Feng Dai, Yuan Xu
For a family of weight functions invariant under a finite reflection group, the boundedness of a maximal function on the unit sphere is established and used to prove a multiplier theorem for the orthogonal expansions with respect to the weight function on the unit sphere. Similar results are also established for the weighted space on the unit ball and on the standard simplex.
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We extend an $L^2$ maximal multiplier result of Bourgain to all $L^p$ spaces, $1<p<\infty$.
For r < 2, we prove the boundedness of a maximal operator formed by applying all multipliers m with $\|m\|_{V^r} \leq 1$ to a given function.
We use Oberlin, Nazarov, and Thiele's Multi-Frequency Calderón-Zygmund decomposition to lower estimates on maximal multipliers in $L^p$. We also improve on classical multiplier results of Coifman, Rubio de Francia, and Semmes.
We prove the $L^p$ boundedness of a maximal operator associated with a dyadic frequency decomposition of a Fourier multiplier, under a weak regularity assumption.
Given Mikhlin-Hörmander multipliers $m_i$, $i=1,..., N$, with uniform estimates we prove an optimal $\sqrt{\log(N+1)}$ bound in $L^p$ for the maximal function $\sup_i|\cF^{-1}[m_i\hat f]|$ and related bounds for maximal functions generated by dilations.
We prove a multiplier version of the Bernstein inequality on the complex sphere. Included in this is a new result relating a bivariate sum involving Jacobi polynomials and Gegenbauer polynomials, which relates the sum of reproducing kernels on spaces of polynomials irreducibly invariant under the unitary group, with the reproducing kernel of the sum of these spaces, which is irreducibly invariant under the action of the orthogonal group.
We show that maximal operators formed by dilations of Mikhlin-H"ormander multipliers are typically not bounded on $L^p(R^d)$. We also give rather weak conditions in terms of the decay of such multipliers under which $L^p$ boundedness of the maximal operators holds.
Several estimates for singular integrals, maximal functions and the spherical summation operator are given in the spaces $L^p_{\text{rad}}L^2_{\text{ang}}(\mathbb{R}^n)$, $n\geq 2$.
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 $Q_0$.
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.
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。
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.
Under the assumption that orthogonal polynomials of several variables admit an addition formula, we can define a convolution structure and use it to study the Fourier orthogonal expansions on a homogeneous space. We define a maximal function via the convolution structure induced by the addition formula and use it to establish a Marcinkiewicz multiplier theorem. For the homogeneous space defined by a family of weight functions on conic domains, we show that the maximal function is bounded by the Hardy-Littlewood maximal function so that the multiplier theorem holds on conic domains.
We study the linearized maximal operator associated with dilates of the hyperbolic cross multiplier in dimension two. Assuming a Lipschitz condition and a lower bound on the linearizing function, we obtain $L^{p}(\mathbb{R}^{2}) \to L^{p}(\mathbb{R}^{2})$ bounds for all $1<p <\infty$. We discuss various related results.
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 $n\ge 4$.
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.
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。
We prove endpoint bounds for derivatives of fractional maximal functions with either smooth convolution kernel or lacunary set of radii in dimensions $n \geq 2$. We also show that the spherical fractional maximal function maps $L^{p}$ into a first order Sobolev space in dimensions $n \geq 5$.
We control a broad class of singular (or "rough") Fourier multipliers by geometrically-defined maximal operators via general weighted $L^2(\mathbb{R})$ norm inequalities. The multipliers involved are related to those of Coifman--Rubio de Francia--Semmes, satisfying certain weak Marcinkiewicz-type conditions that permit highly oscillatory factors of the form $e^{i|ξ|^α}$ for both $α$ positive and negative. The maximal functions that arise are of some independent interest, involving fractional averages associated with tangential approach regions (related to those of Nagel and Stein), and more no…
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 $1<p<\infty$.
The Interplay of Shifted Square and Maximal Function Estimates in the Context of Multilinear Fourier Multipliers
本文通过移位正方形和最大函数的结合，改进了多线性傅里叶乘子的有界性定理，并应用于奇异积分在Orlicz空间中的有界性研究。
We introduce a maximal operator for the natural analogue of the disc multiplier in the framework of Jacobi analysis and determine the mapping properties of said maximal operator, including weak type endpoint results. In particular we obtain sharp information on the almost everywhere convergence of the inverse Jacobi transform.
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
In this paper, we study the $L^p$ boundedness of a class of oscillating multiplier operator for the Dunkl transform, $T_{m_α}=\mathcal{F}_k^{-1}(m_α\mathcal{F}_k(f))$ with $m(ξ)=|ξ|^{-α}e^{\pm i|ξ|}ϕ(ξ)$. We obtain an $L^p$-bound result for the corresponding maximal functions. As a specific applications, we give an extension of the $L^p$ estimate for the wave equation and of Stein's theorem for the analytic family of maximal spherical means \cite{Stein}
In this paper we introduce Stein's square function associated with bilinear Bochner-Riesz means and investigate its $L^p$ boundedness properties. Further, we discuss several applications of the square function in the context of bilinear multipliers. In particular, we obtain results for maximal function associated with generalised bilinear Bochner-Riesz means. This extends the results proved in~\cite{JS}. Another application concerns the $L^p$ estimates for bilinear fractional Schrödinger multipliers. Finally, we improve upon a result of Grafakos, He and Honzik~\cite{GHH} in the context of bili…
In the paper we study the multipliers of Fourier series in polynomials orthogonal in continuous-discrete Sobolev's spaces. Multiplier Theorem for Fourier-Sobolev series is obtained. This result is based on the representation of the Fejér kernel, on the construction of the "humpbacked majorant" and weighted estimates of maximal functions.
We prove endpoint bounds for the square function associated with radial Fourier multipliers acting on $L^{p}$ radial functions. This is a consequence of endpoint bounds for a corresponding square function for Hankel multipliers. We obtain a sharp Marcinkiewicz-type multiplier theorem for multivariate Hankel multipliers and $L^p$ bounds of maximal operators generated by Hankel multipliers as corollaries. The proof is built on techniques developed by Garrigós and Seeger for characterizations of Hankel multipliers.
Nikodym sets and maximal functions associated with spheres
本文研究球面相关的Nikodym集与极大函数，证明尖锐L^p估计，得出Nikodym集须有完整Hausdorff维数，并利用波动方程局部光滑估计推导一般极大函数的L^p界。
On maximal functions generated by Hörmander-type spectral multipliers
本文利用 Doob 变换和 Grafakos-Honzík-Seeger 技巧，证明 Hörmander 型谱乘子有限族极大函数的最优 $L^p$ 对数平方根界，并给出连续尺度极大函数有界的充分条件。
We prove a characterization of the Sobolev spaces $H^α$ on the unit sphere $\mathbb{S}^{d-1}$, where the smoothness index $α$ is any positive real number and $d\geq 2$. This characterization does not use differentiation and it is given in terms of $([α/2]+1)$-multidimensional square functions $S_α$. For $[α/2]=0,$ a function $f\in L^2(\mathbb{S}^{d-1})$ belongs to $H^α(\mathbb{S}^{d-1})$ if and only if $S_α(f)\in L^2(\mathbb{S}^{d-1})$. If $n=[α/2]>0$, the membership of $f$ is equivalent to the existence of $g_1,\cdots,g_n$ in $L^2(\mathbb{S}^{d-1})$ such that $S_α(f,g_1,\ldots,g_n)\in L^2(\ma…
We prove a maximal Fourier restriction theorem for the sphere $\mathbb{S}^{d-1}$ in $\mathbb{R}^{d}$ for any dimension $d\geq 3$ in a restricted range of exponents given by the Stein-Tomas theorem. The proof consists of a simple observation. When $d=3$ the range corresponds exactly to the full Stein-Tomas one, but is otherwise a proper subset when $d>3$. We also present an application regarding the Lebesgue points of functions in $\mathcal{F}(L^p)$ when $p$ is sufficiently close to 1.
We study a.e. convergence on $L^p$, and Lorentz spaces $L^{p,q}$, $p>\tfrac{2d}{d-1}$, for variants of Riesz means at the critical index $d(\tfrac 12-\tfrac 1p)-\tfrac12$. We derive more general results for (quasi-)radial Fourier multipliers and associated maximal functions, acting on $L^2$ spaces with power weights, and their interpolation spaces. We also include a characterization of boundedness of such multiplier transformations on weighted $L^2$ spaces, and a sharp endpoint bound for Stein's square-function associated with the Riesz means.
We prove the existence of $L^2$ solutions to the Helmholtz equation $(-Δ- 1)u = f$ in ${\mathbb R}^n$ assuming the given data $f$ belongs to $L^{(2n+2)/(n+5)}({\mathbb R}^n)$ and satisfies the "Fredholm condition" that $\hat{f}$ vanishes on the unit sphere. This problem, and similar results for the perturbed Helmholtz equation $(-Δ-1)u = -Vu + f$, are connected to the Limiting Absorption Principle for Schrödinger operators. The same techniques are then used to prove that a wide range of $L^p \mapsto L^q$ bounds for Bochner-Riesz multipliers are improved if one considers their action on the clo…
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。
We give a new characterisation of inner product spaces amongst normed vector spaces in terms of the maximal cirumradius of spheres. It turns out that a normed vector space $(X,\norm{\cdot})$ with $\dim X\geq 2$ is an inner product space if and only if all spheres are not degenerate, i.e. the maximal circumradius of points on the sphere equals the radius of the sphere.
The bilinear cone multiplier on $\mathbb{R}^2\times \mathbb{R}^2$
本文研究了二维空间上的双线性锥乘法子，通过将双线性算子分解为与线性锥乘法子及其变体相关的平方函数，并利用合适的强极大值估计和几何方法，得到了双线性锥乘法子的$ L^p $有界性。
We investigate Fourier multipliers with smooth symbols defined over locally compact Hausdorff groups. Our main results in this paper establish new Hörmander-Mikhlin criteria for spectral and non-spectral multipliers. The key novelties which shape our approach are three. First, we control a broad class of Fourier multipliers by certain maximal operators in noncommutative $L_p$ spaces. This general principle ---exploited in Euclidean harmonic analysis during the last 40 years--- is of independent interest and might admit further applications. Second, we replace the formerly used cocycle dimensio…
On high dimensional maximal functions associated to Gaussians, balls, and spheres
研究当维度\(d\to\infty\)时，与高斯函数、球和球面相关的极大函数在\(L^p(\mathbb{R}^d)\)上范数的收敛情况，借助傅里叶乘子符号估计等方法证明收敛到\(\lambda(p)\)，并给出\(\lambda(p)\)的相关不等式。
A few years ago, Bourgain proved that the centered Hardy-Littlewood maximal function for the cube has dimension free $L^p$-bounds for $p>1$. We extend his result to products of Euclidean balls of different dimensions. In addition, we provide dimension free $L^p$-bounds for the maximal function associated to products of Euclidean spheres for $p > \frac{N}{N-1}$ and $N \ge 3$, where $N-1$ is the lowest occurring dimension of a single sphere. The aforementioned result is obtained from the latter one by applying the method of rotations from Stein's pioneering work on the spherical maximal function…
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。
Dimension-free estimates for discrete maximal functions over cubes in $\mathbb Z^d$
本文通过将离散乘子解释为连续乘子族并应用分数次积分与复插值，证明了立方体上离散 Hardy--Littlewood 极大函数的无维数 $\ell^p$ 界，并给出 Bourgain 结果的初等证明。
Dimension-free estimates for discrete maximal functions and lattice points in high-dimensional spheres and balls with small radii
本文证明小半径欧几里得球面相关的离散Hardy-Littlewood极大函数在ℓ^p(Z^d)上对p∈[2,∞)具有无维数估计，并推广到球体，首次不依赖连续对应给出完全离散极大函数的无维数估计，关键是小半径高维球面和球体中格点的均匀计数。
On those Weights Satisfying a Weak-Type Inequality for the Maximal Operator and Fractional Maximal Operator
本文针对Muckenhoupt和Wheeden提出的加权弱$(p,p)$不等式公开问题，证明$p>1$时其给出的必要条件对极大算子具有充分性，并得到分数次极大算子的对应结果，完善了权的刻画。
We prove that the discrete spherical maximal functions (in the spirit of Magyar, Stein and Wainger) corresponding to the Euclidean spheres in $\mathbb Z^d$ with dyadic radii have $\ell^p(\mathbb Z^d)$ bounds for all $p\in[2, \infty]$ independent of the dimensions $d\ge 5$. An important part of our argument is the asymptotic formula in the Waring problem for the squares with a dimension-free multiplicative error term. By considering new approximating multipliers we will show how to absorb an exponential in dimension (like $C^d$ for some $C>1$) growth in norms arising from the sampling principle…
Weighted estimates for lacunary maximal functions on homogeneous groups
该研究聚焦齐次群上广义缺项极大函数的加权估计问题，推导得出两类缺项极大函数的改进加权估计，为相关调和分析问题提供了新的结果。
A theorem of Strichartz for multipliers on homogeneous trees
本文将欧氏空间上的Strichartz定理推广至齐次树，用缓增分布和乘子算子替代有界函数与Laplacian，并探讨了热算子、Schrödinger算子及球面平均等具体乘子算子情形。
We prove $L^p\to L^q$ estimates for the local maximal operator associated with dilates of the Kóranyi sphere in Heisenberg groups. These estimates are sharp up to endpoints and imply new bounds on sparse domination for the corresponding global maximal operator. We also prove sharp $L^p\to L^q$ estimates for spherical means over the Korányi sphere, which can be used to improve the sparse domination bounds for the associated lacunary maximal operator.
Dimension-free estimates for full discrete maximal functions associated with Euclidean balls and spheres
本文证明欧氏球全离散 Hardy-Littlewood 极大算子对所有 $p>1$ 及球面极大算子对 $d\geq5, p\geq2$ 满足无维数界，通过离散高斯乘子逼近解决 Stein 问题并加强 MSW 定理。
Let $A$ be a generator of an analytic semigroup having a H{ö}rmander functional calculus on $X = L^p(Ω,Y)$, where $Y$ is a UMD lattice. Using methods from Banach space geometry in connection with functional calculus, we show that for H{ö}rmander spectral multipliers decaying sufficiently fast at $\infty$, there holds a maximal estimate $\| \sup_{t \geq 0} |m(tA)f|\, \|_{L^p(Ω,Y)} \lesssim \|f\|_{L^p(Ω,Y)}$. We also show square function estimates $\left\| \left( \sum_k \sup _{t \geq 0} |m_k(tA)f_k|^2 \right)^{\frac12} \right\|_{L^p(Ω,Y)} \lesssim \left\| \left( \sum _k |f_k|^2 \right)^{\frac12}…
Analytic Schur multipliers
本文研究复双上半平面与双圆盘上的解析Schur乘子，证明具有正则性的边界值函数可嵌入Haagerup张量积空间，对算子扰动理论有重要应用。
In this paper, we establish a noncommutative analogue of Calderón's transference principle, which allows us to deduce noncommutative ergodic maximal inequalities from the special case---operator-valued maximal inequalities. As applications, we deduce dimension-free estimates of noncommutative Wiener's maximal ergodic inequality and noncommutative Stein-Calderón's maximal ergodic inequality over Euclidean spaces. We also show the corresponding individual ergodic theorems. To show Wiener's pointwise ergodic theorem, we construct a dense subset on which pointwise convergence holds following a som…
Generic norm growth of powers of homogeneous unimodular Fourier multipliers
该研究针对齐次幺模傅里叶乘子算子族，证明通用相位函数下其范数随参数t达最大增长阶，说明此前反例中的最大增长是普遍现象。
We prove sparse bounds for the spherical maximal operator of Magyar, Stein and Wainger. The bounds are conjecturally sharp, and contain an endpoint estimate. The new method of proof is inspired by ones by Bourgain and Ionescu, is very efficient, and has not been used in the proof of sparse bounds before. The Hardy-Littlewood Circle method is used to decompose the multiplier into major and minor arc components. The efficiency arises as one only needs a single estimate on each element of the decomposition.
We prove spectral multiplier theorems for Hörmander classes $\mathcal{H}^α\_p$ for 0-sectorial operators A on Banach spaces assuming a bounded $H^\infty(Σ\_σ)$ calculus for some $σ\in (0,π)$ and norm and certain R-bounds on one of the following families of operators: the semigroup $e^{--zA}$ on $\mathbb{C}\_+$, the wave operators $e^{isA}$ for $s \in \mathbb{R}$, the resolvent $(λ-- A)^{-1}$ on $\mathbb{C} \backslash \mathbb{R}$, the imaginary powers $A^{it}$ for $t \in \mathbb{R}$ or the Bochner-Riesz means $(1-A/u)^α\_+$ for $u > 0.$ In contrast to the existing literature we neither assume t…
We provide a new direct proof of the $\ell^2$-boundedness of the Discrete Spherical Maximal Function that neither relies on abstract transference theorems (and hence Stein's Spherical Maximal Function Theorem) nor on delicate asymptotics for the Fourier transform of discrete spheres.
Spherical maximal functions on two step nilpotent Lie groups
该研究聚焦两步幂零李群上的球面极大函数，去掉Métivier群相关结果的非退化条件，证明了所有$d\ge3$的两步幂零李群上该极大函数的精确$L^p$有界性。
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