Multi-linear operators given by singular multipliers

Notes on lattice bump Fourier multiplier operators on $L^2 \times L^2$

Given a smooth bump function, we consider the multiplier formed by taking the linear combination of the translations of the bump function and the corresponding bilinear Fourier multiplier operator. Under certain condition on the bump function, we give a complete characterization of the coefficients of the linear combination for which the corresponding bilinear operator defines a bounded operator from $L^2\times L^2$ to $L^2$-based amalgam spaces.

On Conic Fourier Multipliers

We prove a weighted inequality which controls conic Fourier multiplier operators in terms of lacunary directional maximal operators. By bounding the maximal operators, this enables us to conclude that the multiplier operators are bounded on $L^p(\mathbb{R}^3)$ with $1<p<\infty$.

Hankel Multipliers And Transplantation Operators

Connections between Hankel transforms of different order for $L^p$-functions are examined. Well known are the results of Guy [Guy] and Schindler [Sch]. Further relations result from projection formulae for Bessel functions of different order. Consequences for Hankel multipliers are exhibited and implications for radial Fourier multipliers on Euclidean spaces of different dimensions indicated.

The ε-Maximal Operator and Haar Multipliers on Variable Lebesgue Spaces

C. Stockdale, P. Villarroya, and B. Wick introduced the $ε$-maximal operator to prove the Haar multiplier is bounded on the weighted spaces $L^p(w)$ for a class of weights larger than $A_p$. We prove the $ε$-maximal operator and Haar multiplier are bounded on variable Lebesgue spaces $\Lpp(\R^n)$ for a larger collection of exponent functions than the log-Holder continuous functions used to prove the boundedness of the maximal operator on $\Lpp(\R^n)$. We also prove that the Haar multiplier is compact when restricted to a dyadic cube $Q_0$.

由具有分数维膨胀的 Fourier 乘子给定的极大算子

Maximal operators given by Fourier multipliers with dilation of fractional dimensions

本文研究了分数维膨胀下极大 Fourier 乘子算子的 $L^p$ 界，提出了基于膨胀集维数的准则，确保对任意 $p$ 均有 $L^p$ 界，并涵盖 Mikhlin 型、有限衰减与慢衰减乘子。

Two weight inequalities for individual Haar multipliers and other well localized operators

In this paper we are proving that Sawyer type condition for boundedness work for the two weight estimates of individual Haar multipliers, as well as for the Haar shift and other "well localized" operators.

双线性粗糙奇异积分算子在临界可积性附近的界

Bilinear Rough Singular Integrals near the Critical Integrability via Sharp Fourier Multiplier Criteria

本文研究了双线性粗糙奇异积分算子在临界可积性附近的有界性问题，提出了一种新的傅里叶乘子算子有界性准则，改进了已有条件并接近线性理论推测的端点估计。

Multipliers and imaginary powers of the schrödinger operators characterizing UMD Banach spaces

In this paper we establish $L^p$-boundedness properties for Laplace type transform spectral multipliers associated with the Schrödinger operator $\mathcal{L}=-Δ+V$. We obtain for this type of multipliers pointwise representation as principal value integral operators. We also characterize the UMD Banach spaces in terms of the $L^p$-boundedness of the imaginary powers $\mathcal{L}^{iγ}$, $γ\in \mathbb{R}$, of $\mathcal{L}$.

Maximal operators associated with Fourier multipliers and applications

In this paper, we introduce a criterion for maximal operators associated with Fourier multipliers to be bounded on $L^p(\mathbb{R}^d)$. Noteworthy examples satisfying the criterion are multipliers of the Mikhlin type or limited decay which are not necessarily radial. To do so, we make use of modified square function estimates and bilinear interpolation. In result, we obtain convergence results for fractional half-wave equations and surface averages as well as the $L^p$ boundedness for the maximal operators.

Square functions and maximal operators associated with radial Fourier multipliers

We begin with an overview on square functions for spherical and Bochner-Riesz means which were introduced by Eli Stein, and discuss their implications for radial multipliers and associated maximal functions. We then prove new endpoint estimates for these square functions, for the maximal Bochner-Riesz operator, and for more general classes of radial Fourier multipliers.

Strong converse inequalities for the weighted multivariate Bernstein-Durrmeyer operator on the simplex via multipliers

It is demonstrated that multiplier methods naturally yield better constants in strong converse inequalities for the Bernstein-Durrmeyer operator. The absolute constants obtained in some of the inequalities are independent of the weight and the dimension. The estimates are stated in terms of the $K$-functional that is naturally associated to the operator.

Boundedness of some multi-parameter fiber-wise multiplier operators

We prove $L^p$ estimates for various multi-parameter bi- and trilinear operators with symbols acting on fibers of the two-dimensional functions. In particular, this yields estimates for the general bi-parameter form of the twisted paraproduct studied in arXiv:1011.6140.

$L^2\times L^2\times L^2\to L^{2/3}$ boundedness for trilinear multiplier operator

This paper discusses the boundedness of the trilinear multiplier operator $T_{m}(f_1,f_2,f_3)$, when the multiplier satisfies a certain degree of smoothness but with no decaying condition and is $L^{q}$-integrable with an admissible range of $q$. The boundedness is stated in the terms of $\|m\|_{L^{q}}$. In particular, \begin{equation*}\|T_{m}\|_{L^2\times L^2\times L^2\to L^{2/3}}\lesssim\|m\|_{L^{q}}^{q/3}.\end{equation*}

A characterization of maximal operators associated with radial Fourier multipliers

We give a simple necessary and sufficient condition for maximal operators associated with radial Fourier multipliers to be bounded on $L^p_{rad}$ and $L^p$ for certain $p$ greater than $2$. The range of exponents obtained for the $L^p_{rad}$ characterization is optimal for the given condition. The $L^p$ characterization is derived from an inequality of Heo, Nazarov, and Seeger regarding a characterization of radial Fourier multipliers.

Generalised differences and a class of multiplier operators in Fourier analysis

The ranges of a certain type of second order differential operator, on a Sobolev subspace of the Lebesgue space $L^2$ of the circle group, can be characterised by the vanishing of the Fourier coefficients at (generally) two integers that are the zeros of the multiplier of the operator. It is proved here that the range of any such operator may be alternatively described as comprising those functions in $L^2$ that are the sum of five generalised second order differences, each such difference involving the zeros of the multiplier. In fact, higher order operators and differences are considered. Th…

Maximal operators associated with bilinear multipliers of limited decay

Results analogous to those proved by Rubio de Francia are obtained for a class of maximal functions formed by dilations of bilinear multiplier operators of limited decay. We focus our attention to $L^2\times L^2\to L^1$ estimates. We discuss two applications: the boundedness of the bilinear maximal Bochner-Riesz operator and of the bilinear spherical maximal operator. For the latter we improve the known results by reducing the dimension restriction from $n\ge 8$ to $n\ge 4$.

Notes on bilinear lattice bump Fourier multipliers

We consider the bilinear Fourier multiplier operator with the multiplier written as a linear combination of a fixed bump function. For those operators we prove two transference theorems, one in amalgam spaces and the other in Wiener amalgam spaces.

通过平方函数估计最大傅里叶乘子算子在R²上的估计

Estimates for maximal Fourier multiplier operators on $\Bbb R^2$ via square functions

本文通过平方函数估计，推导出Bochner-Riesz型傅里叶乘子在R²上的最大函数L^p有界性，扩展了Carbery 1983年的结果。

An optimal multiplier theorem for Grushin operators in the plane, II

In a previous work we proved a spectral multiplier theorem of Mihlin--Hörmander type for two-dimensional Grushin operators $-\partial_x^2 - V(x) \partial_y^2$, where $V$ is a doubling single-well potential, yielding the surprising result that the optimal smoothness requirement on the multiplier is independent of $V$. Here we refine this result, by replacing the $L^\infty$ Sobolev condition on the multiplier with a sharper $L^2$ condition. As a consequence, we obtain the sharp range of $L^1$ boundedness for the associated Bochner--Riesz means. The key new ingredient of the proof is a precise po…

Iterated trilinear Fourier integrals with arbitrary symbols

We prove $L^p$ estimates for trilinear multiplier operators with singular symbols. These operators arise in the study of iterated trilinear Fourier integrals, which are trilinear variants of the bilinear Hilbert transform. Specifically, we consider trilinear operators determined by multipliers that are products of two functions ${m}_1(ξ_1, ξ_2)$ and ${m}_2(ξ_2, ξ_3)$, such that the singular set of $m_1$ lies in the hyperplane $ ξ_1=ξ_2$ and that of $m_2$ lies in the hyperplane $ξ_2=ξ_3$. While previous work \cite{MTT2} requires that the multipliers satisfy $χ_{ξ_1 <ξ_2} \cdot χ_{ξ_2<ξ_3}$, our…

Fourier-like multipliers and applications for integral operators

Timelimited functions and bandlimited functions play a fundamental role in signal and image processing. But by the uncertainty principles, a signal cannot be simultaneously time and bandlimited. A natural assumption is thus that a signal is almost time and almost bandlimited. The aim of this paper is to prove that the set of almost time and almost bandlimited signals is not excluded from the uncertainty principles. The transforms under consideration are integral operators with bounded kernels for which there is a Parseval Theorem. Then we define the wavelet multipliers for this class of operat…

多参数多线性Fourier乘子算子的一个精确Hörmander估计

A sharp Hörmander estimate for multi-parameter and multi-linear Fourier multiplier operators

本文研究多线性多参数Fourier乘子的Hörmander型定理，给出了基于L^u的Sobolev范数刻画下的光滑性假设结果，在多参数双线性情形下精确、多线性情形下几乎精确，且改进了单参数多线性情形的已有结论。

Weighted norm inequalities for multilinear operators and applications to multilinear Fourier multipliers

Let $T$ be a multilinear operator which is bounded on certain products of unweighted Lebesgue spaces of $\mathbb R^n$. We assume that the associated kernel of $T$ satisfies some mild regularity condition which is weaker than the usual Hölder continuity of those in the class of multilinear Calderón-Zygmund singular integral operators. We then show the boundedness for $T$ and the boundedness of the commutator of $T$ with BMO functions on products of weighted Lebesgue spaces of $\mathbb R^n$. As an application, we obtain the weighted norm inequalities of multilinear Fourier multipliers and of the…

Littlewood-Paley equivalence and homogeneous Fourier multipliers

We consider certain Littlewood-Paley operators and prove characterization of some function spaces in terms of those operators. When treating weighted Lebesgue spaces, a generalization to weighted spaces will be made for Hörmander's theorem on the invertibility of homogeneous Fourier multipliers. Also, applications to the theory of Sobolev spaces will be given.

Limited smoothness conditions with mixed norms for bilinear Fourier multipliers

In this paper, the $L^2 \times L^{\infty} \to L^2$ and $L^2 \times L^2 \to L^1$ boundedness of bilinear Fourier multiplier operators is discussed under weak smoothness conditions on multipliers. As an application, we prove the $L^2 \times BMO \to L^2$ and $L^2 \times L^2 \to H^1$ boundedness of bilinear operators with multipliers of limited smoothness satisfying vanishing conditions.

Estimates for Schur Multipliers and Double Operator Integrals -- A Wavelet Approach

We discuss the work of Birman and Solomyak on the singular numbers of integral operators from the point of view of modern approximation theory, in particular with the use of wavelet techniques. We are able to provide a simple proof of norm estimates for integral operators with kernel in $B^{\frac{1}{p}-\frac{1}{2}}_{p,p}(\mathbb R,L_2(\mathbb R))$. This recovers, extends and sheds new light on a theorem of Birman and Solomyak. We also use these techniques to provide a simple proof of Schur multiplier bounds for double operator integrals, with bounded symbol in $B^{\frac{1}{p}-\frac{1}{2}}_{\fr…

Quantitative weighted estimates for the Littlewood-Paley square function and Marcinkiewicz multipliers

Quantitative weighted estimates are obtained for the Littlewood-Paley square function $S$ associated with a lacunary decomposition of ${\mathbb R}$ and for the Marcinkiewicz multiplier operator. In particular, we find the sharp dependence on $[w]_{A_p}$ for the $L^p(w)$ operator norm of $S$ for $1<p\le 2$.

多线性乘法算子和伪微分算子加权估计的必要条件

Necessary conditions for weighted estimates of Multilinear Multipliers and Pseudo-Differential Operators

本文研究了多线性傅里叶乘法算子和伪微分算子加权估计的必要条件，证明了多重权重假设的最佳性，并改进了经典线性情况下的结果。

Generalised differences and multiplier operators in $L^2({\mathbb R})$

Given two real numbers, the $L^2$ functions whose Fourier transforms vanish with a certain rapidity near the given numbers are characterised as those that are expressible as the sum of a certain number of generalised finite differences that is independent of the function. These generalised differences can be regarded as approximating the appropriate powers of first order ordinary differential operators. The upshot of this is that for operators in a certain class of ordinary differential operators that have polynomial multipliers, their ranges on the Sobolev spaces corresponding to the operator…

Weighted Estimates for Operator-Valued Fourier Multipliers

We establish Littlewood-Paley decompositions for Muckenhoupt weights in the setting of UMD spaces. As a consequence we obtain two-weight variants of the Mikhlin multiplier theorem for operator-valued multipliers. We also show two-weight estimates for multipliers satisfying Hörmander type conditions.

Heisenberg-Pauli-Weyl and Donoho-Stark's uncertainty principle for the Weinstein $L^2$-multiplier operators

The aim of this paper is establish the Heisenberg-Pauli-Weyl uncertainty principle and Donho-Stark's uncertainty principle for the Weinstein $L^2$-multiplier operators.

Fourier-Wigner乘子与Schatten类算子的Bochner-Riesz猜想

Fourier-Wigner multipliers and the Bochner-Riesz conjecture for Schatten class operators

本文引入Schatten类算子上的Fourier-Wigner乘子，利用秩一算子卷积和Werner-Young不等式建立相关结果，并证明紧支撑Fourier乘子的等价关系，从而用Schatten类算子重新表述Bochner-Riesz猜想。

Maximal multiplier operators in $L^{p(\cdot)}(\mathbb{R}^{n})$ spaces

In this paper we study some estimates of norms in variable exponent Lebesgue spaces for maximal multiplier operators.We will consider the case when multiplier is the Fourier transform of a compactly supported Borel measure

Spectral multipliers for Schroedinger operators with Poeschl-Teller potential

We prove a sharp Mihlin-Hormander multiplier theorem for Schroedinger operators $H$ on $\R^n$. The method, which allows us to deal with general potentials, improves Hebisch's method relying on heat kernel estimates for positive potentials. Our result applies to, in particular, the negative Poeschl-Teller potential $V(x)= -ν(ν+1) \sech^2 x $, $ν\in \N$, for which $H$ has a resonance at zero.

Hermite multiplier sequences and their associated operators

We provide an explicit formula for the coefficient polynomials of a Hermite diagonal differential operator. The analysis of the zeros of these coefficient polynomials yields the characterization of generalized Hermite multiplier sequences which arise as Taylor coefficients of real entire functions with finitely many zeros. We extend our result to functions in $\mathcal{L-P}$ with infinitely many zeros, under additional hypotheses.

多圆盘中的对数 Bloch 空间、Hankel 算子的端点结果与逐点乘子

Logarithmic Bloch spaces in the polydisc, endpoint results for Hankel operators and pointwise multipliers

本文研究单位多圆盘中的两类对数 Bloch 空间，利用有界 Hankel 算子的符号给出等价刻画，并完整描述不同 Bloch 空间之间的逐点乘子。

谱乘子III：端点界、缠结算子与扭曲Hardy空间

Spectral multipliers III: Endpoint bounds, intertwining operators, and twisted Hardy spaces

该研究将自由拉普拉斯谱乘子的基本估计推广到带实标量势的受扰哈密顿量，得到Mihlin乘子精确界等结果，部分证实相关猜想并刻画扭曲Hardy空间。

Sharp Lower bound estimates for vector-valued and matrix-valued multipliers in $L^p$

We generalize the idea of a multiplier in two different ways and generalize a recent result of Geiss, Montomery-Smith and Saksman. First of all, we consider multipliers in the form of a vector acting on a scalar function. Using this technique we compute the sharp lower bound estimate for $L^p$ operator norm of a quadratic perturbation of the real part of the Ahlfors-Beurling operator. Secondly, we consider matrix-valued multipliers to obtain a new proof showing that the $L^p$ operator norm of the Ahlfors-Beurling operator is bounded below by p^*-1.

与Grushin算子相关的伪乘子的稀疏界，II

Sparse bounds for pseudo-multipliers associated to Grushin operators, II

本文作为前期研究的延续，建立了对应多种符号类的Grushin伪乘子的逐点稀疏控制结果，并由此推导出这些伪乘子的定量加权估计。

Spectral multipliers for Schrödinger operators

We prove a sharp Hörmander multiplier theorem for Schrödinger operators $H=-Δ+V$ on $\mathbb{R}^n$. The result is obtained under certain condition on a weighted $L^\infty$ estimate, coupled with a weighted $L^2$ estimate for $H$, which is a weaker condition than that for nonnegative operators via the heat kernel approach. Our approach is elaborated in one dimension with potential $V$ belonging to certain critical weighted $L^1$ class. Namely, we assume that $\int (1+|x|) |V(x)|dx$ is finite and $H$ has no resonance at zero. In the resonance case we assume $\int (1+|x|^2) |V(x)| dx$ is finite.

与Grushin算子相关的伪乘子的稀疏界，I

Sparse bounds for pseudo-multipliers associated to Grushin operators, I

本文利用稀疏算子的逐点控制方法，证明了满足Hörmander条件的Grushin伪乘子的尖锐定量加权$L^p$估计。

Fock-Sobolev空间上的Fourier乘子与伪微分算子

Fourier multipliers and pseudo-differential operators on Fock-Sobolev spaces

本文研究了$L^2(\mathbb{R}^n)$上有界算子经Bargmann变换导出的Fock空间算子，确定了对应Fourier乘子与伪微分算子的算子，并分析了其在Fock-Sobolev空间上的有界性。

极大次椭圆算子的谱乘子

Spectral multipliers for maximally subelliptic operators

本文研究紧流形上非负自伴极大次椭圆算子的谱乘子，证明 Mihlin–Hörmander 型条件下其为奇异积分算子，并建立相关非各向同性函数空间等价及 $L^p$ Sobolev 有界性条件。

关于与Grushin算子相关的一些算子值Fourier伪乘子

On some operator-valued Fourier pseudo-multipliers associated to Grushin operators

本文延续先前对Grushin伪乘子稀疏控制和定量加权估计的研究，进一步将分析扩展到与Grushin算子相关的一族算子值Fourier伪乘子的类似估计。

Damek-Ricci空间上区分拉普拉斯算子的里斯变换及算子值多变量谱乘子

Riesz transforms for the distinguished Laplacian on Damek-Ricci spaces and operator-valued multivariate spectral multipliers

本文研究了Damek-Ricci空间上区分拉普拉斯算子的里斯变换的L^p有界性，并通过算子值谱乘子定理证明了其在p∈(1,∞)范围内的有界性。

On singular integral and martingale transforms

Linear equivalences of norms of vector-valued singular integral operators and vector-valued martingale transforms are studied. In particular, it is shown that the UMD(p)-constant of a Banach space X equals the norm of the real (or the imaginary) part of the Beurling-Ahlfors singular integral operator, acting on the X-valued L^p-space on the plane. Moreover, replacing equality by a linear equivalence, this is found to be the typical property of even multipliers. A corresponding result for odd multipliers and the Hilbert transform is given.

Approximation by multivariate quasi-projection operators and Fourier multipliers

Multivariate quasi-projection operators $Q_j(f,φ, \widetildeφ)$, associated with a function $φ$ and a distribution/function $\widetildeφ$, are considered. The function $φ$ is supposed to satisfy the Strang-Fix conditions and a compatibility condition with $\widetildeφ$. Using technique based on the Fourier multipliers, we studied approximation properties of such operators for functions $f$ from anisotropic Besov spaces and $L_p$ spaces with $1\le p\le \infty$. In particular, upper and lower estimates of the $L_p$-error of approximation in terms of moduli of smoothness and best approximations a…

General Hormander and Mikhlin conditions for multipliers of Besov spaces

Here a new condition for the geometry of Banach spaces is introduced and the operator--valued Fourier multiplier theorems in weighted Besov spaces are obtained. Particularly, connections between the geometry of Banach spaces and Hormander-Mikhlin conditions are established. As an application of main results the regularity properties of degenerate elliptic differential operator equations are investigated.

加权谱簇界与超球Grushin算子的尖锐乘子定理

Weighted spectral cluster bounds and a sharp multiplier theorem for ultraspherical Grushin operators

本文研究d维球面上在k维球面处奇异的Grushin型退化椭圆算子，通过建立加权谱簇界和超球多项式精确估计，证明了在2k≤d时最优的Mihlin-Hörmander型谱乘子定理及Bochner-Riesz可求和性结果。

Fourier multipliers for nonlocal Laplace operators

Fourier multiplier analysis is developed for nonlocal peridynamic-type Laplace operators, which are defined for scalar fields in $\mathbb{R}^n$. The Fourier multipliers are given through an integral representation. We show that the integral representation of the Fourier multipliers is recognized explicitly through a unified and general formula in terms of the hypergeometric function $_2F_3$ in any spatial dimension $n$. Asymptotic analysis of $_2F_3$ is utilized to identify the asymptotic behavior of the Fourier multipliers $m(ν)$ as $\|ν\|\rightarrow \infty$. We show that the multipliers are …

舒尔乘子在调和分析与算子代数中的影响

The impact of Schur multipliers in harmonic analysis and operator algebras

本文综述舒尔乘子在过去15年的影响，关注其与傅里叶乘子的联系及在调和分析和算子代数间的桥梁作用，介绍舒尔滕p类最新界在相关领域的应用，以及非托普利茨舒尔乘子等方面的新进展。

A robust approach to sharp multiplier theorems for Grushin operators

We prove a multiplier theorem of Mihlin-Hörmander type for operators of the form $-Δ_x - V(x) Δ_y$ on $\mathbb{R}^{d_1}_x \times \mathbb{R}^{d_2}_y$, where $V(x) = \sum_{j=1}^{d_1} V_j(x_j)$, the $V_j$ are perturbations of the power law $t \mapsto |t|^{2σ}$, and $σ\in (1/2,\infty)$. The result is sharp whenever $d_1 \geq σd_2$. The main novelty of the result resides in its robustness: this appears to be the first sharp multiplier theorem for nonelliptic subelliptic operators allowing for step higher than two and perturbation of the coefficients. The proof hinges on precise estimates for eigenv…