Related papers: Restricted testing for the Hardy-Littlewood maxima…
The best known upper estimates for the constants of the Hardy--Littlewood inequality for $m$-linear forms on $\ell_{p}$ spaces are of the form $\left(\sqrt{2}\right) ^{m-1}.$ We present better estimates which depend on $p$ and $m$. An…
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…
We introduce and study the median maximal function \mathcal{M} f, defined in the same manner as the classical Hardy-Littlewood maximal function, only replacing integral averages of f by medians throughout the definition. This change has a…
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 show that the Hardy-Littlewood maximal operator is bounded on a reflexive variable Lebesgue space $L^{p(\cdot)}$ over a space of homogeneous type $(X,d,\mu)$ if and only if it is bounded on its dual space $L^{p'(\cdot)}$, where…
We give necessary and sufficient conditions for two weight norm inequalities for Haar multipliers operators and for square functions. We also give sufficient conditions for two weight norm inequalities for the Hilbert transform.
We prove that, given a constant $K> 2$ and a bounded linear operator $T$ from a JB$^*$-triple $E$ into a complex Hilbert space $H$, there exists a norm-one functional $\psi\in E^*$ satisfying $$\|T(x)\| \leq K \, \|T\| \, \|x\|_{\psi},$$…
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 $\mu$ be a $d$-Ahlfors regular measure…
The aim of this paper is to study two-weight norm inequalities for fractional maximal functions and fractional Bergman operator defined on the upper-half space. Namely, we characterize those pairs of weights for which these maximal…
Let $\omega$ be a B\'ekoll\'e-Bonami weight. We give a complete characterization of the positive measures $\mu$ such that $$\int_{\mathcal H}|M_\omega f(z)|^qd\mu(z)\le C\left(\int_{\mathcal H}|f(z)|^p\omega(z)dV(z)\right)^{q/p}$$ and…
We precisely evaluate the operator norm of the uncentered Hardy-Littlewood maximal function on $L^p(\Bbb R^1)$. We also compute the operator norm of the uncentered Hardy-Littlewood maximal function over rectangles on $L^p(\Bbb R^n)$, and we…
In this work we develop a weight theory in the setting of hyperbolic spaces. Our starting point is a variant of the well-known endpoint Fefferman-Stein inequality for the centered Hardy-Littlewood maximal function. This inequality…
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…
The Hardy--Littlewood inequalities for $m$-linear forms have their origin with the seminal paper of Hardy and Littlewood (Q.J. Math, 1934). Nowadays it has been extensively investigated and many authors are looking for the optimal estimates…
The main objective of this work is to bring together two well known and, a priori, unrelated theories dealing with weighted inequalities for the Hardy-Littlewood maximal operator $M$, and thus, we consider the boundedness of $M$ in the…
In the case (4/3)<p<4, and assuming a pair of locally finite positive Borel measures on the real line have no common point masses, we prove variants of two conjectures of T. Hyt\"onen and E. Vuorinen from 2018 on two weight testing theorems…
In recent work by Reguera and Thiele and by Reguera and Scurry, two conjectures about joint weighted estimates for Calder\'on-Zygmund operators and the Hardy-Littlewood maximal function have been refuted in the one-dimensional case. One of…
We consider the question of how the doubling characteristic of a measure determines the regularity of its support. The question was considered by David, Kenig, and Toro for codimension-1 under a crucial assumption of flatness, and later by…
Weighted inequality on the Hardy-Littlewood maximal function is completely understood while it is not well understood for the spherical maximal function. For the power weight $|x|^{\alpha}$, it is known that the spherical maximal operator…
We repurpose tools from the theory of quantitative rectifiability to study the qualitative rectifiability of measures in $\Bbb{R}^n$, $n\geq 2$. To each locally finite Borel measure $\mu$, we associate a function $\widetilde J_2(\mu, x)$…