Optimal dimension-dependent $\ell^p$ and $\ell^{1,\infty}$ estimates of the discrete Riesz Transforms

Abstract: In this paper, we are concerned with the optimal dimension-dependent $\ell<sup>p$ norm of the discrete Riesz Transforms $R_{\text{dis}}<sup>{(k)}$ on $\mathbb{Z}<sup>d$ given by the singular convolution kernel $K_k(m)=c_d m_k/|m|<sup>{d+1}$, where $c_d=Γ(\frac{d+1}{2})/π<sup>{(d+1)/2}$. We show that for fixed $1&lt;p&lt;\infty$, when $d\to \infty$ $$|R_{dis}^{\left( k \right)}|<em>{\ell ^p\left( \mathbb{Z}^d \right) \rightarrow \ell ^p\left( \mathbb{Z}^d \right)}=2c_d\left( 1+\frac{\left( \sqrt{2}+o\left( 1 \right) \right) d}{2^{{\frac{d}{2}}}} \right) .$$ The operator norm of $R</em>{\text{dis}}<sup>{(k)}$ grows super-exponentially as $d\to\infty$ since $$c_d\sim(\frac{d-1}{2eπ})<sup>{\frac{d-1}{2}}\sqrt{\frac{d-1}π}$$ by Stirling's formula, which gives a negative answer to the conjecture proposed by Bañuelos, Kim and Kwaśnicki in \cite{BKK}. The optimal dimension-dependent $\ell<sup>{1,\infty}$ estimate of $R_{\text{dis}}<sup>{(k)}$ is also established.

• The paper establishes that discrete Riesz transforms have super-exponential ℓ^p norm growth with increasing dimension, refuting the conjectured dimension-free bounds.
• Using a continuous-discrete operator framework and detailed Fourier multiplier analysis, the authors derive sharp asymptotic estimates for ℓ^p, ℓ^{1,∞}, and ℓ^2 spaces.
• The findings have significant implications for discrete harmonic analysis, impacting both theoretical understanding and high-dimensional applications such as signal processing.

Dimension-Dependent $\ell^p$ and $\ell^{1,\infty}$ Estimates for Discrete Riesz Transforms

Background and Motivation

Discrete analogues of classical singular integral operators, such as the Riesz transforms, are fundamental objects in harmonic analysis on groups like $\mathbb{Z}^d$. The continuous Riesz transforms $R^{(k)}$ on $\mathbb{R}^d$ are known to have dimension-free operator norms in $L^p$, specifically $\cot(\frac{\pi}{2p^\ast})$ as proven by Iwaniec and Martin, echoing the result for the Hilbert transform in $d=1$. Recent advances have resolved long-standing issues regarding the $\ell^p$-norm of the discrete Hilbert transform $H_{dis}$, showing it matches its continuous counterpart [BK]. In higher dimensions, Baños, Kim, and Kwaśnicki extended this to probabilistic discrete Riesz transforms and conjectured that the canonical discrete Riesz transforms $\ell^{1,\infty}$0 had dimension-independent $\ell^{1,\infty}$1 norms, i.e., $\ell^{1,\infty}$2 for $\ell^{1,\infty}$3 [BKK]. This paper definitively disproves that conjecture, rigorously characterizing the precise dimension dependence of these norms.

Main Results

The authors establish sharp asymptotic dimension dependence for the operator norms of discrete Riesz transforms. The central finding is that for fixed $\ell^{1,\infty}$4, as $\ell^{1,\infty}$5,

$\ell^{1,\infty}$6

where $\ell^{1,\infty}$7, which itself grows super-exponentially with $\ell^{1,\infty}$8 (Stirling's formula reveals $\ell^{1,\infty}$9).

For $\mathbb{Z}^d$0, they obtain similarly optimal dimension-dependent bounds: $\mathbb{Z}^d$1 as $\mathbb{Z}^d$2.

For $\mathbb{Z}^d$3, strong two-sided estimates are obtained: $\mathbb{Z}^d$4 for absolute constant $\mathbb{Z}^d$5.

The explicit asymptotics and estimates provided definitively show:

• The operator norm of $\mathbb{Z}^d$6 diverges super-exponentially as $\mathbb{Z}^d$7, demonstrating that both the dimension-free conjecture and any upper bound independent of $\mathbb{Z}^d$8 are false, contrary to previous speculation [BKK].

Proof Strategy and Technical Approach

The proofs leverage the "continuous-discrete operator" framework, linking operator norms on $\mathbb{Z}^d$9 with those of certain convolution operators on $R^{(k)}$0. The central device is the reduction to Fourier multiplier analysis for these operators, and careful evaluation of the resulting kernels in both small and large $R^{(k)}$1 regimes.

For $R^{(k)}$2 estimates, detailed analysis of the Fourier multipliers is performed. This utilizes

• The identity involving the gamma function,
• Poisson summation formula for kernel representations,
• Explicit summation and estimation for small and large $R^{(k)}$3 regions,
• Asymptotic expansions to capture the dominant dimension-dependent terms.

For $R^{(k)}$4 and $R^{(k)}$5, optimality is shown by constructing test functions (e.g., mass at origin) and exhibiting the matching upper and lower bounds using advanced harmonic analytic tools and moment asymptotics.

A significant technical point is that the continuous-discrete approach enables transfer of norm bounds between discrete and continuous settings, and can be expected to extend to higher-order discrete Riesz transforms.

Numerical and Contradictory Claims

• The operator norm increases super-exponentially with $R^{(k)}$6, specifically, no dimension-free upper bound exists for $R^{(k)}$7 or $R^{(k)}$8 norms of the discrete Riesz transforms.
• The canonical dimension-independent norm conjecture is invalid not only in its strong form but also for all weaker forms where $R^{(k)}$9 is independent of $\mathbb{R}^d$0.
• The error terms in previous upper bounds are tightly quantified, showing their sharp $\mathbb{R}^d$1-dependence.
• The interpolation across $\mathbb{R}^d$2, $\mathbb{R}^d$3, and $\mathbb{R}^d$4 does not yield optimal constants; explicit computation is necessary.

Implications and Future Directions

The results firmly establish the structural limitations of discrete harmonic analysis on $\mathbb{R}^d$5, revealing inherent dimensional amplification absent in continuous settings. Practically, this means discrete analogues of Calderón-Zygmund theory must account for super-exponential norm scaling when designing discrete singular integrals, especially in high-dimensional applications (e.g., signal processing, lattice data analysis).

Theoretically, the methodology extends to second and higher-order discrete Riesz transforms, suggesting precise dimension-dependent norm characterization can be pursued in broader classes of discrete operators. The explicit bounds obtained also point to fundamental limits on probabilistic representations and convolution kernel constructions in discrete harmonic analysis.

Conclusion

This paper provides a definitive resolution to the optimal norm scaling of discrete Riesz transforms in $\mathbb{R}^d$6 and $\mathbb{R}^d$7 spaces on $\mathbb{R}^d$8. The norms grow super-exponentially in $\mathbb{R}^d$9, disproving prior conjectures of dimension-independence and establishing precise asymptotic bounds. These findings set theoretical and methodological standards for discrete harmonic analysis, with practical ramifications for the analysis of discrete data and operators in high dimensions.

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References

• Baños, Kim, Kwaśnicki, "Sharp $L^p$0 inequalities for discrete singular integrals on the lattice $L^p$1" [BKK]
• Baños, Kwaśnicki, "On the $L^p$2-norm of the discrete Hilbert transform" [BK]