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

In this paper, we are concerned with the optimal dimension-dependent ${\mathrm{\ell }}^p$ norm of the discrete Riesz Transforms $R_{\text{dis}}^{(k)}$ on ${\mathbb{Z}}^d$ given by the singular convolution kernel $K_k(m)=c_d{m}_k\mathrm{/}\mathrm{\mid }m{\mathrm{\mid }}^{d+1}$, where $c_d=\mathrm{\Gamma }(\frac{d+1}{2})\mathrm{/}{\pi }^{(d+1)\mathrm{/}2}$ . We show that for fixed $1<p<\mathrm{\infty }$, when $d\to \mathrm{\infty }$ $$\mathrm{\parallel }{R}_{dis}^{\left(k\right)}{\mathrm{\parallel }}_{{\mathrm{\ell }}^p\left({\mathbb{Z}}^d\right)\to {\mathrm{\ell }}^p\left({\mathbb{Z}}^d\right)}=2c_d(1+\frac{(\sqrt{2}+o\left(1\right))d}{2^{\frac{d}{2}}})\mathrm{.}$$ The operator norm of $R_{\text{dis}}^{(k)}$ grows super-exponentially as $d\to \mathrm{\infty }$ since $c_d\sim (\frac{d-1}{2e\pi }{)}^{\frac{d-1}{2}}\sqrt{\frac{d-1}{\pi }}$ by Stirling’s formula, which gives a negative answer to the conjecture proposed by Bañuelos, Kim and Kwaśnicki in \cite{BKK}. The optimal dimension-dependent ${\mathrm{\ell }}^{1,\mathrm{\infty }}$ estimate of $R_{\text{dis}}^{(k)}$ is also established.