Related papers: The Best constant for the almost uncentered maxima…

In this paper we study the $L^p$ boundedness of the centred and the uncentred Hardy--Littlewood maximal operators on certain Riemannian manifolds with bounded geometry. Our results complement those of various authors. We show that, under…

Lebesgue space estimates are obtained for the circular maximal function on the Heisenberg group $\mathbb{H}^1$ restricted to a class of Heisenberg radial functions. Under this assumption, the problem reduces to studying a maximal operator…

In this paper we study semi-stable, radially symmetric and decreasing solutions $u\in W^{1,p}(B_1)$ of $-\Delta_p u=g(u)$ in $B_1\setminus\{0\}$, where $B_1$ is the unit ball of $\mathbb{R}^N$, $p>1$, $\Delta_p$ is the $p-$Laplace operator…

We establish weak-type $(1,1)$ bounds for the maximal function associated with ergodic averaging operators modeled on a wide class of thin deterministic sets $B$. As a corollary we obtain the corresponding pointwise convergence result on…

We find sharp conditions for the maximal operator associated with generalized spherical mean Radon transform on radial functions $M^{\a,\b}_t$ to be bounded on power weighted Lebesgue spaces. Moreover, we also obtain the corresponding…

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 $d\ge 3$,…

Radial convolution operators on free groups with nonnegative kernel of weak type $(2,2)$ and of restricted weak type $(2,2)$ are characterized. Estimates of weak type $(p,p)$ are obtained as well for $1<p<2.$

We prove the weak type (1,1) estimate for maximal function of the truncated rough Hilbert transform considered in [9] and [10]

We consider the maximal operator with respect to uncentered cubes on Euclidean space with arbitrary dimension. We prove that for any function with bounded variation, the variation of its maximal function is bounded by the variation of the…

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…

We investigate weak-type $(1, 1)$ boundedness of sparse operators with respect to Lebesgue measure. Specifically, we find the Bellman function maximizing level sets of sparse operators (localized to an interval) and use this to find the…

In the context of radial weights we study the dimension dependence of some weighted inequalities for maximal operators. We study the growth of the $A_1$-constants for radial weights and show the equivalence between the uniform boundedness…

We construct a slightly new noncommutative Calder\'on-Zygmund decomposition by further splitting the bad function. Using this tool, we prove the weak type (1,1) boundedness of noncommutative Calder\'on-Zygmund operators under a class of…

Consider a normal Ornstein--Uhlenbeck semigroup in $\Bbb{R}^n$, whose covariance is given by a positive definite matrix. The drift matrix is assumed to have eigenvalues only in the left half-plane. We prove that the associated maximal…

It is well known that the weak ($1,1$) bounds doesn't hold for the strong maximal operators, but it still enjoys certain weak $L\log L$ type norm inequality. Let $\Phi_n(t)=t(1+(\log^+t)^{n-1})$ and the space $L_{\Phi_n}({\mathbb R^{n}})$…

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…

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,…

In this work, we study the minimization of nonlinear functionals in dimension $d\geq 1$ that depend on a degenerate radial weight $w$. Our goal is to prove the existence of minimizers in a suitable functional class here introduced and to…

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…

We identity the optimal non-infinitesimal direction of descent for a convex function. An algorithm is developed that can theoretically minimize a subset of (non-convex) functions.