Related papers: The maximal function of the Devil's staircase is a…

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…

The regularity of the Hardy-Littlewood maximal function, in both discrete and continuous contexts, and for both centered and noncentered variants, has been subjected to intense study for the last two decades. But efforts so far have…

We give a survey, known and new results on the beingness of fixed points of the maximal operator in the more general settings of metric measure space. In particular, we prove that the fixed points of the uncentered one must be the constant…

In this note we describe some recent advances in the area of maximal function inequalities. We also study the behaviour of the centered Hardy-Littlewood maximal operator associated to certain families of doubling, radial decreasing…

We establish that the map $f\mapsto |\nabla \mathcal{M}_{\alpha}f|$ is continuous from $W^{1,1}(\mathbb{R}^d)$ to $L^{q}(\mathbb{R}^d)$, where $\alpha\in (0,d)$, $q=\frac{d}{d-\alpha}$ and $\mathcal{M}_{\alpha}$ denotes either the centered…

Dependencies of the optimal constants in strong and weak type bounds will be studied between maximal functions corresponding to the Hardy--Littlewood averaging operators over convex symmetric bodies acting on $\mathbb R^d$ and $\mathbb…

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…

For a real-valued function $f$ on a metric measure space $(X,d,\mu)$ the Hardy-Littlewood maximal-function of $f$ is given by the following `supremum-norm':…

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…

Let $(X,\mathcal{B}, \mu, T)$ be an ergodic dynamical system on a non-atomic finite measure space. We assume without loss of generality that $\mu(X)=1.$ Consider the maximal function $\dis R^*:(f, g) \in L^p\times L^q \to R^*(f, g)(x) =…

The Hardy-Littlewood maximal function $\mathcal{M}$ and the trigonometric function $\sin{x}$ are two central objects in harmonic analysis. We prove that $\mathcal{M}$ characterizes $\sin{x}$ in the following way: let $f \in…

In this note, we give a new characterisation of Sobolev $W^{1,1}$ functions among $BV$ functions via Hardy-Littlewood maximal function. Exploiting some ideas coming from the proof of this result, we are also able to give a new…

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,…

We study regularity of the centered Hardy--Littlewood maximal function $M f$ of a function $f$ of bounded variation in $\mathbb R^d$, $d\in \mathbb N$. In particular, we show that at $|D^c f|$-a.e. point $x$ where $f$ has a non-concave…

We prove a rearrangement inequality for the uncentered Hardy-Littlewood maximal function $M_{\mu}$ associate to general measure $\mu$ on $\mathbb{R}$. This inequality is analogous to the Stein's result $cf^{**}(t)\leq(Mf)^{*}(t)\leq C…

Let $M_d$ be the centered Hardy-Littlewood maximal function associated to cubes in $\mathbb{R}^d$ with Lebesgue measure, and let $c_d$ denote the lowest constant appearing in the weak type (1,1) inequality satisfied by $M_d$. We show that…

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 $||Mf||_p\ge (1+\epsilon(p,K))||f||_p$. For $d\ge 3$,…

This is a survey article about recent developments in dimension-free estimates for maximal functions corresponding to the Hardy--Littlewood averaging operators associated with convex symmetric bodies in $\mathbb R^d$ and $\mathbb Z^d$.

We show that the uncentered Hardy-Littlewood maximal operators associated with the Radon measure $\mu$ on $\mathbb{R}^d$ have the uniform lower $L^p$-bounds (independent of $\mu$) that are strictly greater than $1$, if $\mu$ satisfies a…

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.