Aug 2026· 2 citations· ⚡ 1 influential· 14 references
Mathematics
Abstract
We determine the sharp constant in the nearest-neighbor Hardy inequality on $\Z^3$ with the Euclidean inverse-square weight. For every finitely supported function $u:\Z^3\to\C$, we prove \[ \sum_{x\in\Z^3}\sum_{j=1}^3 |u(x+e_j)-u(x)|^2 \geq \frac14\sum_{x\in\Z^3\setminus\{0\}} \frac{|u(x)|^2}{|x|^2}. \] The coefficient $1/4$ is sharp, and equality is not attained by a nonzero finitely supported function. The proof uses an explicit reciprocal edge field and an edgewise completion of squares, together with a concavity argument for the associated vertex weight.
Let $\Omega\subset\mathbb{R}^d$, $d\ge2$, be a bounded connected domain with boundary of class $C^{1,\alpha}$, where $0<\alpha<1$. For the adjoint Neumann--Poincar\'e operator $K^*_{\partial\Omega}$, normalised so that its distinguished eigenvalue is $1/2$, let $\lambda_j^+(\Omega)$ denote the upper min--max values on...
Let \[ \xi\!\left(\frac12+z\right) =\sum_{n\geq 0}\frac{\gamma(n)}{n!}z^{2n}, \qquad J^{d,n}(X) =\sum_{j=0}^{d}\binom dj\gamma(n+j)X^j . \] The Riemann hypothesis is equivalent to the hyperbolicity of $J^{d,n}$ for every $d,n\geq0$. We prove that there is an absolute constant $K>0$ such that \[ n^3\log^2(n+2)\geq Kd^5...
We introduce an explicit deterministic collection of $N$ spherical points, $\cP_N\subset\mathbb S^2$, that we call the {\em deterministic Diamond points}. For every $0<\alpha<2$ we prove that there exists a constant $C_\alpha>0$ such that \[ 0\leq \frac{2^{\alpha+1}}{\alpha+2}N^2-\sum_{x,y\in\cP_N}|x-y|^\alpha \le C_\a...
Carlos Beltrán, J. Marzo, J. Ortega-Cerdà· 0 citations
Let $\mathcal R$ be a normalized root system in $\mathbb R^N$ with a nonnegative multiplicity function $k$, and let $\mathcal F$ be the associated Dunkl transform. We prove a H\"ormander multiplier theorem on the Hardy spaces $H^p_{\mathrm{Dunkl}}$, $0<p\le1$, defined by conical Littlewood--Paley square functions. Let...
Jacek Dziubański, Agnieszka Hejna-Łyżwa· 0 citations
We prove two dimension-free estimates in Gaussian space. The first is an optimal Meyer-type inequality for the Gaussian divergence: for $p\geq2$, \[ \|\delta V\|_{L^p(\gamma_d)} \leq \sqrt p\,|\mathbb E V| +Cp\,\|DV\|_{L^p(\gamma_d;\mathrm{HS}_d)}\, . \] Here $\mathrm{HS}_d$ is the space of $d\times d$ matrices with it...
Fan Chen, Sinho Chewi, Jian-Feng Lu et al.· 1 citation
Let $\mathscr{H}^2$ denote the Hilbert space of Dirichlet series with square-summable coefficients. It follows from a theorem of Bohr that if $f$ in $\mathscr{H}^2$ has a bounded analytic continuation to the right half-plane, then its norm can be computed from the mean values \[\|f\|_{\mathscr{H}^2}^2 = \lim_{\sigma\to...
Ole Fredrik Brevig, Athanasios Kouroupis· 0 citations
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