Dimension-free bounds for Riesz transforms on the Hamming cube via a Bellman function

Abstract: We give a Bellman-function proof of the dimension-free estimate [ \Big| \vec{R} f \Big|{L^p(Ω;\,\ell^2)} \lesssim (p-1) \,|f|{L^p(Ω)}, \qquad 2\le p<\infty, ] for the vector of Riesz transforms associated with the Walsh number operator on the Hamming cube $Ω={-1,1}<sup>n$, as well as for locally compact abelian groups, in particular $Ω=\mathbb{Z}<sup>n$. The argument is based on a Poisson semigroup representation, symmetrized estimates along edges of $Ω$, and a two-point inequality. This is the first non noncommutative proof of this result, after the seminal papers of Lust-Piquard and later Junge-Mei-Parcet. According to an example of Lamberton, for $1<p<2$ such a dimension-free bound is known to be false.

• The paper establishes dimension-free Lp bounds for discrete Riesz transforms using a Bellman function approach.
• It introduces a novel Bellman function construction that simplifies noncommutative techniques into a deterministic framework.
• The results confirm optimal linear p-dependence for p ≥ 2 and demonstrate the impossibility of similar bounds for 1 < p < 2.

Dimension-Free Bounds for Riesz Transforms on the Hamming Cube: Bellman Function Approach

Overview and Motivation

The paper "Dimension-free bounds for Riesz transforms on the Hamming cube via a Bellman function" (2606.20289) addresses the estimation of discrete Riesz transforms in product spaces (specifically the Hamming cube and $\mathbb{Z}^n$), providing explicit, dimension-independent $L^p$ bounds for $p \geq 2$ using a Bellman function framework. These results are significant in harmonic analysis and theoretical probability, particularly given their noncommutative origins and the transition to commutative, deterministic proofs. The paper also establishes the optimality of linear dependence on $p$ in these estimates and clarifies the impossibility of similar dimension-free bounds for $1 < p < 2$.

Mathematical Setting

Discrete Riesz Transforms

On the Hamming cube $\{-1, 1\}^n$, the discrete gradient as finite differences and the Walsh number operator are central:

• Discrete Derivative: $D_j f(x) = f(x) - f(x^{(j)})$, with $x^{(j)}$ denoting the $j$-th coordinate flipped.
• Laplacian: $A f = \sum_{j=1}^n D_j^2 f$.
• Riesz Transform: $L^p$0.

Analogous definitions apply for $L^p$1, utilizing the standard basis and forward/backward differences.

Dimension-Free Inequalities

For functions $L^p$2 on $L^p$3 (or $L^p$4), the main theorem asserts:

$L^p$5

where $L^p$6 is the vector of Riesz transforms, $L^p$7 is a universal constant independent of $L^p$8, and $L^p$9 denotes either $p \geq 2$0 or $p \geq 2$1.

The sharpness of the linear $p \geq 2$2 dependence is proven, matching previously known lower bounds and disproving the feasibility of dimension-independent bounds for $p \geq 2$3.

Methodological Innovations

Bellman Function Construction

Unlike prior works reliant on noncommutative techniques, the authors adopt a Bellman function extending the Littlewood-Paley Bellman methods. The construction leverages:

• Poisson Semigroup Representation: Facilitates the integration by parts identity and provides harmonic extension analogs on discrete domains.
• Two-Point Inequality: For discrete settings, the authors derive a novel segment-average inequality, overcoming obstacles due to discrete derivative support.

The Bellman function $p \geq 2$4 is defined as $p \geq 2$5, with $p \geq 2$6 specifically crafted to accommodate convexity and dissipation properties in both continuous and discrete cases.

Bilinear Embedding and Dissipation Estimates

The analysis exploits both space and time derivatives in Poisson extensions, using segment-average Hessian estimates validated via explicit calculations and leveraging inequalities for averages of powers along edge segments. This approach enables direct control over both the discrete gradients and Poisson time derivatives within the Bellman framework.

Main Results

Dimension-Free and Sharp Bounds

• For $p \geq 2$7: There exists $p \geq 2$8 such that

$p \geq 2$9

for all $p$0.

• Sharpness: Linear dependence in $p$1 is optimal on both $p$2 and $p$3, as demonstrated via explicit majority and binomial examples.
• Failure for $p$4: Using Lamberton's construction, it is shown that no dimension-free estimate is possible in this range.

Bellman Function Technical Properties

• Convexity, Size, and Dissipation: Detailed derivatives and Hessian computations ensure that $p$5 satisfies all necessary properties for the embedding and averaging arguments.
• Extension to Locally Compact Abelian Groups: Proofs for the Hamming cube and $p$6 are generalizable to product groups and other discrete settings.

Implications and Future Directions

The results bridge the gap between noncommutative and commutative proofs for dimension-free Riesz bounds, providing conceptually simpler deterministic tools and sharpening understanding of the interplay between discrete and continuous harmonic analysis. The Bellman function methodology offers a template for analogous dimension-free estimates in other discrete structures and operator frameworks.

Theoretical implications include re-contextualization of noncommutative techniques as not strictly necessary for dimension-free bounds in discrete spaces, suggesting broader applicability of convex analytic approaches. Practically, these estimates inform discrete harmonic analysis applications in combinatorics, theoretical computer science (e.g., analysis of Boolean functions), and probabilistic functional inequalities.

Future research may explore:

• Bellman Function Extensions: Broader classes of discrete operators on general product spaces or graphs.
• Nonlinear and Higher-Order Riesz Transforms: Dimension-free bounds for more complex discrete analogs.
• Sharp Constants and $p$7 Regimes: Improved characterizations of lower bounds and critical behavior in the obstacle range.

Conclusion

This paper delivers a comprehensive dimension-free deterministic proof for discrete Riesz transform bounds on the Hamming cube and $p$8, relying on a Bellman function tailored to the discrete setup. It establishes the optimality of linear $p$9 dependence and clarifies the limitations for $1 < p < 2$0. The methodological innovations invite further development of Bellman function techniques in discrete harmonic analysis and related fields.