Optimal dimension-dependent ${\ell }^p$ and ${\ell }^{1,\infty }$ estimates of the second-order discrete Riesz transforms

Abstract

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 $j\ne k$ 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 $c_d\sim \sqrt{\pi d}{\left(\frac{d}{2\pi e}\right)}^{d/2}$ 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 \[ 4c_d \leq \|R_{\mathrm{dis}}^{(jj-kk)}\|_{\ell^1\to\ell^{1,\infty}} \leq 4c_d\left[ 1+(1+o(1))\frac{d}{2^{d/2}} \right]. \]

This is a follow-up study to our previous work arXiv:2606.19841