Harmonic Analysis associated with a discrete Laplacian

TL;DR

This work develops a comprehensive discrete harmonic-analysis framework for the one-dimensional lattice, examining the discrete Laplacian $\Delta_{ ext{d}}$ via its heat and Poisson semigroups, fractional powers, square functions, and Riesz transforms. By leveraging the heat kernel $G(n,t)= e^{-2t} I_n(2t)$ and vector-valued Calderón–Zygmund theory on weighted $\ell^p$ spaces, the authors establish maximum principles for $(-\Delta_{\text{d}})^\sigma$, $\ell^p(w)$-boundedness of maximal operators and square functions, and boundedness of the discrete Riesz transforms, which are shown to align with the discrete Hilbert transform. They further connect conjugate harmonic functions to Riesz transforms, proving convergence results and Cauchy–Riemann-type relations that mirror the continuous theory. The technical backbone relies on detailed properties and representations of the modified Bessel functions $I_k$, enabling precise kernel estimates and CN-type decompositions in the discrete setting. Overall, the paper extends classical Calderón–Zygmund and harmonic-analysis tools to the discrete lattice, with implications for discrete PDEs and signal processing on $\mathbb{Z}$, including weighted $\ell^p$-theory and connections to discrete Hilbert transforms.

Abstract

It is well-known that the fundamental solution of $$ u_t(n,t)= u(n+1,t)-2u(n,t)+u(n-1,t), \quad n\in\mathbb{Z}, $$ with $u(n,0) =δ_{nm}$ for every fixed $m \in\mathbb{Z}$, is given by $u(n,t) = e^{-2t}I_{n-m}(2t)$, where $I_k(t)$ is the Bessel function of imaginary argument. In other words, the heat semigroup of the discrete Laplacian is described by the formal series $$ W_tf(n) = \sum_{m\in\mathbb{Z}} e^{-2t} I_{n-m}(2t) f(m). $$ By using semigroup theory, this formula allows us to analyze some operators associated with the discrete Laplacian. In particular, we obtain the maximum principle for the discrete fractional Laplacian, weighted $\ell^p(\mathbb{Z})$-boundedness of conjugate harmonic functions, Riesz transforms and square functions of Littlewood-Paley. Interestingly, it is shown that the Riesz transforms coincide essentially with the so called discrete Hilbert transform defined by D. Hilbert at the beginning of the XX century. We also see that these Riesz transforms are limits of the conjugate harmonic functions. The results rely on a careful use of several properties of Bessel functions.

Theorems & Definitions (24)

• Theorem 1
• Theorem 2
• Theorem 3
• Proposition 1
• proof
• Remark 1
• Proposition 2
• proof
• Remark 2
• proof : Proof of Theorem \ref{['thm:fractional-Laplacian']}