Sharp Fractional Riesz Estimates on the Hypercube

Abstract

Let ${\Omega }_n=\{-1,1{\}}^n$ be the $n$-dimensional hypercube equipped with the normalized uniform measure, let $\nabla$ be the Walsh gradient and let $\Delta$ be the Walsh Laplacian. For every $1<p\le 2$ we prove the following estimate \[ \|\nabla f\|_{L_p(Ω_n;\ell_2^n)} \leq c_{\rm abs}(p-1)^{-2}\|Δ^{1/p}f\|_{L_p(Ω_n)}. \] The exponent $\frac{1}{p}$ is optimal, thus this settles the open problem on the sharp fractional Riesz estimate by Efraim and Lust-Piquard \cite{E-LP2008} which was subsequently highlighted by Ivanisvili and Volberg \cite{I-V2022}. We also establish the higher-order counterpart. As applications of our results, we obtain simpler proofs of the optimal short-time estimate for $\nabla e^{-t\Delta }$, and the Bernstein-Markov type inequality for $d$-bounded degree functions.