Related papers: On unboundedness of maximal operators for directio…

We prove that the maximal operator associated with variable homogeneous planar curves $(t, u t^{\alpha})_{t\in \mathbb{R}}$, $\alpha\not=1$ positive, is bounded on $L^p(\mathbb{R}^2)$ for each $p>1$, under the assumption that…

For a Schwartz function $f$ on the plane and a non-zero $v\in\ZR^2$ define the Hilbert transform of $f$ in the direction $v$ to be $$ H_vf(x)=\text{p.v.}\int_\ZR f(x-vy) \frac{dy}y $$ Let $\zeta$ be a Schwartz function with frequency…

For any dimension $n \geq 2$, we consider the maximal directional Hilbert transform $\mathscr{H}_U$ on $\mathbb R^n$ associated with a direction set $U \subseteq \mathbb S^{n-1}$: \[ \mathscr{H}_Uf(x) := \frac{1}{\pi} \sup_{v \in U} \Bigl|…

For any nonempty set $U\subset\R^+$, we consider the maximal operator $\h^U$ defined as $\h^Uf=\sup_{u\in U}|H^{(u)} f|$, where $H^{(u)}$ represents the Hilbert transform along the monomial curve $u\gamma(s)$. We focus on the…

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,…

Let $H^{(u)}$ be the Hilbert transform along the parabola $(t, ut^2)$ where $u\in \mathbb R$. For a set $U$ of positive numbers consider the maximal function $\mathcal{H}^U \!f= \sup\{|H^{(u)}\! f|: u\in U\}$. We obtain an (essentially)…

Let $\Omega $ be any set of directions (unit vectors) on the plane. We study maximal operators defined by \md0 M_\Omega f(x)=\sup_{\delta >0, \omega \in \Omega} \frac{1}{2\delta}\int_{-\delta}^\delta |f(x+t\omega)|dt. \emd for the…

Let $M^{(u)}$, $H^{(u)}$ be the maximal operator and Hilbert transform along the parabola $(t, ut^2) $. For $U\subset(0,\infty)$ we consider $L^p$ estimates for the maximal functions $\sup_{u\in U}|M^{(u)} f|$ and $\sup_{u\in U}|H^{(u)}…

Assume that $Au=f,\quad (1)$ is a solvable linear equation in a Hilbert space $H$, $A$ is a linear, closed, densely defined, unbounded operator in $H$, which is not boundedly invertible, so problem (1) is ill-posed. It is proved that the…

We prove that the lacunary spherical maximal operator, defined on the $n$-dimensional real hyperbolic space, is bounded on $L^p(\mathbb{H}^n)$ for all $n\ge2$ and $1<p\le\infty$. In particular, the lacunary set is significantly larger than…

We prove the boundedness of the maximal operator and Hilbert transform along certain variable parabolas in $L^p$ for $p>p_0$ with some $p_0\in (1, 2)$. Connections with the Hilbert transform along vector fields and the polynomial Carleson's…

We study the boundedness of the Hilbert transform $H$ and the Hilbert maximal operator $H^*$ on weighted Lorentz spaces $\Lambda^p_u(w)$. We start by giving several necessary conditions that, in particular, lead us to the complete…

We show that maximal operators formed by dilations of Mikhlin-H"ormander multipliers are typically not bounded on $L^p(R^d)$. We also give rather weak conditions in terms of the decay of such multipliers under which $L^p$ boundedness of the…

We investigate the Hilbert transform and the maximal operator along a class of variable non-flat polynomial curves $(P(t),u(x)t)$ with measurable $u(x)$, and prove uniform $L^p$ estimates for $1<p<\infty$. In particular, via the change of…

In this paper we prove that maximal H-monotone operators $T:H^n\rightrightarrows V_1$ whose domain is all the Heisenberg group $H^n$ are locally bounded. This implies that they are upper semicontinuous. As a consequence, maximal…

We study the maximal operator on the variable exponent H\"older spaces in the setting of metric measure spaces. The boundedness is proven for metric measure spaces satisfying an annular decay property. Let us stress that there are no…

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 prove $L^p$ bounds for the maximal operators associated to an Ahlfors-regular variant of fractal percolation. Our bounds improve upon those obtained by I. {\L}aba and M. Pramanik and in some cases are sharp up to the endpoint. A…

Suppose $L(H)$ is the space of all bounded linear operators on a complex Hilbert space $H.$ This article deals with the problem of characterizing the extreme contractions of $L(H)$ with respect to the numerical radius norm on $L(H).$ In…

We analyze spectral properties of the Hilbert $L$-matrix $$\left(\frac{1}{\max(m,n)+\nu}\right)_{m,n=0}^{\infty}$$ regarded as an operator $L_{\nu}$ acting on $\ell^{2}(\mathbb{N}_{0})$, for $\nu\in\mathbb{R}$, $\nu\neq0,-1,-2,\dots$. The…