Discrete analogues of second-order Riesz transforms

Discrete analogues of classical operators in harmonic analysis have been widely studied, revealing deep connections with areas such as ergodic theory and analytic number theory. This line of research is commonly known as \emph{Discrete Analogues in Harmonic Analysis (DAHA)}. In this paper, we study the $\ell^p$ norms of discrete analogues of second-order Riesz transforms. Using probabilistic methods, we construct a new class of second-order discrete Riesz transforms $\mathcal{R}^{(jk)}$ on the lattice $\mathbb{Z}^d$, $d \ge 2$. We show that for $1<p<\infty$, their $\ell^p(\mathbb{Z}^d)$ norms coincide with those of the classical second-order Riesz transforms $R^{(jk)}$ on $L^p(\mathbb{R}^d)$ when $j \neq k$, and are comparable up to dimensional constants when $j = k$. The operators $\mathcal{R}^{(jk)}$ differ from the discrete analogue $R^{(jk)}{\mathrm{dis}}$ by convolution with an $\ell^1(\mathbb{Z}^d)$ function. Applications are given to the DAHA of the Beurling–Ahlfors operator. We also show that $\mathcal{R}^{(jk)}$ arise as discrete analogues of certain Calderón–Zygmund operators $\mathbf{R}^{(jk)}$, which differ from $R^{(jk)}$ by convolution with an $L^1(\mathbb{R}^d)$ function. Finally, we conjecture that the $L^p$ norms of $\mathcal{R}^{(jk)}$, $R^{(jk)}{\mathrm{dis}}$, and $\mathbf{R}^{(jk)}$ agree with those of the classical Riesz transforms, known to equal the corresponding martingale transform norms.

💡 Research Summary

The paper investigates discrete analogues of second‑order Riesz transforms on the lattice ℤ^d (d ≥ 2) and establishes sharp ℓ^p‑norm estimates that match those of the classical continuous operators on L^p(ℝ^d). Using a probabilistic framework, the authors construct a new family of discrete operators 𝓡^{(jk)}. The construction starts with the periodic heat kernel H_t(x)=∑_{n∈ℤ^d}P_t(x−n), where P_t is the Euclidean heat kernel. By applying a Doob h‑transform (h=H_t) to a space‑time Brownian motion and then performing a martingale transform with a predictable ±1‑valued sequence, they obtain a space‑time martingale. Conditional expectation of this martingale yields the operator

 𝓡^{(jk)}f(n)=∫_0^∞ ∂_j∂_k H_t(n) dt * f(n),

where ∂_j denotes the discrete difference in the j‑th coordinate. This kernel is essentially the sampled version of the continuous second‑order Riesz kernel K^{(jk)}(x)=c_d x_j x_k/|x|^{d+2}, but with a periodic correction.

The authors then apply Burkholder’s sharp martingale inequalities (for j≠k) and Choi’s non‑symmetric version (for j=k) to obtain upper bounds:

• For j≠k, ‖2 𝓡^{(jk)}‖_{ℓ^p→ℓ^p}=p^−1, where p^=max(p, p/(p−1)).
• For j=k, ‖𝓡^{(jj)}‖_{ℓ^p→ℓ^p}=γ(p), the Choi constant.

Lower bounds are proved via dilation and approximation arguments that compare the discrete kernel with its continuous counterpart. Consequently, for j≠k the ℓ^p‑norms coincide exactly with those of the classical Riesz transforms R^{(jk)} on L^p(ℝ^d); for j=k the norms are comparable up to dimension‑dependent constants C_1, C_2.

A further key result is the relationship between 𝓡^{(jk)} and the traditional discrete Riesz transform R^{(jk)}_{dis} defined by direct sampling of the continuous kernel:

 𝓡^{(jk)} = R^{(jk)}_{dis} + J^{(jk)} *,

where J^{(jk)}∈ℓ^1(ℤ^d) and ‖J^{(jk)}‖{ℓ^1}≤C_d, a constant depending only on the dimension. This shows that the two discrete operators differ by convolution with an ℓ^1 function, providing a simple way to bound ‖R^{(jk)}{dis}‖ by ‖𝓡^{(jk)}‖ plus a dimensional constant, without invoking Calderón‑Zygmund theory.

The paper also defines continuous probabilistic operators 𝔅R^{(jk)} by reversing the discrete construction. These operators satisfy the Calderón‑Zygmund kernel conditions and differ from the classical R^{(jk)} only by convolution with an L^1(ℝ^d) function. Hence 𝓡^{(jk)} can be viewed as the discrete analogue (DAHA) of 𝔅R^{(jk)}.

An application to the Beurling–Ahlfors operator is presented: using the second‑order Riesz transforms, a discrete version of the Beurling–Ahlfors transform is constructed, and its ℓ^p‑norm is shown to align with the known continuous bounds.

The main theorems (Theorem 1.1 and 1.2) summarize these findings:

• For j≠k, the ℓ^p‑norms of 𝓡^{(jk)} equal (p^*−1) and match those of R^{(jk)} and R^{(jk)}_{dis}.
• For j=k, the ℓ^p‑norm of 𝓡^{(jj)} is bounded by the Choi constant γ(p), and is comparable (up to C_1, C_2) with the norm of R^{(jj)}_{dis}.

The authors conjecture that all four families of operators (continuous R^{(jk)}, discrete R^{(jk)}_{dis}, probabilistic discrete 𝓡^{(jk)}, and probabilistic continuous 𝔅R^{(jk)}) have identical ℓ^p/L^p norms for every p and every pair (j,k). They also discuss open problems, notably whether the ℓ^1‑norm of J^{(jk)} can be bounded independently of dimension, which would lead to dimension‑free ℓ^p‑norm estimates for the discrete transforms.

In summary, the paper provides a probabilistic construction of discrete second‑order Riesz transforms, proves sharp ℓ^p‑norm results that parallel the continuous theory, elucidates the precise relationship between various discrete and continuous formulations via ℓ^1/L^1 convolution errors, and opens pathways for further research on dimension‑free bounds and higher‑order discrete singular integrals.