Randomized second order Riesz projections on the Hamming cube

Abstract

In this paper, we improve the arbitrary Banach space \(n \log n\) bound of Ivanisvili--Volberg \cite{IvanisviliVolberg2022} for the second order projection bound to the order \(\sqrt{n}\) bound. Moreover, we study the lower Riesz estimate with the pointwise square gradient, and prove a fixed chaos characterization: on every fixed homogeneous Walsh chaos $H_k$, the dimension free estimate \[ \|Δ^{1/2}f\|_{L^p(Ω_n;X)} \lesssim_{p,k,X} \||\nabla f|_X\|_{L^p(Ω_n)} \] holds for all $n$ if and only if $X$ has Rademacher type $2$. We also consider an exact tail space norm of the analytic paraproduct $T_{\ddot{I}}†g(z)={\int }_0^{z}g(\hat{I}\P )\ddot{I}{†}^{\prime }(\hat{I}\P )d\hat{I}\P$ on Banach valued \(H^\infty\) spaces. A matching lower bound of Volberg \cite{Volberg2024} \[ \|T_φ:H_d^\infty(\mathbb D;Y)\to H^\infty(\mathbb D;Y)\| \asymp_{α,φ} d^{-α} \] under a nondegenerate boundary singularity assumption is established.