Harmonic analysis associated with a discrete Laplacian

DOI10.1007/S11854-017-0015-6zbMATH Open1476.39003arXiv1401.2091OpenAlexW2963694963MaRDI QIDQ2408403

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Publication date: 12 October 2017

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Abstract: It is well-known that the fundamental solution of ${\displaystyle }$ u_t(n,t)= u(n+1,t)-2u(n,t)+u(n-1,t), quad ninmathbb{Z}, ${\displaystyle }$ with $u(n,0)=delt{a}_{nm}$ for every fixed $minmathbbZ$, 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 ${\displaystyle }$ W_tf(n) = sum_{minmathbb{Z}} e^{-2t} I_{n-m}(2t) f(m). ${\displaystyle }$ 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 $el{l}^p(mathbbZ)$-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.

Full work available at URL: https://arxiv.org/abs/1401.2091

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