Joan Mateu, Joan Verdera
We estimate in Lp the maximal Riesz transform in terms of the Riesz transform itself for p greater than 1. In the limiting case p=1 the weak L1 inequality is shown to fail. Surprisingly, the weak L1 inequality for the maximal Beurling transform in terms of the Beurling transform does hold.
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We prove that the operator norm on weighted Lebesgue space L2(w) of the commutators of the Hilbert, Riesz and Beurling transforms with a BMO function b depends quadratically on the A2-characteristic of the weight, as opposed to the linear dependence known to hold for the operators themselves. It is known that the operator norms of these commutators can be controlled by the norm of the commutator with appropriate Haar shift operators, and we prove the estimate for these commutators. For the shift operator corresponding to the Hilbert transform we use Bellman function methods, however there is n…
It is known that the improved Cotlar's inequality $B^{*}f(z) \le C M(Bf)(z)$, $z\in\mathbb C$, holds for the Beurling transform $B$, the maximal Beurling transform $B^{*}f(z)=$ $\displaystyle\sup_{\varepsilon >0}\left|\int_{|w|>\varepsilon}f(z-w) \frac{1}{w^2} \,dw\right|$, $z\in\mathbb C$, and the Hardy--Littlewood maximal operator $M$. In this note we consider the maximal Beurling transform associated with squares, namely, $B^{*}_Sf(z)=\displaystyle\sup_{\varepsilon >0}\left|\int_{w\notin Q(0,\varepsilon)}f(z-w) \frac{1}{w^2} \,dw \right|$, $z\in\mathbb C$, $Q(0,\varepsilon)$ being the squar…
W prove a dimension-free estimate for the $L^2(\mathbb{R}^d)$ norm of the maximal truncated Riesz transform in terms of the $L^2(\mathbb{R}^d)$ norm of the Riesz transform. Consequently, the vector of maximal truncated Riesz transforms has a dimension-free estimate on $L^2(\mathbb{R}^d)$. We also show that the maximal function of the vector of truncated Riesz transforms has a dimension-free estimate on all $L^p(\mathbb{R}^d)$ spaces, $1<p<\infty.$
We prove some new $L^p$ estimates for maximal Bochner-Riesz operator in the plane.
We prove that the bilinear maximal Bochner-Riesz operator $T_*^λ$ is bounded from $L^{p_1}(\mathbb R^n)\times L^{p_2}(\mathbb R^n)$ to $L^p(\mathbb R^n)$ for appropriate $(p_1,p_2,p)$ when $λ>(4n+3)/5$.
We establish improved and sharp $L^p$ estimates for the maximal bilinear Bochner-Riesz means in all dimensions $n\geq 1$. This work extends the results proved by Jeong and Lee \cite{JL}. We also recover the known results for the bilinear Bochner-Riesz means. The method of proof involves a new decomposition of the bilinear Bochner-Riesz multiplier $(1-|ξ|^2-|η|^2)_+^α$ and delicate analysis in proving $L^p$ estimates for frequency localized square functions.
$L^{p}$ estimates for multilinear maximal Bochner--Riesz means and square function
本文研究多线性极大Bochner–Riesz平均及其平方函数的$L^{p}$有界性，借鉴Jotsaroop和Shrivastava关于双线性Bochner–Riesz平均极大估计的方法完成证明。
We study the maximal estimates for the bilinear spherical average and the bilinear Bochner-Riesz operator. Firstly, we obtain $L^p\times L^q \to L^r$ estimates for the bilinear spherical maximal function on the optimal range. Thus, we settle the problem which was previously considered by Geba, Greenleaf, Iosevich, Palsson and Sawyer, later Barrionevo, Grafakos, D. He, Honzík and Oliveira, and recently Heo, Hong and Yang. Secondly, we consider $L^p\times L^q \to L^r$ estimates for the maximal bilinear Bochner-Riesz operators and improve the previous known results. For the purpose we draw a conn…
We study weighted $(L^p, L^q)$-boundedness properties of Riesz potentials and fractional maximal functions for the Dunkl transform. In particular, we obtain the weighted Hardy-Littlewood-Sobolev type inequality and weighted week $(L^1, L^q)$ estimate. We find a sharp constant in the weighted $L^p$-inequality, generalizing the results of W. Beckner and S. Samko.
A weighted decoupling inequality and its application to the maximal Bochner-Riesz problem
该研究证明了p=2n/(n-1)时的加权L^pℓ^p解耦估计，将其应用于极大Bochner-Riesz问题，得到三维空间中超出实插值指数的结果，并改进了平面情形的相关结论。
Bilinear Bochner-Riesz means for convex domains and Kakeya Maximal function
该研究引入平面凸域相关的双线性Bochner-Riesz平均，借助双线性Kakeya极大函数建立其$L^p$估计以证明有界性，并揭示该极大函数与Lacey双线性极大函数的联系。
The sharp $L^p$ bound for the imaginary part of the Beurling-Ahlfors operator on the lattice
本文证明了二维整数格上离散Beurling--Ahlfors算子虚部的sharp $\ell^p$ 范数界为 $p^\star-1$，通过鞅表示但需克服强微分从属缺失的困难。
We prove boundedness of Calderón-Zygmund operators acting in Banach functions spaces on domains, defined by the $L_1$ Carleson functional and $L_q$ ($1<q<\infty$) Whitney averages. For such bounds to hold, we assume that the operator maps towards the boundary of the domain. We obtain the Carleson estimates by proving a pointwise domination of the operator, by sparse operators with a causal structure. The work is motivated by maximal regularity estimates for elliptic PDEs and is related to one-sided weighted estimates for singular integrals.
Hardy-Littlewood maximal, generalized Bessel-Riesz and generalized fractional integral operators in generalized Morrey and $BMO_ϕ$ spaces associated with Dunkl operator on the real line
本文研究了与Dunkl算子相关的广义Morrey和$BMO_ϕ$空间中Hardy-Littlewood极大函数、Bessel-Riesz算子和广义分数积分算子的有界性问题。
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…
$L^p$ norm of truncated Riesz transform and an improved dimension-free $L^p$ estimate for maximal Riesz transform
该研究利用径向Fourier乘子分部积分法，证明极大截断Riesz变换的$L^p$范数界无维数依赖，给出具体估计式，同时得到截断Riesz变换的范数收缩性与精确范数等式。
Maximal Riesz transform in terms of Riesz transform on quantum tori and Euclidean space
本文在量子环面和量子欧几里得空间上建立了最大Riesz变换的$L_p$有界性，并证明当$2\leq p<\infty$时范数常数与维数无关。
The paper concerns the magnetic Schrödinger operator on $R^n$. Under certain conditions, given in terms of the reverse Hölder inequality on the magnetic field and the electric potential, we prove some $L^p$ estimates on the Riesz transforms and we establish some related maximal inequalities.
Dimension-free $L^p$ estimates for higher order maximal Riesz transforms in terms of the Riesz transforms
本文证明了高阶极大Riesz变换向量可以用对应的Riesz变换表示的无维数$L^p(\mathbb{R}^d)$估计，并给出了常数对$p$的显式依赖。
The paper concerns the magnetic Schrödinger operator on $R^n$. We prove some $L^p$ estimates on the Riesz transforms and we establish some related maximal inequalities. The conditions that we arrive at, are essentially based on the control of the magnetic field by the electric potential.
Bounds for the maximal and Riesz potential operators with variable fractionality
针对可变分数阶的分数阶极大算子和Riesz势算子，证明了L^{p(·)}到L^{q(·)}的有界性，通过三指数Muckenhoupt条件和堆积条件给出界。
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.
The main aim of this paper is to investigate $\left( H_{p},L_{p}\right) $ and $\left( H_{p},L_{p,\infty }\right) $ type inequalities for maximal operators of Riesz logarithmic means of one-dimensional Vilenkin-Fourier series.
Let $\{A_t\}_{t>0}$ be the dilation group in ${\Bbb R}^n$ generated by the infinitesimal generator $M$ where $A_t=\exp(M\log t)$, and let $\varrho\in C^{\infty}({\Bbb R}^n\setminus\{0\})$ be a $A_t$-homogeneous distance function defined on ${\Bbb R}^n$. For $f\in {\frak S}({\Bbb R}^n)$, we define the maximal quasiradial Bochner-Riesz operator ${\frak M}^δ_{\varrho}$ of index $δ>0$ by $${\frak M}^δ_{\varrho} f(x)=\sup_{t>0}|{\Cal F}^{-1}[(1-\varrho/t)_+^δ\hat f ](x)|.$$ If $A_t=t I$ and $\{ξ\in {\Bbb R}^n| \varrho(ξ)=1\}$ is a smooth convex hypersurface of finite type, then we prove in an extre…
Shorter proof of dimension-free $L^p$ estimates for maximal Riesz transforms
本文提出最大Riesz变换$L^p$估计的简洁直接证明，常数不依赖维数，改进了先前依赖维数的结果。
We show various $L^p$ estimates for Schrödinger operators $-Δ+V$ on $\RR^n$ and their square roots. We assume reverse Hölder estimates on the potential, and improve some results of Shen \cite{Sh1}. Our main tools are improved Fefferman-Phong inequalities and reverse Hölder estimates for weak solutions of $-Δ+V$ and their gradients.
On pointwise convergence of multilinear Bochner-Riesz means
本文改进了多线性Bochner-Riesz平均逐点收敛的指标范围，通过归纳建立k线性极大算子的L^p与加权估计（高维p<2/k时为新结果），并借助Stein平方函数变体及其多线性推广证明了该估计。
In this work, we describe several results exhibited during a talk at the El Escorial 2012 conference. We aim to pursue the development of a multi-frequency Calderon-Zygmund analysis introduced in [9]. We set a definition of general multi-frequency Calderon-Zygmund operator. Unweighted estimates are obtained using the corresponding multi-frequency decomposition of [9]. Involving a new kind of maximal sharp function, weighted estimates are obtained.
In this paper, we develop via real variable methods various characterisations of the Hardy spaces in the multi-parameter flag setting. These characterisations include those via, the non-tangential and radial maximal function, the Littlewood--Paley square function and area integral, Riesz transforms and the atomic decomposition in the multi-parameter flag setting. The novel ingredients in this paper include (1) establishing appropriate discrete Calderón reproducing formulae in the flag setting and a version of the Plancherel--Pólya inequalities for flag quadratic forms; (2) introducing the maxi…
In this paper we study maximal and square functions associated with bilinear Bochner-Riesz means at the critical index. In particular, we prove that they satisfy weighted estimates from $L^{p_1}(w_1)\times L^{p_2}(w_2)\rightarrow L^p(v_w)$ for bilinear weights $(w_1,w_2)\in A_{\vec{P}}$ where $p_1,p_2>1$ and $\frac{1}{p_1}+\frac{1}{p_2}=\frac{1}{p}$. Also, we show that both the operators fail to satisfy weak-type estimates at the end-point $(1,1,\frac{1}{2})$.
Let H be the Hermite operator -Δ+|x|^2 on \mathbb{R}^n. We prove a weighted L^2 estimate of the maximal commutator operator \sup_{R>0}|[b, S_R^λ(H)](f)|, where [b, S_R^λ(H)](f) = bS_R^λ(H) f - S_R^λ(H)(bf) is the commutator of a BMO function b and the Bochner-Riesz means S_R^λ(H) for the Hermite operator H. As an application, we obtain the almost everywhere convergence of [b, S_R^λ(H)](f) for large λand f\in L^p(\mathbb{R}^n).
We prove an almost everywhere convergence result for Bochner-Riesz means of $L^p$ functions on Heisenberg-type groups, yielding the existence of a $p>2$ for which convergence holds for means of arbitrarily small order. The proof hinges on a reduction of weighted $L^2$ estimates for the maximal Bochner-Riesz operator to corresponding estimates for the non-maximal operator, and a `dual Sobolev trace lemma', whose proof is based on refined estimates for Jacobi polynomials.
On almost everywhere convergence of planar Bochner-Riesz means
本文研究了在$5/3\leq p\leq 2$范围内，$L^p$函数的平面Bochner-Riesz均值几乎处处收敛性，通过建立与Bochner-Riesz乘法算子相关的最大算子的精确$L^{5/3}$估计来实现，该估计依赖于一个新的精细$L^2$估计。
Let $w$ be a Muckenhoupt weight and $WH^p_w(\mathbb R^n)$ be the weighted weak Hardy spaces. In this paper, by using the atomic decomposition of $WH^p_w(\mathbb R^n)$, we will show that the maximal Bochner-Riesz operators $T^δ_*$ are bounded from $WH^p_w(\mathbb R^n)$ to $WL^p_w(\mathbb R^n)$ when $0<p\le1$ and $δ>n/p-{(n+1)}/2$. Moreover, we will also prove that the Bochner-Riesz operators $T^δ_R$ are bounded on $WH^p_w(\mathbb R^n)$ for $0<p\le1$ and $δ>n/p-{(n+1)}/2$. Our results are new even in the unweighted case.
Let $L$ be a homogeneous divergence form higher order elliptic operator with complex bounded measurable coefficients on $\mathbb{R}^n$ and $X$ a ball quasi-Banach function space on $\mathbb{R}^n$ satisfying some mild assumptions. Denote by $H_{X,\, L}(\mathbb{R}^n)$ the Hardy space, associated with both $L$ and $X$, which is defined via the Lusin area function related to the semigroup generated by $L$. In this article, the authors establish both the maximal function and the Riesz transform characterizations of $H_{X,\, L}(\mathbb{R}^n)$. The results obtained in this article have a wide range o…
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 give a new proof of the sharp weighted $L^2$ inequality ||T||_{L^2(w)} \leq c [w]_{A_2} where $T$ is the Hilbert transform, a Riesz transform, the Beurling-Ahlfors operator or any operator that can be approximated by Haar shift operators. Our proof avoids the Bellman function technique and two weight norm inequalities. We use instead a recent result due to A. Lerner to estimate the oscillation of dyadic operators.
For a family of weight functions, $h_κ$, invariant under a finite reflection group on $\RR^d$, analysis related to the Dunkl transform is carried out for the weighted $L^p$ spaces. Making use of the generalized translation operator and the weighted convolution, we study the summability of the inverse Dunkl transform, including as examples the Poisson integrals and the Bochner-Riesz means. We also define a maximal function and use it to prove the almost everywhere convergence.
We will explain how to compute the exact $L^p$ operator norm of a "quadratic perturbation" of the real part of the Ahlfors--Beurling operator. For the lower bound estimate we use a new approach of constructing a sequence of laminates (probability measures for which Jensen's inequality holds, but for rank one concave functions) to give an almost extremal sequence to approximate the operator. The upper bound estimate is given by extending the estimates of the quadratic perturbation of the martingale transform to continuous martingales. The use of "heat martingales" then allow us to connect the R…
We prove sharp power-weighted $L^p$, weak type and restricted weak type inequalities for the heat semigroup maximal operator and Riesz transforms associated with the Bessel operator $B_ν$ in the exotic range of the parameter $-\infty < ν< 1$. Moreover, in the same framework, we characterize basic mapping properties for other fundamental harmonic analysis operators, including the heat semigroup based vertical $g$-function and fractional integrals (Riesz potential operators).
We introduce some new functions spaces to investigate some problems at or beyond endpoint. First, we prove that Bochner-Riesz means $B_R^λ$ are bounded from some subspaces of $L^p_{|x|^α}$ to $L^p_{|x|^α}$ for $ \frac{n-1}{2(n+1)}<λ\leq \frac{n-1}{2}, 0 < p\leq p'_λ=\frac{2n}{n+1+2λ}, n(\frac{p}{p_λ}-1)< α<n(\frac{p}{p'_λ}-1)$, and $0<R<\infty,$ and so are the maximal Bochner-Riesz means $B_*^λ$ for $ \frac{n-1}{2}\leq λ< \infty, 0 < p\leq 1$ and $-n< α<n(p-1)$. From these we obtain the $L^p_{|x|^α}$-norm convergent property of $B_R^λ$ for these $λ,p,$ and $α$. Second, let $n\geq 3,$ we prove …
Discrete analogues of second-order Riesz transforms
本文研究了第二阶里斯茨变换的离散类比的\ell^p范数，构造了新的离散里斯茨变换并探讨其与经典算子的关系。
Restriction and Kakeya maximal estimates in $\mathbb{R}^4$
该研究结合平面刷论证与解耦- incidence方法，推导出R^4中Fourier限制和Bochner-Riesz问题的新界，并获得维度3.054的凯凯ya极大估计。
Weak type $(1, 1)$ of Riesz transform on some direct product manifolds with exponential volume growth
本文研究了直积流形 ${\mathbb{H}}^n \times M$ 上的 Riesz 变换，在体积加倍和热核上估计条件下建立了其弱型 $(1,1)$ 性质，并推广至热极大算子及更广泛的指数体积增长流形。
As a corollary to our main result we deduce sharp A_p$ inequalities for T being either the Hilbert transform in dimension d=1, the Beurling transform in dimension d=2, or a Riesz transform in any dimension d\ge 2. For T_{\ast} the maximal truncations of these operators, we prove the sharp A_p weighted weak and strong-type L ^{p} (w) inequalities, for all 1<p<\infty. Key elements of the proof are (1) extrapolation (2) a recent argument for the A_2 bound in the untruncated case, an argument of Lacey-Petermichl-Reguera. (3) a weak-L^1 estimate for duals of maximal truncations. And (4) recent char…
This is an expository paper on the characterization of the even (or odd) smooth homogeneous convolution Calderón-Zygmund operators in R^n such that the maximal singular integral can be controlled in the L^2 norm by the singular integral. We focuss attention on special cases of the general statements to convey the main ideas of the proofs in a transparent way, as free as possible of the technical complications inherent to the general case. Particular attention is devoted to higher Riesz transforms. The exposition is based on two papers by J.Mateu, J.Orobitg, C.Pérez and J.Verdera published in 2…
We prove that bilinear forms associated to the rough homogeneous singular integrals $T_Ω$ on $\mathbb R^d$, where the angular part $Ω\in L^q (S^{d-1})$ has vanishing average and $1<q\leq \infty$, and to Bochner-Riesz means at the critical index in $\mathbb R^d$ are dominated by sparse forms involving $(1,p)$ averages. This domination is stronger than the weak-$L^1$ estimates for $T_Ω$ and for Bochner-Riesz means, respectively due to Seeger and Christ. Furthermore, our domination theorems entail as a corollary new sharp quantitative $A_p$-weighted estimates for Bochner-Riesz means and for homog…
Let $p(\cdot):\ \mathbb R^n\to(0,\infty)$ be a variable exponent function satisfying that there exists a constant $p_0\in(0,p_-)$, where $p_-:=\mathop{\mathrm {ess\,inf}}_{x\in \mathbb R^n}p(x)$, such that the Hardy-Littlewood maximal operator is bounded on the variable exponent Lebesgue space $L^{p(\cdot)/p_0}(\mathbb R^n)$. In this article, via investigating relations between boundary valued of harmonic functions on the upper half space and elements of variable exponent Hardy spaces $H^{p(\cdot)}(\mathbb R^n)$ introduced by E. Nakai and Y. Sawano and, independently, by D. Cruz-Uribe and L.-A…
Cotlar martingale transforms and related singular integrals
本文探讨Cotlar鞅变换及其相关奇异积分的结构，揭示其与Hilbert变换的相似性，并研究Riesz变换的范数渐进行为，同时解决两个长期未解的问题。
We give a new proof of the sharp one weight $L^p$ inequality for any operator $T$ that can be approximated by Haar shift operators such as the Hilbert transform, any Riesz transform, the Beurling-Ahlfors operator. Our proof avoids the Bellman function technique and two weight norm inequalities. We use instead a recent result due to A. Lerner to estimate the oscillation of dyadic operators. Our method is flexible enough to prove the corresponding sharp one-weight norm inequalities for some operators of harmonic analysis: the maximal singular integrals associated to $T$, Dyadic square functions …
Muckenhoupt-Type Weights and Quantitative Weighted Estimate in the Bessel Setting
本文在 Bessel 环境中引入新的 Muckenhoupt 型权类，刻画 Hardy–Littlewood 极大算子的加权有界性，并建立 Riesz 交换子的加权有界性、紧性、端点弱型估计及定量加权界。
We present a general approach for proving the optimality of the exponents on weighted estimates. We show that if an operator $T$ satisfies a bound like $$ \|T\|_{L^{p}(w)}\le c\, [w]^β_{A_p} \qquad w \in A_{p}, $$ then the optimal lower bound for $β$ is closely related to the asymptotic behaviour of the unweighted $L^p$ norm $\|T\|_{L^p(\mathbb{R}^n)}$ as $p$ goes to 1 and $+\infty$, which is related to Yano's classical extrapolation theorem. By combining these results with the known weighted inequalities, we derive the sharpness of the exponents, without building any specific example, for a w…
$\mathrm{L}^p$ bounds for parabolic Riesz transforms with rough coefficients: The case $1<p \leq 2$
研究非自治二阶抛物型微分算子相关里斯变换的$\mathrm{L}^p$界，通过$\mathrm{L}^p$预解式界确定复系数下$1<p \leq2$的最大开区间，实系数时证明可外推到全范围，基于两种几何的时空非对角界论证。
This paper provides a deeper study of the Hardy and $\rm BMO$ spaces associated to the Neumann Laplacian $Δ_N$. For the Hardy space $H^1_{Δ_N}(\mathbb{R}^n)$ (which is a proper subspace of the classical Hardy space $H^1(\mathbb{R}^n)$) we demonstrate that the space has equivalent norms in terms of Riesz transforms, maximal functions, atomic decompositions, and weak factorizations. While for the space ${\rm BMO}_{Δ_N}(\mathbb{R}^n)$ (which contains the classical $\rm BMO(\mathbb{R}^n)$) we prove that it can be characterized in terms of the action of the Riesz transforms associated to the Neuman…
Weak type estimates for Bochner--Riesz means on Hardy-type spaces associated with ball quasi-Banach function spaces
本文在球拟巴拿赫函数空间关联的哈代型空间框架下，建立了 Bochner–Riesz 平均及其极大算子的弱型有界性，并应用于加权勒贝格、Herz、洛伦兹、变指标勒贝格及 Morrey 空间。
We establish a Maz'ya type capacitary inequality which resolves a special case of a conjecture by David R. Adams. As a consequence, we obtain several equivalent norms for Choquet integrals associated to Bessel or Riesz capacities. This enables us to obtain bounds for the Hardy-Littlewood maximal function in a sublinear setting.
The scalar $T1$ theorem for pairs of doubling measures fails for Riesz transforms when p not 2
本文构造反例证明倍增测度对下单个 Riesz 变换的标量 T1 定理在 p ≠ 2 时失效，同时改进 Sawyer-Wick 的向量值 T1 定理，并澄清极大函数双权不等式不能仅由 A_p 条件刻画。
We prove a weighted norm inequality for the maximal Bochner--Riesz operator and the associated square-function. This yields new $L^p(R^d)$ bounds on classes of radial Fourier multipliers for $p\ge 2+4/d$ with $d\ge 2$, as well as space-time regularity results for the wave and Schrödinger equations.
Capacitary Maximal Inequalities and Applications
本文引入基于Hausdorff容量和Riesz容量的Hardy-Littlewood极大函数容量类比，证明其在容量积分空间上的强型与弱型有界性，并由此改进Lebesgue型微分定理及Sobolev空间中Lebesgue点的估计。
The purpose of this paper is to describe the smooth homogeneous Calderon-Zygmund operators for which the maximal singular integral T*f may be controlled by the singular integral Tf. We consider two types of control. The first is the L2 estimate of T*f by Tf, namely the estimate of the L2 norm of T*f by a constant times the L2 norm of Tf. The second is the pointwise estimate of T*f(x) by a constant times M(Tf)(x), where M denotes the Hardy-Littlewood maximal operator. Notice that this is an improved variant of Cotlar's inequality, because the term Mf(x) is missing on the right hand side. Our ma…
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