发表机构
Seoul National University; Macquarie University; National Taiwan University（首尔大学; 麦考瑞大学; 台湾大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文针对高度复合 regime 下的 lacunary 离散球面极大算子，建立了适配序列的 Orlicz 弱型端点估计，还给出了特定序列下的 $\ell(\log\ell)^2\to\ell^{1,\infty}$ 估计，解决了相关端点行为问题。
AI 中文摘要
设维数 $d\ge5$。对于高度复合 regime 下的半径 lacunary 序列 $\{\lambda^{1/2}_k\}$，即 $\lambda_k=\mu_k!$，且当 $k\to\infty$ 时 $\log \mu_k/\log k\to\infty$，我们考虑与离散球面平均相关的 lacunary 离散球面极大算子 $A_\star f:=\sup_k |A_{\lambda_k}f|$，其中离散球面平均定义为 $A_\lambda f(x):=\frac{1}{s_\lambda}\sum_{\substack{n\in\mathbb{Z}^d\\\\|n|^2=\lambda}} f(x-n)$，$s_\lambda:=\\#\{n\in\mathbb{Z}^d:|n|^2=\lambda\}$。Kesler、Lacey 和 Mena 证明了 $A_\star$ 在 $\ell^p(\mathbb{Z}^d)$ 上对每个 $p>1$ 均有界，并提出了其在 Orlicz 空间尺度上的端点行为问题。本文通过建立如下适配序列的端点估计解决该问题：定义 $\mathcal{N}_\mu(N)=\\#\{k:\mu_k\leq N\}$ 和 $\mathcal{C}_\mu(\theta)=\sup_{N\geq2}\frac{\mathcal{N}_\mu(N)}{N^{\theta}}$，则对任意 $\alpha>0$，有 $\\#\{x\in\mathbb{Z}^d:A_\star f(x)>\alpha\} \lesssim_d \sum_{x\in\mathbb{Z}^d}\frac{|f(x)|}{\alpha} \left(1+\log^+\frac{|f(x)|}{\alpha}\right)^2 \Theta_\mu\\!\left(1+\log^+\frac{|f(x)|}{\alpha}\right)$，其中 $\Theta_\mu(L)=1+\mathcal{C}_\mu(c_d/L)^{3/2}$，$c_d$ 为依赖于维数的常数。我们还在 $\lambda_k=(2^k)!$ 时去掉 $\Theta_\mu$，给出 $\ell(\log\ell)^2\to\ell^{1,\infty}$ 估计。据我们所知，这是首次得到 Kesler、Lacey 和 Mena 提出的 $A_\star f$ 的 Orlicz 弱型端点估计。
英文摘要
Let $d\ge5$, and let $A_λ$ denote the normalized discrete spherical average on $\mathbb{Z}^d$ with squared radius $λ$. For a strictly increasing sequence $(μ_k)$ of positive integers, set $λ_k=μ_k!$ and consider the lacunary maximal operator $A_\star f:=\sup_k|A_{λ_k}f|$. For each fixed $γ>0$, we prove that the estimate \[ \#\{x:A_\star f(x)>α\} \le K\sum_x\frac{|f(x)|}α \left(1+\log^+\frac{|f(x)|}α\right)^γ\] holds with some finite $K$ for all $α>0$ and finitely supported $f$ if and only if \[ \sup_{N\ge2}\frac{\#\{k:μ_k\le N\}}{(1+\log N)^γ}<\infty. \] The summand is interpreted as zero when $f(x)=0$. The case $γ=1$ answers the $\ell\log\ell$ endpoint question of Kesler, Lacey and Mena in the formulation stated above. For $μ_k=2^{2^k}$, these estimates hold for every fixed $γ>0$, but $A_\star$ is not of weak type $(1,1)$. For factorial blocks formed from arbitrary subsets of dyadic grids at widely separated scales, we characterize restricted weak type $(p,p)$, $1<p\le d/(d-1)$, by a uniform covering condition. In particular, for each $1<p_*\le d/(d-1)$, we construct a set of squared radii whose maximal operator is of restricted weak type $(p_*,p_*)$, of strong type $(p,p)$ for $p>p_*$, and unbounded on $\ell^p$ for $1\le p<p_*$.
CommentsTypos are fixed and comments are welcome!
