Related papers: On directional maximal operators associated with g…

Let $\Omega $ be any set of directions (unit vectors) on the plane. In this paper we study maximal operator of the one dimensional maximal function computed in the directions of $\Omega$ We are interested in extensions of lacunary sets of…

We completely characterize the boundedness of planar directional maximal operators on L^p. More precisely, if Omega is a set of directions, we show that M_Omega, the maximal operator associated to line segments in the directions Omega, is…

We study a two-dimensional discrete directional maximal operator along the set of the prime numbers. We show existence of a set of vectors, which are lattice points in a sufficiently large annulus, for which the $\ell^2$ norm of the…

We introduce a notion of (finite order) lacunarity in higher dimensions for which we can bound the associated directional maximal operators in $L^p(\mathbb{R}^n)$, with $p>1$. In particular, we are able to treat the classes previously…

We develop a notion of finite order lacunarity for direction sets in $\mathbb R^{d+1}$. Given a direction set $\Omega$ that is sublacunary according to this definition, we construct random examples of Euclidean sets that contain unit line…

We introduce the notion of \textit{Perron capacity} of a set of slopes $\Omega \subset \mathbb{R}$. Precisely, we prove that if the Perron capacity of $\Omega$ is finite then the directional maximal operator associated $M_\Omega$ is not…

We show that for any infinite set of unit vectors $U$ in $\ZR^2$ the maximal operator defined by $$ H_Uf(x)=\sup_{u\in U}\bigg|\pv\int_{-\infty}^\infty \frac{f(x-tu)}{t}dt\bigg|,\quad x\in \ZR^2, $$ is not bounded in $L^2(\ZR^2)$.

Given a discrete function $f:\Z^d \to \R$ we consider the maximal operator $$Mf(\vec{n}) = \sup_{r\geq0} \frac{1}{N(r)} \sum_{\vec{m} \in \bar{\Omega}_r} \big|f(\vec{n} + \vec{m})\big|,$$ where $\big\{\bar{\Omega}_r\big\}_{r \geq 0}$ are…

Given any finite direction set $\Omega$ of cardinality $N$ in Euclidean space, we consider the maximal directional Hilbert transform $H_{\Omega}$ associated to this direction set. Our main result provides an essentially sharp uniform bound,…

We prove a weighted inequality which controls conic Fourier multiplier operators in terms of lacunary directional maximal operators. By bounding the maximal operators, this enables us to conclude that the multiplier operators are bounded on…

We prove three results concerning convolution operators and lacunary maximal functions associated to dilates of measures. First, we obtain an $H^1$ to $L^{1,\infty}$ bound for lacunary maximal operators under a dimensional assumption on the…

We prove the validity of maximum principles for a class of fully nonlinear operators on unbounded subdomains $\Omega \subset \mathbb R^n$ of cylindrical type. The main structural assumption is the uniform ellipticity of the operator along…

We establish the sharp growth rate, in terms of cardinality, of the $L^p$ norms of the maximal Hilbert transform $H_\Omega$ along finite subsets of a finite order lacunary set of directions $\Omega \subset \mathbb R^3$, answering a question…

We obtain positive and negative results concerning lacunary discrete maximal operators defined by dilations of sufficiently nonsingular hypersurfaces arising from Diophantine equations in many variables. Our negative results show that this…

We study the lacunary analogue of the $\delta$-discretised spherical maximal operators introduced by Hickman and Jan\v{c}ar, for $\delta \in (0, 1/2)$, and establish the boundedness on $L^p$ for all $1 < p < \infty$, along with the endpoint…

Let $\mathfrak{n}$ be a nonempty, proper, convex subset of $\mathbb{C}$. The $\mathfrak{n}$-maximal operators are defined as the operators having numerical ranges in $\mathfrak{n}$ and are maximal with this property. Typical examples of…

We first prove De Giorgi type level estimates for functions in $W^{1,t}(\Omega)$, $\Omega\subset\mathbb{R}^N$, with $t>N\geq 2$. This augmented integrability enables us to establish a new Harnack type inequality for functions which do not…

We provide a condition on a set of directions $\Omega \subset \mathbb{S}^1$ ensuring that the associated directional maximal operator $M_\Omega$ is unbounded on $L^p(\mathbb{R}^2)$ for every $1 \leq p < \infty$. The techniques of proof…

Let $A$ be a square matrix and let $\Omega$ be an open set in the plane containing the spectrum of $A$. We consider the problem of maximizing the operator norm $\|f(A)\|$ amongst all holomorphic functions $f$ from $\Omega$ into the closed…

We prove new endpoint bounds for the lacunary spherical maximal operator and as a consequence obtain almost everywhere pointwise convergence of lacunary spherical means for functions locally in $L\log\log\log L(\log\log\log\log…