Optimal dimension-dependent ${\mathrm{\ell }}^p$ and ${\mathrm{\ell }}^{1,\mathrm{\infty }}$ 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 $$R_{\mathrm{d}\mathrm{i}\mathrm{s}}^{(jk)}f(n)=c_d\sum _{m\in {\mathbb{Z}}^d\setminus \{0\}}\frac{m_j{m}_k}{\mathrm{\mid }m{\mathrm{\mid }}^{d+2}}f(n-m),c_d=\frac{\mathrm{\Gamma }\left(\frac{d+2}{2}\right)}{{\pi }^{d\mathrm{/}2}}\mathrm{.}$$ For $j\mathrm{\ne }k$ and every fixed $1<p<\mathrm{\infty }$, we prove that $$\mathrm{\parallel }{R}_{\mathrm{d}\mathrm{i}\mathrm{s}}^{(jk)}{\mathrm{\parallel }}_{{\mathrm{\ell }}^p\to {\mathrm{\ell }}^p}=c_d[\frac{2}{2^{d\mathrm{/}2}}+(\frac{8}{3}+o(1))\frac{d}{3^{d\mathrm{/}2}}]$$ and $$\frac{2c_d}{2^{d\mathrm{/}2}}\le \mathrm{\parallel }{R}_{\mathrm{d}\mathrm{i}\mathrm{s}}^{(jk)}{\mathrm{\parallel }}_{{\mathrm{\ell }}^1\to {\mathrm{\ell }}^{1,\mathrm{\infty }}}\le c_d[\frac{2}{2^{d\mathrm{/}2}}+(\frac{8}{3}+o(1))\frac{d}{3^{d\mathrm{/}2}}]\mathrm{.}$$ Since $c_d\sim \sqrt{\pi d}{\left(\frac{d}{2\pi e}\right)}^{d\mathrm{/}2}$ by Stirling’s formula, thl ${\mathrm{\ell }}^p$ 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 $1<p<\mathrm{\infty }$, $R_{\mathrm{d}\mathrm{i}\mathrm{s}}^{(jj)}$ is neither bounded from ${\mathrm{\ell }}^p({\mathbb{Z}}^d)$ to ${\mathrm{\ell }}^p({\mathbb{Z}}^d)$ nor of weak type $(1,1)$. Cancellation is restored for the operators $R_{\mathrm{d}\mathrm{i}\mathrm{s}}^{(jj-kk)}=R_{\mathrm{d}\mathrm{i}\mathrm{s}}^{(jj)}-R_{\mathrm{d}\mathrm{i}\mathrm{s}}^{(kk)}$. For every fixed $1<p<\mathrm{\infty }$, $$\mathrm{\parallel }{R}_{\mathrm{d}\mathrm{i}\mathrm{s}}^{(jj-kk)}{\mathrm{\parallel }}_{{\mathrm{\ell }}^p\to {\mathrm{\ell }}^p}=4c_d[1+(1+o(1))\frac{d}{2^{d\mathrm{/}2}}],$$ and $$4c_d\le \mathrm{\parallel }{R}_{\mathrm{d}\mathrm{i}\mathrm{s}}^{(jj-kk)}{\mathrm{\parallel }}_{{\mathrm{\ell }}^1\to {\mathrm{\ell }}^{1,\mathrm{\infty }}}\le 4c_d[1+(1+o(1))\frac{d}{2^{d\mathrm{/}2}}]\mathrm{.}$$