Schauder estimates for discrete fractional integrals

TL;DR

This paper extends the theory of discrete fractional integrals by establishing Schauder estimates, analyzing their regularity on discrete H"older spaces, and building on previous work on nonlocal fractional derivatives.

Contribution

It introduces a new approach to defining discrete fractional integrals via semigroup theory and derives regularity results analogous to continuous Schauder estimates.

Findings

Discrete fractional integrals are shown to have regularity properties on discrete H"older spaces.

The paper establishes discrete Schauder estimates for these integrals.

It connects the theory of discrete fractional integrals with nonlocal fractional derivatives.

Abstract

In this note we focus on the discrete fractional integrals as a natural continuation of our previous work about nonlocal fractional derivatives, discrete and continuous. We define the discrete fractional integrals by using the semigroup theory and we study the regularity of {discrete fractional integrals} on the discrete H\"older spaces, which it is known in the differential equations field as the discrete Schauder estimates.