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

Abstract

In this paper, we are concerned with the optimal dimension-dependent ${\ell }^p$ norm of the discrete Riesz Transforms $R_{\text{dis}}^{(k)}$ on ${\mathbb{Z}}^d$ given by the singular convolution kernel $K_k(m)=c_d{m}_k/|m{|}^{d+1}$, where $c_d=\Gamma (\frac{d+1}{2})/{\pi }^{(d+1)/2}$ . We show that for fixed $1<p<\infty$, when $d\to \infty$ $$\Vert R_{dis}^{\left(k\right)}{\Vert }_{{\ell }^p\left({\mathbb{Z}}^d\right)\to {\ell }^p\left({\mathbb{Z}}^d\right)}=2c_d\left(1+\frac{\left(\sqrt{2}+o\left(1\right)\right)d}{2^{\frac{d}{2}}}\right).$$ The operator norm of $R_{\text{dis}}^{(k)}$ grows super-exponentially as $d\to \infty$ 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 ${\ell }^{1,\infty }$ estimate of $R_{\text{dis}}^{(k)}$ is also established.