Related papers: On the maximal directional Hilbert transform

Let $n\ge 2$ be an integer. To each irreducible representation $\sigma$ of $\mathrm O(1)$, an $\mathrm {O}(1)$-Kepler problem in dimension $n$ is constructed and analyzed. This system is super integrable and when $n=2$ it is equivalent to a…

Let $1<p\not=2<\infty$ and let $S^p_n$ be the associated Schatten von Neumann class over $n\times n$ matrices. We prove new characterizations of unital positive Schur multipliers $S^p_n\to S^p_n$ which can be dilated into an invertible…

Given Mikhlin-H\"ormander multipliers $m_i$, $i=1,..., N$, with uniform estimates we prove an optimal $\sqrt{\log(N+1)}$ bound in $L^p$ for the maximal function $\sup_i|\cF^{-1}[m_i\hat f]|$ and related bounds for maximal functions…

Kusner asked if $n+1$ points is the maximum number of points in $\mathbb{R}^n$ such that the $\ell_p$ distance $(1<p<\infty)$ between any two points is $1$. We present an improvement to the best known upper bound when $p$ is large in terms…

We prove the $L^p$ boundedness of the circular maximal function on the Heisenberg group $\mathbb{H}^1$ for $2<p\le \infty$. The proof is based on the square sum estimate associated with the $2\times 2$ cone $|(\xi_1',\xi_2')|=…

Let $f\colon X\to Y$ be a $\sigma$-perfect $k$-dimensional surjective map of metrizable spaces such that $\dim Y\leq m$. It is shown that, for every positive integer $p\geq 1$ there exists a dense $G_{\delta}$-subset ${\mathcal H}(k,m,p)$…

We consider a conjecture attributed to Muckenhoupt and Wheeden which suggests a positive relationship between the continuity of the Hardy-Littlewood maximal operator and the Hilbert transform in the weighted setting. Although continuity of…

In recent articles it was proved that when $\mu$ is a finite, radial measure in $\real^n$ with a bounded, radially decreasing density, the $L^p(\mu)$ norm of the associated maximal operator $M_\mu$ grows to infinity with the dimension for a…

We investigate the connection between maximal directional derivatives and differentiability for Lipschitz functions defined on Laakso space. We show that maximality of a directional derivative for a Lipschitz function implies…

The classical Hopf's lemma can be reformulated as uniqueness of continuation result. We aim in the present work to quantify this property. We show precisely that if a solution $u$ of a divergence form elliptic equation attains its maximum…

The goal of this note is to establish non-tangential convergence results for Schr\"{o}dinger operators along restricted curves. We consider the relationship between the dimension of this kind of approach region and the regularity for the…

In this paper we address the following question: given $ p\in (1,\infty)$, $n \geq 1$, does there exists a constant $A(p,n)>1$ such that $\| M f\|_{L^{p}}\geq A(n,p) \| f\|_{L^{p}}$ for any nonnegative $f \in L^{p}(\mathbb{R}^{n})$, where…

For every vector $\overline \alpha\in \RR^n$ and for every rational approximation $(\overline p,q)\in \RR^n\times\RR$ we can associate the displacement vector $q\alpha-\overline p$. We focus on algebraic vectors, namely $\overline…

Let $n\ge 2$ be a positive integer. To each irreducible representation $\sigma$ of $\mathrm{Sp}(1)$, an $\mathrm{Sp}(1)$-Kepler problem in dimension $(4n-3)$ is constructed and analyzed. This system is super integrable and when $n=2$ it is…

We prove that for every $p > 1$ and for every potential $V \in L^p$, any nonnegative function satisfying $-\Delta u + V u \ge 0$ in an open connected set of $\mathbb{R}^N$ is either identically zero or its level set $\{u = 0\}$ has zero…

Let $X$ be a toric variety and $u$ be a normalized symplectic potential of the corresponding polytope $P$. Suppose that the Riemannian curvature is bounded by 1 and $ \int_{\partial P} u ~ d \sigma < C_1, $ then there exists a constant…

Let $\Omega$ be a bounded, smooth domain of $\mathbb{R}^{N},$ $N\geq2.$ For $p>N$ and $1\leq q(p)<\infty$ set \[ \lambda_{p,q(p)}:=\inf\left\{ \int_{\Omega}\left\vert \nabla u\right\vert ^{p}\mathrm{d}x:u\in W_{0}^{1,p}(\Omega)\text{ \ and…

We prove that for all fixed $p > 2$, the translative packing density of unit $\ell_p$-balls in $\mathbb{R}^n$ is at most $2^{(\gamma_p + o(1))n}$ with $\gamma_p < - 1/p$. This is the first exponential improvement in high dimensions since…

The principle of optimizing inequalities, or their equivalent operator theoretic formulation, is well established in analysis. For an operator, this corresponds to extending its action to larger domains, hopefully to the largest possible…

Let $x \in [0,1)$ be an irrational number with continued fraction expansion $[a_1(x),a_2(x), \cdots,a_n(x),\cdots]$ and $q_n(x)$ be the denominator of its $n$-th convergent. We establish, for any $\alpha,\beta$ in $[0,+\infty]$, the…