesgue space inequalities are established, which give decay as the oscillation becomes stronger; this extends the well-known linear theory of oscillatory integral in some directions. The proof relies on a combination of time-frequency analysis of Coifman-Meyer type with stationary and non-stationary phase estimates. As a consequence of this analysis, we obtain Lebesgue estimates for new bilinear multipliers defined by non-smooth symbols.     

Plancherel-type estimates and sharp spectral multipliers

We study general spectral multiplier theorems for self-adjoint positive definite operators on L²(X,μ), where X is any open subset of a space of homogeneous type. We show that the sharp Hörmander-type spectral multiplier theorems follow from the appropriate estimates of the L² norm of the kernel of spectral multipliers and the Gaussian bounds for the corresponding heat kernel. The sharp Hörmander-type spectral multiplier theorems are motivated and connected with sharp estimates for the critical exponent for the Riesz means summability, which we also study here. We discuss several examples, which include sharp spectral multiplier theorems for a class of scattering operators on R³ and new spectral multiplier theorems for the Laguerre and Hermite expansions.     

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.     

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.     

Extrapolation from A∞ weights and applications

We generalize the A_p extrapolation theorem of Rubio de Francia to A_∞ weights in the context of Muckenhoupt bases. Our result has several important features. First, it can be used to prove weak endpoint inequalities starting from strong-type inequalities, something which is impossible using the classical result. Second, it provides an alternative to the technique of good-λ inequalities for proving L^p norm inequalities relating operators. Third, it yields vector-valued inequalities without having to use the theory of Banach space valued operators. We give a number of applications to maximal functions, singular integrals, potential operators, commutators, multilinear Calderón-Zygmund operators, and multiparameter fractional integrals. In particular, we give new proofs, which completely avoid the good-λ inequalities, of Coifman's inequality relating singular integrals and the maximal operator, of the Fefferman-Stein inequality relating the maximal operator and the sharp maximal operator, and the Muckenhoupt-Wheeden inequality relating the fractional integral operator and the fractional maximal operator.     

Functional Gabor frame multipliers

A Gabor frame multiplier is a bounded operator that maps normalized tight Gabor frame generators to normalized tight Gabor frame generators. While characterization of such operators is still unknown, we give a complete characterization for the functional Gabor frame multipliers. We prove that a L^∞ -function h is a functional Gabor frame multiplier (for the time-frequency lattice aℤ × bℤ) if and only if it is unimodular and is a-periodic. Along the same line, we also characterize all the Gabor frame generators g (resp. frame wavelets ψ) for which there is a function ∈ L^∞(ℝ) such that {wg_mn} (resp. ωψ_k,ℝ) is a normalized tight frame.     

Paraproducts and Hankel operators of Schatten class via p-John-Nirenberg Theorem

We give an interpolation-free proof of the known fact that a dyadic paraproduct is of Schatten-von Neumann class S_p, if and only if its symbol is in the dyadic Besov space B_p^d. Our main tools are a product formula for paraproducts and a “p-John-Nirenberg-Theorem” due to Rochberg and Semmes.We use the same technique to prove a corresponding result for dyadic paraproducts with operator symbols.Using an averaging technique by Petermichl, we retrieve Peller's characterizations of scalar and vector Hankel operators of Schatten-von Neumann class S_p for 1<p<∞. We then employ vector techniques to characterise little Hankel operators of Schatten-von Neumann class, answering a question by Bonami and Peloso.Furthermore, using a bilinear version of our product formula, we obtain characterizations for boundedness, compactness and Schatten class membership of products of dyadic paraproducts.     

广义Bochner-Riesz乘子的等价性及其应用——献给陆善镇教授75华诞

本文研究下面这两个函数,（1-｜x｜^ρ1）＋^α和（1-｜x｜^ρ2）＋^α,其中ρ1,ρ2和α均为正实数,文献上称这类函数为广义Bochner-Riesz乘子.本文将证明,当α给定时,对任意的ρ1〉0和ρ2〉0,这两个函数作为乘子,其乘子算子的L^p有界性和Hp有界性是等价的.     

Heat-diffusion maximal operators for Laguerre semigroups with negative parameters

We prove L^p boundedness for the maximal operator of the heat semigroup associated to the Laguerre functions, , when the parameter α is greater than -1. Namely, the maximal operator is of strong type (p,p) if p>1 and , when -1<α<0. If α?0 there is strong type for 1<p?∞. The behavior at the end points is studied in detail.     

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     

Oscillating multipliers for some eigenfunction expansions

Let P be a non-negative, self-adjoint differential operator of degree d on ℝⁿ. Assume that the associated Bochner-Riesz kernel s _R ^δ satisfies the estimate, |s _R ^δ (x, y)| ≤ C R^n/d(1+R^1/d|x - y|^-αδ+β)for some fixed constants a>0 and β. We study L^p boundedness of operators of the form m(P), m coming from the symbol class S _p ^−α . We prove that m(P) is bounded on L^P if . We also study multipliers associated to the Hermite operator H on ℝⁿ and the special Hermite operator L on ℂⁿ given by the symbols . As a special case we obtain L^p boundedness of solutions to the Wave equation associated to H and L.     

The Weyl transform and bounded operators on Lp(Rn)

A multiplier theorem for the Weyl transform is proved. This theorem is used to derive sufficient conditions for the boundedness of a general operator on L^p($\text{R}$ⁿ). An application to multipliers of the Hermite expansion is given.     

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.     

Generalized dirac operators on nonsmooth manifolds and Maxwell's equations

We develop a function theory associated with Dirac type operators on Lipschitz subdomains of Riemannian manifolds. The main emphasis is on Hardy spaces and boundary value problems, and our aim is to identify the geometric and analytic assumptions guaranteeing the validity of basic results from complex function theory in this general setting. For example, we study Plemelj-Calderón-Seeley-Bojarski type splittings of Cauchy boundary data into traces of ‘inner’ and ‘outer’ monogenics and show that this problem has finite index. We also consider Szegö projections and the corresponding L^p-decompositions. Our approach relies on an extension of the classical Calderón-Zygmund theory of singular integral operators which allow one to consider Cauchy type operators with variable kernels on Lipschitz graphs. In the second part, where we explore connections with Maxwell's equations, the main novelty is the treatment of the corresponding electro-magnetic boundary value problem by recasting it as a ‘half’ Dirichlet problem for a suitable Dirac operator.     

Multiplier Theorems for the Short-Time Fourier Transform

So-called short-time Fourier transform multipliers (also called Anti-Wick operators in the literature) arise by applying a pointwise multiplication operator to the STFT before applying the inverse STFT. Boundedness results are investigated for such operators on modulation spaces and on L _p -spaces. Because the proofs apply naturally to Wiener amalgam spaces the results are formulated in this context. Furthermore, a version of the Hardy-Littlewood inequality for the STFT is derived. This paper was written while the author was researching at University of Vienna (NuHAG) supported by Lise Meitner fellowship No M733-N04. This research was also supported by the Hungarian Scientific Research Funds (OTKA) No K67642.     

Maximal functions for multipliers on stratified groups

In this paper we obtain a refined $L^p$ bound for maximal functions of the multiplier operators on stratified groups and maximal functions of the multi‐parameter multipliers on product spaces of stratified groups. As an application we find a refined $L^p$ bound for maximal functions of joint spectral multipliers on Heisenberg group.     

On operator-multipliers for mixed-norm L^{\bar p} spaces

We use a bootstrapping property of R-boundedness to show that some (known) n-dimensional Fourier multiplier theorems on UMD spaces with property α are immediate corollaries of their one-dimensional versions. The method also gives easy extensions of these results to mixed-norm spaces.Received: 15 September 2004     

L-Fourier multipliers for the Dunkl operator on the real line

We consider Fourier multipliers for L^p associated with the Dunkl operator on and establish a version of Hörmander's multiplier theorem. In applying this version, we come up with some results regarding the oscillating multipliers, partial sum operators and generalized Bessel potentials.     

Vector-valued Littlewood-Paley-Stein theory for semigroups

We develop a generalized Littlewood-Paley theory for semigroups acting on L^p-spaces of functions with values in uniformly convex or smooth Banach spaces. We characterize, in the vector-valued setting, the validity of the one-sided inequalities concerning the generalized Littlewood-Paley-Stein g-function associated with a subordinated Poisson symmetric diffusion semigroup by the martingale cotype and type properties of the underlying Banach space. We show that in the case of the usual Poisson semigroup and the Poisson semigroup subordinated to the Ornstein-Uhlenbeck semigroup on Rⁿ, this general theory becomes more satisfactory (and easier to be handled) in virtue of the theory of vector-valued Calderón-Zygmund singular integral operators.