Optimal dimension-dependent $\ell^p$ and $\ell^{1,\infty}$ estimates of the second-order discrete Riesz transforms
In this paper we investigate the optimal dimension-dependent estimates of the second-order discrete Riesz transforms \[ R_{\mathrm{dis}}^{(jk)}f(n) = c_d\sum_{m\in\mathbb Z^d\setminus\{0\}} \frac{m_jm_k}{|m|^{d+2}}f(n-m), \qquad c_d=\frac{\Gamma\left(\frac{d+2}{2}\right)}{\pi^{d/2}}. \] 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{\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 Ba\~nuelos 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 \[ 4c_d \leq \|R_{\mathrm{dis}}^{(jj-kk)}\|_{\ell^1\to\ell^{1,\infty}} \leq 4c_d\left[ 1+(1+o(1))\frac{d}{2^{d/2}} \right]. \]