We determine the sharp asymptotic behavior of the optimal constants in the discrete Hardy-Rellich inequalities on the lattice $\mathbb{Z}^d$. For every fixed integer $m\ge 1$, let ${\mathcal C}_{m,d}$ be the best constant in the $m$-th order Hardy-Rellich inequality. We prove that $$\lim_{d\to\infty}\frac{\mathcal C_{m,d}}{d^m}=2^m.$$ Our approach combines a Fourier reduction to weighted inequalities with the flat torus and general weighted Hardy-Rellich identities of first and second order. A key novelty is the use of probabilistic concentration estimates, specifically Hoeffding's inequality and entropy methods, to handle estimates involving the anisotropic weight $\omega^\gamma$ $(\gamma\geq 1)$ where $$\omega(x)=\sum_{j=1}^{d} \Big(\sin\frac{x_j}{2}\Big)^2$$ in a dimension-uniform manner. These tools yield asymptotically sharp weighted estimates on the torus, from which the lattice inequalities follow by iteration.
The well-known characterization of Hardy spaces $\mathrm{H}_{p}(\mathbb D)$, $0<p<\infty$, in terms of the Littlewood-Paley g-function $$ S f (\zeta) = \left(\int_0^1 |f'(r \zeta)|^2 (1 - r) \mathrm dr\right)^{1/2} \in \mathrm{L}_{p} $$ is generalized to Hardy-type spaces $X_A$ corresponding to quasi-Banach lattices $X...
E. Abakumov, Evgueni Doubtsov, D. Rutsky· 0 citations
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...
Let $\mu$ be a log-concave probability measure on $\mathbb R^n$ and let $f\colon\mathbb R^n\to\mathbb R^k$ be a polynomial mapping of degree at most $d$. We show that \[ \mu(f\in A) \le C\bigl(\lambda_k(A)\bigr)^{\frac{1}{k(d-1)+1}} \] for every Borel set $A\subset\mathbb R^k$ whenever the image measure $\mu\circ f^{-1...
For a countable family of hyperplanes $H_j\subset \mathbb{P}^m$, $j\in\mathbb{N}$, in general position and a linearly nondegenerate holomorphic curve $f\colon \mathbb{C}\to \mathbb{P}^m$ of finite lower order, we prove that the Nevanlinna defects $\delta_f(H_j)$ satisfy $$ \sum_{j=1}^{\infty}\delta_f(H_j)^{1/3}<\infty....
Let $1\leq k\leq n$. We prove that $k$-dimensional intrinsic Lipschitz graphs in the Heisenberg group $\mathbb{H}^n$ satisfy a geometric lemma $\mathrm{GLem}(\beta_{2,\mathcal{V}_k},p)$ for horizontal $\beta$-numbers with an exponent $p=p(k)$. Previously, this result was known only in the case $k=1$; our proof recovers...
Yi-Bo Chen, Katrin Fässler, Kilian Zambanini University of Jyväskylä et al.· 0 citations
For every finite family of Euclidean balls in $\mathbb{R}^n$, $n\ge2$, and every $0<\delta<1/2$, we select a subfamily whose $(1+\delta)$-dilations cover the original union and whose undilated balls have multiplicity at most $C_n\delta^{-(n-1)/2}$. This proves the covering estimate conjectured by Han and Lu \cite{HL}....
Mayukh Mukherjee· 0 citations
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