发表机构
Laboratory of Mathematics and Complex Systems (Ministry of Education), School of Mathematical Sciences, Beijing Normal University（北京师范大学数学科学学院，数学与复杂系统教育部重点实验室）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究二阶离散Riesz变换的最优维数依赖估计，证明非对角情形$\ell^p$范数精确公式并否定Bañuelos-Kim猜想，同时揭示对角情形无界性及差算子范数结果。
AI 中文摘要
本文研究二阶离散Riesz变换的最优维数依赖估计，其中算子定义为 \\[ R_{\mathrm{dis}}^{(jk)}f(n) = c_d\sum_{m\in\mathbb Z^d\setminus\{0\}} \frac{m_jm_k}{|m|^{d+2}}f(n-m), \qquad c_d=\frac{\Gamma\left(\frac{d+2}{2}\right)}{\pi^{d/2}}. \\] 对于 $j\neq k$ 及任意固定的 $1<p<\infty$，我们证明 \\[ \\|R_{\mathrm{dis}}^{(jk)}\\|_{\ell^p\to\ell^p} = c_d\left[ \frac{2}{2^{d/2}} + \left(\frac83+o(1)\right)\frac{d}{3^{d/2}} \right] \\] 且 \\[ \frac{2c_d}{2^{d/2}} \leq \\|R_{\mathrm{dis}}^{(jk)}\\|_{\ell^1\to\ell^{1,\infty}} \leq c_d\left[ \frac{2}{2^{d/2}} + \left(\frac83+o(1)\right)\frac{d}{3^{d/2}} \right]. \\] 由于斯特林公式给出 $ c_d\sim \sqrt{\pi d}\left(\frac{d}{2\pi e}\right)^{d/2} $，这些$\ell^p$估计对Bañuelos和Kim在文献\cite{BK2}中提出的猜想给出了否定答案。对角情形展现出截然不同的现象：对每个 $j$ 和每个 $1<p<\infty$，$R_{\mathrm{dis}}^{(jj)}$ 既不是从 $\ell^p(\mathbb Z^d)$ 到 $\ell^p(\mathbb Z^d)$ 的有界算子，也不具有弱 $(1,1)$ 型。对于算子 $R_{\mathrm{dis}}^{(jj-kk)} =R_{\mathrm{dis}}^{(jj)}-R_{\mathrm{dis}}^{(kk)}$，相消性得以恢复。对任意固定的 $1<p<\infty$，\\[ \\|R_{\mathrm{dis}}^{(jj-kk)}\\|_{\ell^p\to\ell^p} = 4c_d\left[ 1+(1+o(1))\frac{d}{2^{d/2}} \right], \\] 且 \\[ 4c_d \leq \\|R_{\mathrm{dis}}^{(jj-kk)}\\|_{\ell^1\to\ell^{1,\infty}} \leq 4c_d\left[ 1+(1+o(1))\frac{d}{2^{d/2}} \right]. \\]
英文摘要
In this paper we investigate the optimal dimension-dependent estimates of the second-order discrete Riesz transforms \[ R_{\mathrm{dis}}^{(jk)}f(n) = c_d\sum_{m\in\mathbb Z^d\setminus\{0\}} \frac{m_jm_k}{|m|^{d+2}}f(n-m), \qquad c_d=\frac{Γ\left(\frac{d+2}{2}\right)}{π^{d/2}}. \] For $j\neq k$ and every fixed $1<p<\infty$, we prove that \[ \|R_{\mathrm{dis}}^{(jk)}\|_{\ell^p\to\ell^p} = c_d\left[ \frac{2}{2^{d/2}} + \left(\frac83+o(1)\right)\frac{d}{3^{d/2}} \right] \] and \[ \frac{2c_d}{2^{d/2}} \leq \|R_{\mathrm{dis}}^{(jk)}\|_{\ell^1\to\ell^{1,\infty}} \leq c_d\left[ \frac{2}{2^{d/2}} + \left(\frac83+o(1)\right)\frac{d}{3^{d/2}} \right]. \] Since $ c_d\sim \sqrt{πd}\left(\frac{d}{2πe}\right)^{d/2} $ by Stirling's formula, thl $\ell^p$ estimates give a negative answer to the conjecture proposed by Bañuelos and Kim in \cite{BK2}. The diagonal case exhibits quite a different phenomenon: for every $j$ and every $1<p<\infty$, $R_{\mathrm{dis}}^{(jj)}$ is neither bounded from $\ell^p(\mathbb Z^d)$ to $\ell^p(\mathbb Z^d)$ nor of weak type $(1,1)$. Cancellation is restored for the operators $R_{\mathrm{dis}}^{(jj-kk)} =R_{\mathrm{dis}}^{(jj)}-R_{\mathrm{dis}}^{(kk)}$. For every fixed $1<p<\infty$, \[ \|R_{\mathrm{dis}}^{(jj-kk)}\|_{\ell^p\to\ell^p} = 4c_d\left[ 1+(1+o(1))\frac{d}{2^{d/2}} \right], \] and \[ 4c_d \leq \|R_{\mathrm{dis}}^{(jj-kk)}\|_{\ell^1\to\ell^{1,\infty}} \leq 4c_d\left[ 1+(1+o(1))\frac{d}{2^{d/2}} \right]. \]
CommentsThis is a follow-up study to our previous work arXiv:2606.19841
