Discrete H\"older spaces, their characterization via semigroups associated to the discrete Laplacian and kernels estimates

TL;DR

This paper characterizes discrete H"older spaces using heat and Poisson semigroups linked to the discrete Laplacian, enabling direct analysis of fractional powers and potentials without relying on pointwise definitions.

Contribution

It introduces new semigroup-based characterizations of discrete H"older and Zygmund spaces, simplifying regularity analysis of fractional Laplacians and potentials.

Findings

Boundedness of heat and Poisson kernels established

Characterizations facilitate analysis of fractional powers

Results applicable to discrete Zygmund spaces

Abstract

In this paper we characterize the discrete H\"older spaces by means of the heat and Poisson semigroups associated to the discrete Laplacian. These characterizations allow us to get regularity properties of fractional powers of the discrete Laplacian and the Bessel potentials along these spaces and also in the discrete Zygmund spaces in a more direct way than using the pointwise definition of the spaces. To obtain our results, it has been crucial to get boundedness properties of the heat and Poisson kernels and their derivatives in both space and time variables. We believe that these estimates are also of independent interest.