Julian Weigt
Let $\mathcal M$ be the uncentered Hardy-Littlewood maximal operator or the dyadic maximal operator and $d\geq1$. We prove that for a set $E\subset\mathbb R^d$ of finite perimeter the bound $\operatorname{var}\mathcal M1_E\leq C_d\operatorname{var}1_E$ holds. We also prove this for the local maximal operator.
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We prove the continuity of the map $f \mapsto \widetilde{M}f$ from $BV(\mathbb{R})$ to itself, where $\widetilde{M}$ is the uncentered Hardy--Littlewood maximal operator. This answers a question of Carneiro, Madrid and Pierce.
$L^{p}-$estimates for uncentered spherical averages and lacunary maximal functions
本文引入非中心球面平均等算子的双线性类似物，获得了$d\geq2$维度下的$L^p$估计，并在线性情形下证明了单尺度平均算子的$L^p$改进性质与缺项极大函数的$L^p$估计。
The purpose of this note is to find the least weak type $(1,1)$ bound for the almost uncentered maximal operator on radial decreasing functions.
We obtain sharp bounds for the modulus of continuity of the uncentered maximal function in terms of the modulus of continuity of the given function, via integral formulas. Some of the results deduced from these formulas are the following: The best constants for Lipschitz and Hölder functions on proper subintervals of $\mathbb{R}$ are $\operatorname{Lip}_α( Mf) \le (1 + α)^{-1}\operatorname{Lip}_α( f)$, $α\in (0,1]$. On $\mathbb{R}$, the best bound for Lipschitz functions is $ \operatorname{Lip} ( Mf) \le (\sqrt2 -1)\operatorname{Lip}( f).$ In higher dimensions, we determine the asymptotic beha…
The Variation of the Uncentered Maximal Operator with respect to Cubes
本文研究欧几里得空间中非中心方体极大算子的变差有界性，证明有界变差函数的极大函数变差受原函数变差乘以维数常数控制，并推广至满足内锥星条件和平铺性质的更一般集合族。
Uncentered Fractional Maximal functions and mean oscillation spaces associated with dyadic Hausdorff content
本文研究了非中心分数极大函数在关联于二进Hausdorff内容的平均振荡空间上的作用，改进了其在BMO和VMO函数上的现有结果，并证明了多维非中心极大函数在相应空间上的有界性。
In this paper we establish new optimal bounds for the derivative of some discrete maximal functions, both in the centered and uncentered versions. In particular, we solve a question originally posed by Bober, Carneiro, Hughes and Pierce.
We explore some variants of the Boman covering lemma, and their relationship to the boundedness properties of the maximal operator. Let $1 < p < \infty$ and let $q$ be its conjugate exponent. We prove that the strong type $(q,q)$ of the uncentered maximal operator, by itself, implies certain generalizations of the Boman covering lemma for the exponent $p$, and in turn, these generalizations entail the weak type $(q,q)$ of the centered maximal operator. We show by example that it is possible for the uncentered maximal operator to be unbounded for all $1 < s < \infty$, while the conclusion of th…
In this paper we investigate some questions related to the continuity of maximal operators in $W^{1,1}$ and $BV$ spaces, complementing some well-known boundedness results. Letting $\widetilde M$ be the one-dimensional uncentered Hardy-Littlewood maximal operator, we prove that the map $f \mapsto \big(\widetilde Mf\big)'$ is continuous from $W^{1,1}(\mathbb{R})$ to $L^1(\mathbb{R})$. In the discrete setting, we prove that $\widetilde M: BV(\mathbb{Z}) \to BV(\mathbb{Z})$ is also continuous. For the one-dimensional fractional Hardy-Littlewood maximal operator, we prove by means of counterexample…
We study the Hardy-Littlewood maximal operator defined via an unconditional norm, acting on block decreasing functions. We show that the uncentered maximal operator maps block decreasing functions of special bounded variation to functions with integrable distributional derivatives, thus improving their regularity. In the special case of the maximal operator defined by the l_infty-norm, that is, by averaging over cubes, the result extends to block decreasing functions of bounded variation, not necessarily special.
In this paper we study the regularity properties of certain maximal operators of convolution type at the endpoint $p=1$, when acting on radial data. In particular, for the heat flow maximal operator and the Poisson maximal operator, when the initial datum $u_0 \in W^{1,1}( \mathbb{R}^d)$ is a radial function, we show that the associated maximal function $u^*$ is weakly differentiable and $$\|\nabla u^*\|_{L^1(\mathbb{R}^d)} \lesssim_d \|\nabla u_0\|_{L^1(\mathbb{R}^d)}.$$ This establishes the analogue of a recent result of H. Luiro for the uncentered Hardy-Littlewood maximal operator, now in a…
In this article, we prove some total variation inequalities for maximal functions. Our results deal with two possible generalizations of the results contained in Aldaz and Pérez Lázaro's work, one of whose considers a variable truncation of the maximal function, and the other one interpolates the centered and the uncentered maximal functions. In both contexts, we find sharp constants for the desired inequalities, which can be viewed as progress towards the conjecture that the best constant for the variation inequality in the centered context is one. We also provide counterexamples showing that…
Let $w$ denote a weight in $\mathbb{R}^n$ which belongs to the Muckenhoupt class $A_\infty$ and let $\mathsf{M}_w$ denote the uncentered Hardy-Littlewood maximal operator defined with respect to the measure $w(x)dx$. The \emph{sharp Tauberian constant} of $\mathsf M_w$ with respect to $α$, denoted by $\mathsf{C}_w (α)$, is defined by \[ \mathsf{C}_w (α) := \sup_{E:\, 0 < w(E) < \infty}w(E)^{-1}w\big(\big\{x \in \mathbb{R}^n:\, \mathsf{M}_w χ_E (x) > α\big\}\big). \] In this paper, we show that the Solyanik estimate \[ \lim_{α\rightarrow 1^-}\mathsf{C}_w(α) = 1 \] holds. Following the classical…
Rearrangement inequalities of the one-dimensional maximal functions associated with general measures
本研究证明了实数域上与一般测度相关的无中心Hardy-Littlewood极大函数的重排不等式，该不等式与Stein的结果类似，并进一步算出了该极大函数在弱Lp空间上的最优常数。
Sobolev bounds and counterexamples for the second derivative of the maximal function in one dimension
本文研究一维非中心Hardy-Littlewood极大函数二阶导数的$L^1$范数有界性，对递减偶Sobolev函数给出正结果，并构造非偶递减反例。
Study of Bounded Variation on the H-L Nontangential Operator
本文证明了离散一维Hardy-Littlewood非切向极大算子参数在$[\frac{1}{3},+\infty)$时的锐利有界性，推广了Bober等人的结果，并为连续版本提供了替代证明。
In this paper we address the $W^{1,1}$-continuity of several maximal operators at the gradient level. A key idea in our global strategy is the decomposition of a maximal operator, with the absence of strict local maxima in the disconnecting set, into "lateral" maximal operators with good monotonicity and convergence properties. This construction is inspired in the classical sunrise lemma in harmonic analysis. A model case for our sunrise strategy considers the uncentered Hardy-Littlewood maximal operator $\widetilde{M}$ acting on $W^{1,1}_{\rm rad}(\mathbb{R}^d)$, the subspace of $W^{1,1}(\mat…
We characterize the space $BV(I)$ of functions of bounded variation on an arbitrary interval $I\subset \mathbb{R}$, in terms of a uniform boundedness condition satisfied by the local uncentered maximal operator $M_R$ from $BV(I)$ into the Sobolev space $W^{1,1}(I)$. By restriction, the corresponding characterization holds for $W^{1,1}(I)$. We also show that if $U$ is open in $\mathbb{R}^d, d >1$, then boundedness from $BV(U)$ into $W^{1,1}(U)$ fails for the local directional maximal operator $M_T^{v}$, the local strong maximal operator $M_T^S$, and the iterated local directional maximal operat…
On the frequency function of Hardy-Littlewood maximal functions
本文研究了Hardy-Littlewood极大函数的频率函数在离散和连续设置中的行为，分析了其渐进行为和小值密度，并探讨了p>1时的特性差异。
We study the problem of whether the centered Hardy--Littlewood maximal function of a singular function is absolutely continuous. For a parameter $d \in (0,1)$ and a closed set $E\subset [0,1]$, let $μ$ be a $d$-Ahlfors regular measure associated with $E$. We prove that for the cumulative distribution function $f(x)=μ([0,x])$ its maximal function $Mf$ is absolutely continuous. We then adapt our method to the multiparameter case and show that the same is true in the positive cone defined by these functions, i.e., for functions of the form $f(x)=\sum_{i=1}^{n}μ_i([0,x])$ where $\{μ_i\}_{i=1}^{n}$…
We study the Sobolev regularity on the sphere $\mathbb{S}^d$ of the uncentered fractional Hardy-Littlewood maximal operator $\widetilde{\mathcal{M}}_β$ at the endpoint $p=1$, when acting on polar data. We first prove that if $q=\frac{d}{d-β}$, $0<β<d$ and $f$ is a polar $W^{1,1}(\mathbb{S}^d)$ function, we have $$\|\nabla \widetilde{\mathcal{M}}_βf\|_q\lesssim_{d,β}\|\nabla f\|_1.$$ We then prove that the map $$f\mapsto \big | \nabla \widetilde{\mathcal{M}}_βf \big |$$ is continuous from $W^{1,1}(\mathbb{S}^d)$ to $L^q(\mathbb{S}^d)$ when restricted to polar data. Our methods allow us to give …
Let $\mathcal{B}$ denote a collection of open bounded sets in $\mathbb{R}^n$, and define the associated maximal operator $M_{\mathcal{B}}$ by $$ M_{\mathcal{B}}f(x) := \sup_{x \in R \in \mathcal{B}} \frac{1}{|R|}\int_R |f|. $$ The sharp Tauberian constant of $M_{\mathcal{B}}$ associated to $α$, denoted by $C_{\mathcal{B}}(α)$, is defined as $$ C_{\mathcal{B}}(α) := \sup_{E :\, 0 < |E| < \infty}\frac{1}{|E|}\big|\big\{x \in \mathbb{R}^n:\, M_{\mathcal{B}}χ_E (x) > α\big\}\big|.$$ Motivated by previous work of A. A. Solyanik, we show that if $M_{\mathcal{B}}$ is the uncentered Hardy-Littlewood m…
Parabolic Muckenhoupt Weights Characterized by Parabolic Fractional Maximal and Integral Operators with Time Lag
本文引入带时滞的非对角双权抛物 Muckenhoupt 类，刻画带时滞抛物分数次极大算子与新型分数次积分的加权有界性，并给出 Sobolev 嵌入和热方程先验估计。
Characterizations of parabolic Muckenhoupt classes
本文利用带时间滞后的非中心抛物型极大函数刻画极限抛物型 A_∞ 与 A_1 类，并借助 Calderón-Zygmund 分解等论证扩展抛物型 Muckenhoupt 权理论。
Sharp higher order regularity of discrete maximal functions
研究离散非中心极大算子应用于特征函数时\(k = 0,1,2\)阶导数在\(\ell^p(\mathbb{Z})\)中的精确界，\(k = 1,2\)时为Hardy-Littlewood极大函数导数首获精确界，还给出\(k\geq3\)及一般函数的下界。
Let $0<α<d$ and $1\leq p<d/α$. We present a proof that for all $f\in W^{1,p}(\mathbb{R}^d)$ both the centered and the uncentered Hardy-Littlewood fractional maximal operator $\mathcal M_αf$ are weakly differentiable and $ \|\nabla\mathcal M_αf\|_{p^*} \leq C_{d,α,p} \|\nabla f\|_p , $ where $ p^* = (p^{-1}-α/d)^{-1} . $ In particular it covers the endpoint case $p=1$ for $0<α<1$ where the bound was previously unknown. For $p=1$ we can replace $W^{1,1}(\mathbb{R}^d)$ by $\mathrm{BV}(\mathbb{R}^d)$. The ingredients used are a pointwise estimate for the gradient of the fractional maximal function…
We study the problem concerning the variation of the Hardy-Littlewood maximal function in higher dimensions. As the main result, we prove that the variation of the non-centered Hardy-Littlewood maximal function of a radial function is comparable to the variation of the function itself.
We prove sharp local and global variation bounds for the centred Hardy--Littlewood maximal functions of indicator functions in one dimension. We characterise maximisers, treat both the continuous and discrete settings and extend our results to a larger class of functions.
We show that a function $ f $ of bounded variation satisfies $$ \Var Mf \leq C \Var f $$ where $ Mf $ is the centered Hardy-Littlewood maximal function of $ f $. Consequently, the operator $ f \mapsto (Mf)' $ is bounded from $ W^{1,1}(R) $ to $ L^{1}(R) $. This answers a question of Hajlasz and Onninen in the one-dimensional case.
We prove that if $f:I\subset \Bbb R\to \Bbb R$ is of bounded variation, then the noncentered maximal function $Mf$ is absolutely continuous, and its derivative satisfies the sharp inequality $\|DMf\|_1\le |Df|(I)$. This allows us obtain, under less regularity, versions of classical inequalities involving derivatives.
We prove that for the dyadic maximal operator $\mathrm M$ and every locally integrable function $f\in L^1_{\mathrm{loc}}(\mathbb R^d)$ with bounded variation, also $\mathrm M f$ is locally integrable and $\mathop{\mathrm{var}}\mathrm M f\leq C_d\mathop{\mathrm{var}} f$ for any dimension $d\geq1$. It means that if $f\in L^1_{\mathrm{loc}}(\mathbb R^d)$ is a function whose gradient is a finite measure then so is $\nabla \mathrm M f$ and $\|\nabla \mathrm M f\|_{L^1(\mathbb R^d)}\leq C_d\|\nabla f\|_{L^1(\mathbb R^d)}$. We also prove this for the local dyadic maximal operator.
Multiparameter extensions of the Christ-Kiselev maximal theorem: strong variational bounds
本文证明了Christ-Kiselev极大定理的多参数强变差界，并推广到多线性算子情形。
In this paper we study the regularity of the non-centered fractional maximal operator $M_β$. As the main result, we prove that there exists $C(n,β)$ such that if $q=n/(n-β)$ and $f$ is a radial function, then $\|DM_βf\|_{L^{q}(\mathbb{R}^n)}\leq C(n,β)\|Df\|_{L^{1}(\mathbb{R}^n)}$. The corresponding result was previously known only if $n=1$ or $β=0$. Our proofs are almost free from one-dimensional arguments. Therefore, we believe that the new approach may be very useful when trying to extend the result for all $f\in W^{1,1}(\mathbb{R}^n)$.
Variation of the one-dimensional centered maximal operator on simple functions with gaps between pieces
本文研究了一维中心最大算子在具有间隔部分的简单函数上的变化，证明了其变差不等式并提出猜想，同时给出了离散情况下的定理及等价性转移结果。
On the variation and asymptotics of maximal functions
本文针对卷积型极大算子变差的两个猜想取得部分进展，在1维和更高维度分别证明了相关变差不等式，确定了维度相关常数的最优渐近依赖关系，其结果依赖于单位球Hardy-Littlewood极大函数的新渐近估计。
We establish in this paper, an existence and uniqueness result of right continuous with bounded variation solution for a perturbed differential inclusion governed by time-dependent maximal monotone operators.
This paper concerns existence of right-continuous with bounded variation solutions of a perturbed second-order differential inclusion governed by time and state-dependent maximal monotone operators.
This paper is devoted to the study of evolution problems involving time dependent maximal monotone operators, which are right continuous and bounded in variation with respect to the Vladimirov's pseudo distance. Several variants and applications are presented.
We prove weak-type (1,1) estimates for compositions of maximal operators with singular integrals. Our main object of interest is the operator $Δ^*Ψ$ where $Δ^*$ is Bourgain's maximal multiplier operator and $Ψ$ is the sum of several modulated singular integrals; here our method yields a significantly improved bound for the $L^q$ operator norm when $1 < q < 2$. We also consider associated variation-norm estimates.
We establish a higher dimensional counterpart of Bourgain's pointwise ergodic theorem along an arbitrary integer-valued polynomial mapping. We achieve this by proving variational estimates $V_r$ on $L^p$ spaces for all $1<p<\infty$ and $r>\max\{p, p/(p-1)\}$. Moreover, we obtain the estimates which are uniform in the coefficients of a polynomial mapping of fixed degree.
The second derivative of the discrete Hardy-Littlewood maximal function
本文针对离散非中心 Hardy-Littlewood 极大函数，首次给出了其高阶正则性的正面结果，突破了连续背景下高阶正则性不可能存在的限制。
In this paper we study some questions about the continuity of classical and fractional maximal operators in the Sobolev space $W^{1,1}$, in both continuous and discrete setting, giving a positive answer to two questions posed recently, one of them regarding the continuity of the map $f \mapsto \big(\widetilde M_βf\big)'$ from $W^{1,1}(\mathbb{R})$ to $L^q(\mathbb{R})$, for $q=\frac{1}{1-β}$. Here $\widetilde M_β$ denotes the non-centered fractional maximal operator on $\mathbb{R}$ with $β\in(0,1)$. The second one regarding the continuity of the discrete centered maximal operator in the space o…
In this paper we establish $L^p$ boundedness properties for maximal operators, Littlewood-Paley functions and variation operators involving Poisson semigroups and resolvent operators associated with nonsymmetric Ornstein-Uhlenbeck operators. We consider the Ornstein-Uhlenbeck operators defined by the identity as the covariance matrix and having a drift given by the matrix $-λ(I+R)$, being $λ>0$ and $R$ a skew-adjoint matrix. The semigroup associated with these Ornstein-Uhlenbeck operators are the basic building blocks of all normal Ornstein-Uhlenbeck semigroups.
Sharp variational inequalities for the Hardy-Littlewood maximal operator on finite undirected graphs
本文研究了 Hardy-Littlewood 最大算子在完全图上的尖锐 p-变分不等式，回答了 Liu 和 Xue 的问题，并通过计算找到连接图上最多五个顶点的 1-变分不等式尖锐常数，提出路径图的猜想。
Two-parameter variational estimates for averages over tori
本文针对$\mathbb{R}^3$中环面平均值引入局部双参数$r$-变差范数，建立直至端点的尖锐$L^p$-$L^q$界，并与单参数情形对比，揭示两者有界区域的本质差异。
We prove an extension of the Walsh-analog of the Carleson-Hunt theorem, where the $L^\infty$ norm defining the Carleson maximal operator has been replaced by an $L^q$ maximal-multiplier-norm. Additionally, we consider certain associated variation-norm estimates.
In this paper we study the regularity properties of two maximal operators of convolution type: the heat flow maximal operator (associated to the Gauss kernel) and the Poisson maximal operator (associated to the Poisson kernel). In dimension $d=1$ we prove that these maximal operators do not increase the $L^p$-variation of a function for any $p \geq 1$, while in dimensions $d>1$ we obtain the corresponding results for the $L^2$-variation. Similar results are proved for the discrete versions of these operators.
A variation norm Carleson theorem in higher dimensions
本文将Fefferman的多边形傅里叶部分和极大函数界推广到高维下的强$r$-变分范数界与2-振荡界，并针对嵌套矩形的部分傅里叶和建立了相关变分与振荡界，完善了高维Carleson定理的研究。
In this note we prove the following good-$λ$ inequality, for $r>2$, all $λ> 0$, $δ\in \big(0, \frac{1}{2} \big)$ \[ ν\big\{ V_r(f) > 3 λ; \mathcal{M}(f) \leq δλ\big\} \leq 4 ν\{s(f) > δλ\} + {δ^2 \left(1+\frac{16}{r-2}\right)^2} \cdot ν\big\{ V_r(f) > λ\big\}, \] where $\mathcal{M}(f)$ is the martingale maximal function, $s(f)$ is the conditional martingale square function. This immediately proves that $V_r(f)$ is bounded on $L^p$, $1 < p <\infty$ and moreover is integrable when the maximal function is.
The purpose of this note is to provide a summary of the recent work of the authors on two variations of the pointwise convergence problem for the solutions to the fractional Schrödinger equations; convergence along a tangential line and along a set of lines, as exhibiting some new results in each setting. For the former case, we make a simple observation on a path along a tangential curve of exponential order. We discuss counterexamples for the latter case that show some of the known smooth regularities are essentially optimal.
The centered maximal operator removes the non-concave Cantor part from the gradient
该研究针对d维空间中有界变差函数的中心Hardy–Littlewood极大函数正则性，证明非凹爆破点处的极大函数性质，并推导出满足特定条件时极大函数的BV正则性可提升至Sobolev正则性。
Let S(n,0) be the set of monic complex polynomials of degree $n\ge 2$ having all their zeros in the closed unit disk and vanishing at 0. For $p\in S(n,0)$ denote by $|p|_{0}$ the distance from the origin to the zero set of $p'$. We determine all 0-maximal polynomials of degree $n$, that is, all polynomials $p\in S(n,0)$ such that $|p|_{0}\ge |q|_{0}$ for any $q\in S(n,0)$. Using a second order variational method we then show that although some of these polynomials are linearly inextensible, they are not locally maximal for Sendov's conjecture.
In this paper we prove weighted $\ell^p$-inequalities for variation and oscillation operators defined by semigroups of operators associated with discrete Jacobi operators. Also, we establish that certain maximal operators involving sums of differences of discrete Jacobi semigroups are bounded on weighted $\ell^p$-spaces. $\ell^p$-boundedness properties for the considered operators provide information about the convergence of the semigroup of operators defining them.
In this paper, we establish UMD lattice-valued variational inequalities for differential operators, ergodic averages and analytic semigroups. These results generalize, on the one hand some scalar-valued variational inequalities in ergodic theory, on the other hand Xu's very recent result on UMD lattice-valued maximal inequality. As a consequence, we deduce the jump estimates and obtain quantitative information on the rate of the pointwise convergence.
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