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 $d\to \infty$ $$\|R_{dis}^{\left( k \right)}\|_{\ell ^p\left( \mathbb{Z}^d \right) \rightarrow \ell ^p\left( \mathbb{Z}^d \right)}=2c_d\left( 1+\frac{\left( \sqrt{2}+o\left( 1 \right) \right) d}{2^{\frac{d}{2}}} \right) .$$ 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 Ba\~{n}uelos, Kim and Kwa\'{s}nicki in \cite{BKK}. The optimal dimension-dependent $\ell^{1,\infty}$ estimate of $R_{\text{dis}}^{(k)}$ is also established.