The best constant for the centered maximal operator on radial decreasing functions
J. M. Aldaz, J. Pérez Lázaro
证明了径向可积函数上中心Hardy-Littlewood极大算子所满足的弱型（1，1）不等式中的最小常数为1。
We show that the lowest constant appearing in the weak type (1,1) inequality satisfied by the centered Hardy-Littlewood maximal operator on radial integrable functions is 1.
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We study the regularity properties of the centered fractional maximal function $M_β$. More precisely, we prove that the map $f \mapsto |\nabla M_βf|$ is bounded and continuous from $W^{1,1}(\mathbb{R}^d)$ to $L^q(\mathbb{R}^d)$ in the endpoint case $q=d/(d-β)$ if $f$ is radial function. For $d=1$, the radiality assumption can be removed. This corresponds to the counterparts of known results for the non-centered fractional maximal function. The main new idea consists in relating the centered and non-centered fractional maximal function at the derivative level.
Centered Hardy-Littlewood maximal function on hyperbolic spaces, $p > 1$
本文证明了实或复双曲空间上中心Hardy-Littlewood极大函数的$L^p$（$p > 1$）维数自由界.
We prove the continuity of the map $f \mapsto (Mf)'$ from $W^{1,1}(\mathbb{R})$ to $L^1(\mathbb{R})$, where $M$ is the centered Hardy-Littlewood maximal operator. This solves a question posed by Carneiro, Madrid and Pierce.
Let $1<p<\infty$. We prove that there exists an $\varepsilon_p>0$ such that for each $f\in L^p(\mathbb{R})$, the centered Hardy-Littlewood maximal operator $M$ on $\mathbb{R}$ satisfies the lower bound $\|Mf\|_{L^p(\mathbb{R})}\ge (1+\varepsilon_p)\|f\|_{L^p(\mathbb{R})}$.
The best constant for centered Hardy-Littlewood maximal inequality
我们找到了一维中心Hardy-Littlewood极大算子的弱型$(1,1)$不等式的最佳可能常数$C$的精确值.我们证明了$C$是二次方程$12C^{2}-22C+5=0$的最大根，从而得到$C=1.5675208...$。这是第一次的最佳常数的基本不等式之一，满足中心极大算子精确评估。
For $1<p<\infty$ and $M$ the centered Hardy-Littlewood maximal operator on $\mathbb{R}$, we consider whether there is some $\varepsilon=\varepsilon(p)>0$ such that $\|Mf\|_p\ge (1+\varepsilon)||f||_p$. We prove this for $1<p<2$. For $2\le p<\infty$, we prove the inequality for indicator functions and for unimodal functions.
Regularity of the Hardy-Littlewood maximal operator on block decreasing functions
研究了作用在块减函数上的通过无条件范数定义的Hardy-Littlewood极大算子.我们证明了非中心极大算子将特殊有界变差的块减函数映射到具有可积分布导数的函数，从而改善了它们的正则性。在由l_infty-范数定义的最大算子的特殊情况下，即通过对立方体求平均，结果扩展到有界变差的块减函数，不一定是特殊的。
For all $p>1$ and all centrally symmetric convex bodies $K\subset \mathbb{R}^d$ define $Mf$ as the centered maximal function associated to $K$. We show that when $d=1$ or $d=2$, we have $||Mf||_p\ge (1+ε(p,K))||f||_p$. For $d\ge 3$, let $q_0(K)$ be the infimum value of $p$ for which $M$ has a fixed point. We show that for generic shapes $K$, we have $q_0(K)>q_0(B(0,1))$.
We show that the non-centered maximal function of a BV function is quasicontinuous. We also show that \emph{if} the non-centered maximal functions of an SBV function is a BV function, then it is in fact a Sobolev function. Using a recent result of Weigt, we are in particular able to show that the non-centered maximal function of a set of finite perimeter is a Sobolev function.
Some new $\left(H_{p},L_{p}\right)$ type inequalities of maximal operators of Vilenkin-Nörlund means with non-decreasing coefficients
本文证明并讨论了非减系数Vilenkin-Nörlund平均极大算子的几个新的$\left(H_{p},L_{p}\right)$型不等式.我们也应用这些不等式证明了这样的Vilenkin-Nörlund手段的强收敛定理。这些不等式在特殊意义上是最好的可能。作为应用，给出了一些著名的和新的结果。
We prove that in a metric measure space $X$, if for some $p \in (1,\infty)$ there are uniform bounds (independent of the measure) for the weak type $(p,p)$ of the centered maximal operator, then $X$ satisfies a certain geometric condition, the Besicovitch intersection property, which in turn implies the uniform weak type $(1,1)$ of the centered operator. Thus, the following characterization is obtained: the centered maximal operator satisfies uniform weak type $(1,1)$ bounds if and only if the space $X$ has the Besicovitch intersection property. In $\mathbb{R}^d$ with any norm, the constants c…
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.
Bounds for maximal functions associated to rotational invariant measures in high dimensions
在最近的文章中，证明了当$μ$是$\real^n$中的有限径向测度且密度有界且径向递减时，相关极大算子$M_μ$的$L^p(μ)$范数随着$p$的小范围值的维数接近1而增长到无穷大。证明了当$μ$是限制在单位球和$p<2$上的Lebesgue测度时，极大算子的$L^p$算子范数在维数上是无界的，即使其作用被限制在径向减函数上.尽管如此，这个最大的运营商承认无量纲$L^p$边界的每一个$p>2$，当限制到径向下降的功能。另一方面，当$μ$是高斯测度时，对于任何有限的$p> 1$，即使在径向递减函数的子空间中，极大算子的$L^p$算子范数也随着维数增长到无穷大。
In this article we characterize all possible cases that may occur in the relations between the sets of $p$ for which weak type $(p,p)$ and strong type $(p,p)$ inequalities for the Hardy--Littlewood maximal operators, both centered and non-centered, hold in the context of general metric measure spaces.
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 non-centered maximal operators related to a non-doubling and non-radial exponential measure
本文研究指数测度下非中心 Hardy–Littlewood 极大算子，证明立方体与菱形情形 p>1 的 L^p 有界性并否定弱 (1,1) 型，欧氏球情形在 d≤4 时得到相应肯定结论。
In this article we study a special class of non-doubling metric measure spaces for which there is a significant difference between the incidence of weak and restricted weak type $(p,p)$ inequalities for the centered and non-centered Hardy--Littlewood maximal operators, $M^c$ and $M$. As a corollary we extend the result obtained in [2].
Two-weight norm inequalities for parabolic fractional maximal functions
本文利用抛物型Muckenhoupt权研究抛物型分数次极大函数，证明了双权弱型估计、Fefferman-Stein型不等式，并在抛物型Sawyer条件及更强bump条件下建立强型估计。
The norm of the centered Hardy--Littlewood maximal operator
本文确定了中心 Hardy--Littlewood 极大算子在实数轴上的精确 $L^p$ 范数为 $p/(p-1)$，解决了尖锐强型问题，并否证了相关猜想。
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…
We investigate the magnitude relation of the non-centered Hardy-Littlewood maximal operators and centered one. By using a discretization technique, we prove two facts: the first one is that the space is ultrametric if and only if the two maximal operators are identical for all discrete measure; the second is, the uncentred maximal operator is strictly greater than the centered one if $(M,d_g)$ is a Riemannian manifold and $μ$ is the Riemannian volume measure.
We establish that the map $f\mapsto |\nabla \mathcal{M}_αf|$ is continuous from $W^{1,1}(\mathbb{R}^d)$ to $L^{q}(\mathbb{R}^d)$, where $α\in (0,d)$, $q=\frac{d}{d-α}$ and $\mathcal{M}_α$ denotes either the centered or non-centered fractional Hardy--Littlewood maximal operator. In particular, we cover the cases $d >1$ and $α\in (0,1)$ in full generality, for which results were only known for radial functions.
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…
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。
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 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 investigate a dichotomy property for Hardy--Littlewood maximal operators, non-centered $M$ and centered $M^c$, that was noticed by Bennett, DeVore and Sharpley. We illustrate the full spectrum of possible cases related to the occurrence or not of this property for $M$ and $M^c$ in the context of non-doubling metric measure spaces $(X, ρ, μ)$. In addition, if $X = \mathbb{R}^d$, $d \geq 1$, and $ρ$ is the metric induced by an arbitrary norm on $\mathbb{R}^d$, then we give the exact characterization (in terms of $μ$) of situations in which $M^c$ possesses the dichotomy property provided that …
An example on the maximal function associated to a nondoubling measure
我们证明了存在一个测度$μ$，定义在双曲平面上，且具有多项式增长，使得与$μ$相关的中心极大算子不满足弱型$(1,1)$界。
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。
On high dimensional maximal operators
本文介绍了极大函数不等式领域的一些新进展。我们还研究了中心的Hardy-Littlewood极大算子的行为与某些家庭的加倍，径向减少措施，并采取行动的径向函数。实际上，我们精确地确定了弱型$(1,1)$边界在维数上是一致的。
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…
We establish continuity mapping properties of the non-centered fractional maximal operator $M_β$ in the endpoint input space $W^{1,1}(\mathbb{R}^d)$ for $d \geq 2$ in the cases for which its boundedness is known. More precisely, we prove that for $q=d/(d-β)$ the map $f \mapsto |\nabla M_βf|$ is continuous from $W^{1,1}(\mathbb{R}^d)$ to $L^{q}(\mathbb{R}^d)$ for $ 0 < β< 1$ if $f$ is radial and for $1 \leq β< d$ for general $f$. The results for $1\leq β< d$ extend to the centered counterpart $M_β^c$. Moreover, if $d=1$, we show that the conjectured boundedness of that map for $M_β^c$ implies i…
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 $\{μ_i\}_{i=1}^{n}$…
In this article we investigate a special class of non-doubling metric measure spaces in order to describe the possible configurations of $P_{k,\rm s}^{\rm c}$, $P_{k,\rm s}$, $P_{k,\rm w}^{\rm c}$ and $P_{k,\rm w}$, the sets of all $p \in [1, \infty]$ for which the weak and strong type $(p,p)$ inequalities hold for the centered and non-centered modified Hardy--Littlewood maximal operators, $M^{\rm c}_k$ and $M_k$, $k \geq 1$. For any fixed $k$ we describe the necessary conditions that $P_{k,\rm s}^{\rm c}$, $P_{k,\rm s}$, $P_{k,\rm w}^{\rm c}$ and $P_{k,\rm w}$ must satisfy in general and illu…
On the variation of the Hardy-Littlewood maximal function
证明了有界变差函数$ f $满足$$ \Var Mf \leq C \Var f $$，其中$ Mf $是$ f $的中心Hardy-Littlewood极大函数.因此，算子$ f \mapsto (Mf)' $从$ W^{1,1}(R) $到$ L^{1}(R) $是有界的。这回答了Hajlasz和Onninen在一维情况下的一个问题。
Dimension dependency of the weak type $(1,1)$ bounds for maximal functions associated to finite radial measures
我们证明了中心Hardy-Littlewood极大函数满足的弱型（1，1）不等式中的最佳常数随维数指数快速增长.
Endpoint mapping properties of spherical maximal operators
对于函数$f\in L^p(\Bbb R^d)$，$d\ge 2$，设$A_t f(x)$为$f$在以$x$为中心的半径为$t$的球面上的平均值。给定一个扩张集$E\subset (0,\infty)$，我们证明了由$M_E f(x)=\sup_{t\in E} |A_t f(x)|$定义的极大算子$M_E$的端点界。
On the lack of dimension free estimates in Lp for maximal functions associated to radial measures
在最近的一篇文章中，J. Aldaz证明了与有限径向测度相关的中心极大算子的弱L1界不能独立于维数。我们表明，至少对于小p接近1，同样的结果适用于这种措施的密度递减的Lp界。我们还给出了一些具体的例子，包括高斯测度，其中获得了相对于一般情况的更好的估计。
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。
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…
A note on boundedness of the Hardy-Littlewood maximal operator on Morrey spaces
本文证明了Hardy-Littlewood极大算子在Morrey空间$\mathcal{M}_{1,λ}(\rn)$，$0 \le \la < n$上是有界的，其中径向减函数是$\rn$上的
We study mapping properties of the centered Hardy--Littlewood maximal operator $\mathcal{M}$ acting on Lorentz spaces $L^{p,q}(\mathfrak{X})$ in the context of certain non-doubling metric measure spaces $\mathfrak{X}$. The special class of spaces for which these properties are very peculiar is introduced and many examples are given. In particular, for each $p_0, q_0, r_0 \in (1, \infty)$ with $r_0 \geq q_0$ we construct a space $\mathfrak{X}$ for which the associated operator $\mathcal{M}$ is bounded from $L^{p_0,q_0}(\mathfrak{X})$ to $L^{p_0,r}(\mathfrak{X})$ if and only if $r \geq r_0$.
Let $f\in L^p(\mathbb{R}^d)$, $d\ge 3$, and let $A_t f(x)$ the average of $f$ over the sphere with radius $t$ centered at $x$. For a subset $E$ of $[1,2]$ we prove close to sharp $L^p\to L^q$ estimates for the maximal function $\sup_{t\in E} |A_t f|$. A new feature is the dependence of the results on both the upper Minkowski dimension of $E$ and the Assouad dimension of $E$. The result can be applied to prove sparse domination bounds for a related global spherical maximal function.
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 $f\in W^{1,1}(\mathbb{R}^n)$.
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。
Counterexamples to Hardy--Littlewood maximal inequalities on solvable groups
本文在三种可解群（灯夫群、Baumslag--Solitar 群、Sol 群）上构造反例，证明中心 Hardy--Littlewood 极大算子对任意 $p\ge1$ 均非弱型，首次给出有限生成群中弱型 $(1,1)$ 失败的例子。
We study different geometric properties on infinite graphs, related to the weak-type boundedness of the Hardy-Littlewood maximal averaging operator. In particular, we analyze the connections between the doubling condition, having finite dilation and overlapping indices, uniformly bounded degree, the equidistant comparison property and the weak-type boundedness of the centered Hardy-Littlewood maximal operator. Several non-trivial examples of infinite graphs are given to illustrate the differences among these properties.
In this paper we study a question related to the continuity of maximal operators of convolution type acting on $W^{1,1}(\mathbb{R})$. We prove that the map $u\mapsto (u^{*})'$ is continuous from $W^{1,1}(\mathbb{R})$ to $L^{1}(\mathbb{R})$, where $u^{*}$ is the maximal function associated to the Poisson kernel, the Heat kernel or a family of kernels related to the fractional Laplacian. This is the first result of this type for a centered maximal operator.
The weak type $(1,1)$ bounds for the maximal function associated to cubes grow to infinity with the dimension
设$M_d$为$\mathbb{R}^d$中立方的中心Hardy-Littlewood极大函数，$c_d$为$M_d$满足的弱型（1，1）不等式中的最小常数.我们证明了$c_d \to \infty$是$d\to \infty$，从而回答了E. M. Stein和J.O.斯特伦伯格
In this paper we consider weighted Morrey spaces ${\mathcal M}_{λ, {\mathcal F}}^p(w)$ adapted to a family of cubes ${\mathcal F}$, with norm $$\|f\|_{{\mathcal M}_{λ, {\mathcal F}}^p(w)}:=\sup_{Q\in {\mathcal F}}\left(\frac{1}{|Q|^λ}\int_Q|f|^pw\right)^{1/p},$$ and the question we deal with is whether a Muckenhoupt-type condition characterizes the boundedness of the Hardy--Littlewood maximal operator on ${\mathcal M}_{λ, {\mathcal F}}^p(w)$. In the case of the global Morrey spaces (when ${\mathcal F}$ is the family of all cubes in ${\mathbb R}^n$) this question is still open. In the case of t…
This paper will be devoted to study the regularity and continuity properties of the following local multilinear fractional type maximal operators, $$\mathfrak{M}_{α,Ω}(\vec{f})(x)=\sup\limits_{0<r<{\rm dist}(x,Ω^c)}\frac{r^α}{|B(x,r)|^m}\prod\limits_{i=1}^m\int_{B(x,r)}|f_i(y)|dy,\quad \hbox{for \ }0\leqα<mn,$$ where $Ω$ is a subdomain in $\mathbb{R}^n$, $Ω^c=\mathbb{R}^n\backslashΩ$ and $B(x,r)$ is the ball in $\mathbb{R}^n$ centered at $x$ with radius $r$. Several new pointwise estimates for the derivative of the local multilinear maximal function $\mathfrak{M}_{0,Ω}$ and the fractional maxi…
A Banach space that distinguishes two maximal operators
本文构造了一个局部可积函数的翻译不变的巴拿赫空间，在该空间上极大卷积算子$M^\diamond$有界，但尖锐极大算子$M^\sharp$无界，回答了Maz'ya提出的75个分析问题中的一个问题。
On the dimension dependence of some weighted inequalities
在径向权的背景下，我们研究了极大算子的一些加权不等式的维数依赖性。研究了径向权的$A_1$ -常数的增长性，证明了这些常数的一致有界性、环上极大算子的无量纲弱$L^1$估计和环上权递减且本质不变的条件之间的等价性.这些条件中的每一个示出提供无量纲加权弱型$L^1$估计的中心最大Hardy-Littlewood算子作用于径向函数。最后证明了当考虑$A_1$权时，具有维数一致有界常数的加权$L^n(\real^n)$上的泛极大算子是限制弱型的.
Let $G=(V,E)$ be a finite graph and $M_G$ be the centered Hardy-Littlewood maximal operator defined there. We find the optimal value $\bf{C}_{G,p}$ such that the inequality $$\text{Var}_{p}(M_{G}f)\leq {\textbf{C}}_{G,p}\text{Var}_{p}(f)$$ holds for every $f:V\to \mathbb{R},$ where $\text{Var}_p$ stands for the $p$-variation, when: (i) $G=K_n$ (complete graph) and $p\in [\frac{\log(4)}{\log(6)},\infty)$ or $G=K_4$ and $p\in (0,\infty)$; (ii) $G=S_n$ (star graph) and $1\ge p\ge \frac{1}{2}$; $p\in (0,\frac{1}{2})$ and $n\ge C(p)$ or $G=S_3$ and $p\in (1,\infty).$ We also find the value of the n…
Behavior of weak type bounds for high dimensional maximal operators defined by certain radial measures
如[A1]所示，中心Hardy-Littlewood极大算子满足的弱型（1，1）不等式中的最低常数随维数指数快速增长.在这里，我们将这个结果推广到更广泛的径向措施和一些值的p>1。此外，我们改进了以前已知的p=1的界限。粗略地说，当p \in（1，1.03]时，如果μ定义为满足某种温和增长条件的径向递减密度，则弱型（p，p）不等式中的最佳常数c_{p,d,μ}满足c_{p,d,μ} \ge 1.005 ^d，对所有充分大的d。我们还表明，指数增长的最佳常数发生某些家庭的加倍措施，并为任意高的值的p。有限径向测度极大函数弱（1，1）型界的维数依赖性。布尔。隆德。Math.Soc.39（2007）203-208.也可以在数学ArXiv上找到。
Let $M_{G}$ be the centered Hardy-Littlewood maximal operator on a finite graph $G$. We find $\underset{p\to \infty}{\lim}\|M_{G}\|_{p}^{p }$ when $G$ is the start graph ($S_n$) and the complete graph ($K_n$), and we fully describe $\|M_{S_n}\|_{p}$ and the corresponding extremizers for $p\in (1,2)$. We prove that $\underset{p\to \infty}{\lim}\|M_{S_n}\|_{p}^{p }=\frac{1+\sqrt{n}}{2}$ when $n\ge 25$. Also, we compute the best constant ${\bf C}_{S_n,2}$ such that for every $f:V\to \mathbb{R}$ we have $Var_{2}M_{S_n}f\le {\bf C}_{S_n,2} Var_{2}f$. We prove that ${\bf C}_{S_n,2}=\frac{(n^2-n-1)^{…
Dimension-free L2 maximal inequality for spherical means in the hypercube
我们确定了标题的结果。在组合术语中，这意味着对于足够小的eps > 0，对于所有n，n维超立方体的顶点的eps分数的任何标记必然留下顶点x，使得标记的顶点是以x为中心的每个球体的少数。
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 …
On the endpoint regularity of discrete maximal operators
给定一个离散函数$f:\Z^d \to \R$，我们考虑极大算子$$Mf(\vec{n}) = \sup_{r\geq0} \frac{1}{N(r)} \sum_{\vec{m} \in \barΩ_r} \big|f(\vec{n} + \vec{m})\big|,$$，其中$\big\{\barΩ_r\big\}_{r \geq 0}$是包含原点的凸集$Ω$（开的、有界的、具有Lipschitz边界的）的扩张，$N(r)$是$\barΩ_r$内部的格点数.本文证明了算子$f \mapsto \nabla M f$是有界的，并且是从$l^1(\Z^d)$到$l^1(\Z^d)$连续的.我们也证明了同样的结果，这个离散的最大算子的非中心的版本。
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…
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