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 $T_\varphi g(z)=\int_0^z g(ζ)\varphi'(ζ)\,dζ$ 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_{α,\varphi} d^{-α} ] under a nondegenerate boundary singularity assumption is established.

• The paper establishes a sharp √n bound for randomized second order Riesz projections on the Hamming Cube, improving prior n log n estimates.
• It extends the analysis to weighted, anisotropic settings with explicit dependence on weight distribution, adapting Banach space geometry.
• The study characterizes type 2 conditions for lower Riesz inequalities and outlines an iteration principle for higher order Bernstein–Markov estimates.

Randomized Second Order Riesz Projections on the Hamming Cube

Introduction and Context

The study of harmonic analysis on the Hamming cube $\Omega_n = \{-1,1\}^n$ serves as a model where probability theory, Banach space geometry, and discrete Fourier analysis intersect. A central object in this setting, the Riesz transform associated with the discrete Laplacian (number operator), is fundamental in understanding analytic inequalities and Banach space-valued functional inequalities.

Recent works, notably by Ivanisvili and Volberg, have established $n\log n$ upper bounds for Banach space-valued second order Riesz projections, but a dimension dependence reflecting the worst-case behavior as predicted by examples from Hytönen–Naor remained open. The present paper establishes a $\sqrt{n}$ bound for randomized second order Riesz projections mapping into arbitrary Banach spaces, sharply improving previous results and answering standing conjectures about the optimal dependence on $n$.

Main Results

Sharp $\sqrt{n}$ Upper Bound for Second Order Riesz Projections

The primary result is an operator norm bound for the randomized second order Riesz projection on the discrete cube: $$\left(\mathbb{E}_{x,\delta} \left\|\sum_{j=1}^n \delta_j \Delta^{-1} D_j g(x)\right\|_X^p \right)^{1/p} \leq C_p \sqrt{n} \|g\|_{L^p(\Omega_n; X)}$$ for any Banach space $X$, $1 \leq p < \infty$, and function $g: \Omega_n \to X$. This result supersedes the previous $n\log n$ bound and achieves the $n\log n$0 scaling predicted by the Hytönen–Naor example. The proof leverages a refined heat semigroup representation that removes the Banach space geometry from the probabilistic moment calculation, reducing the analysis to scalar moment estimates.

Weighted and Anisotropic Extensions

The argument extends to anisotropic settings where the Laplacian is replaced with a weighted version $n\log n$1 for positive weights $n\log n$2. The optimal constant in this context depends explicitly on the weight distribution: $n\log n$3 yielding the same $n\log n$4 order in the isotropic case, but reflecting more intricate geometry when weights vary.

Lower Riesz Estimate and the Type 2 Condition

A further contribution is the sharp characterization of Banach-space geometry conditions required for lower Riesz-type inequalities involving square gradients. On each fixed homogeneous Walsh chaos $n\log n$5, the authors establish

$n\log n$6

if and only if $n\log n$7 has Rademacher type $n\log n$8. This result is sharp and is proved through a combination of explicit block constructions and Banach-valued decoupling for Rademacher chaos. Notably, finite cotype or cotype $n\log n$9 are not sufficient for dimension-free square gradient Riesz inequalities on the cube, as demonstrated using $\sqrt{n}$0-spaces for $\sqrt{n}$1.

Paraproduct Norms in Analytic Function Spaces

The paper addresses analytic paraproduct operators on vector-valued Hardy spaces, establishing that under a corner singularity of order $\sqrt{n}$2, the operator norm exhibits $\sqrt{n}$3 decay: $\sqrt{n}$4 This matches lower bounds predicted by sharp one-variable function theory (via the Lehman–Warschawski expansion) and confirms that the analytic tail decay cannot generally be improved to $\sqrt{n}$5 in the presence of a nontrivial boundary singularity.

Iteration Principle for Bernstein–Markov Inequalities

A structural iteration principle is established: given a first-order square gradient Bernstein–Markov estimate on Hilbertian sums $\sqrt{n}$6, higher-order Hilbertian square-function estimates follow by iteration, with the $\sqrt{n}$7th-order estimate incurring a multiplicative cost of the first-order constant to the $\sqrt{n}$8th power. Consequently, no further Banach space geometric assumptions are required to control higher-order derivatives once the first order is established.

Numerical and Structural Sharpness

• The $\sqrt{n}$9 scaling for the operator norm is not only optimal but matches explicit constructions in certain Banach spaces, as shown by the Hytönen–Naor example.
• The $n$0 asymptotics for analytic paraproducts are exactly realized for conformal maps with boundary angles $n$1, excluding the possibility of improvement through analytic means alone when $n$2.
• The type $n$3 characterization is shown to be both necessary and sufficient on fixed homogeneous chaos, while the failure of cotype $n$4 sufficiency is concretely demonstrated for $n$5, $n$6.

Theoretical and Practical Implications

The $n$7 dimension dependence for second order Riesz projections resolves a question underpinning a variety of analytic inequalities on product spaces, such as discrete analogues of Riesz transforms, hypercontractivity, and Bernstein–Markov inequalities. The precise dependence on Banach-space geometry for lower bounds deepens the understanding of how probabilistic inequalities transfer from the scalar to vector-valued framework.

The extension to weighted anisotropic operators provides a fine-grained tool for analyzing functional inequalities in non-uniform product settings, relevant to high-dimensional probability and geometric functional analysis.

The sharp paraproduct norm result has ramifications for the analytic theory underpinning tail space estimates and the structure of regularity in Hardy-type spaces, crucial for problems in vector-valued harmonic analysis.

The iteration principle streamlines the understanding of how first order functional inequalities propagate to higher orders, consolidating the structural role of type/cotype and convexity properties in Banach space valued discrete analysis.

Conclusion

This work establishes the sharp dimension dependence for randomized second order Riesz projections in Banach spaces on the Hamming cube, resolving conjectures about optimality and providing a comprehensive extension to weighted settings. The identification of Rademacher type $n$8 as the precise Banach space geometric threshold for homogeneous chaos lower Riesz bounds, and the precise tail behavior for analytic paraproducts, offers both clarity and critical tools for further advances in analysis on discrete high-dimensional structures. The implications permeate the study of functional inequalities, the geometry of Banach spaces, and the development of analytic methods in high-dimensional discrete harmonic analysis.

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Reference:

"Randomized second order Riesz projections on the Hamming cube" (2606.28793)