Randomized second order Riesz projections on the Hamming cube

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 $$\mathrm{\parallel }{\mathrm{\Delta }}^{1\mathrm{/}2}f{\mathrm{\parallel }}_{L^p({\mathrm{\Omega }}_n;X)}{\lesssim }_{p,k,X}\mathrm{\parallel }\mathrm{\mid }\mathrm{\nabla }f{\mathrm{\mid }}_X{\mathrm{\parallel }}_{L^p({\mathrm{\Omega }}_n)}$$ 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 $T_{\phi }g(z)={\int }_0^{z}g(\zeta ){\phi }^{\mathrm{\prime }}(\zeta )d\zeta$ on Banach valued $H^{\mathrm{\infty }}$ spaces. A matching lower bound of Volberg \cite{Volberg2024} $$\mathrm{\parallel }{T}_{\phi }:H_d^{\mathrm{\infty }}(\mathbb{D};Y)\to H^{\mathrm{\infty }}(\mathbb{D};Y)\mathrm{\parallel }{\asymp }_{\alpha ,\phi }{d}^{-\alpha }$$ under a nondegenerate boundary singularity assumption is established.