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 $$\|\Delta^{1/2}f\|_{L^p(\Omega_n;X)} \lesssim_{p,k,X} \||\nabla f|_X\|_{L^p(\Omega_n)}$$ 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 $T_\varphi g(z)=\int_0^z g(\zeta)\varphi'(\zeta)\,d\zeta$ on Banach valued $H^\infty$ spaces. A matching lower bound of Volberg \cite{Volberg2024} $$\|T_\varphi:H_d^\infty(\mathbb D;Y)\to H^\infty(\mathbb D;Y)\| \asymp_{\alpha,\varphi} d^{-\alpha}$$ under a nondegenerate boundary singularity assumption is established.