# On the translation invariant operators in $\ell^p(\mathbb{Z}^d)$

## TL;DR

This paper investigates the boundedness of translation invariant operators on discrete lattice spaces, providing a Mikhlin type multiplier theorem and analyzing the boundedness of a discrete wave equation.

## Contribution

It introduces a Mikhlin type multiplier theorem for $ ext{ell}^p( ext{Z}^d)$ and studies $ ext{ell}^p- ext{ell}^q$ boundedness of a discrete wave equation, advancing discrete harmonic analysis.

## Key findings

- Established a Mikhlin type multiplier theorem for discrete spaces.
- Proved $ ext{ell}^p- ext{ell}^q$ boundedness for a discrete wave equation.
- Identified conditions for boundedness of translation invariant operators.

## Abstract

In this paper we study boundedness of translation invariant operators in the discrete space $\ell^p(\mathbb{Z}^d)$. In this context a Mikhlin type multiplier theorem is given, yielding boundedness for certain known operators . We also give $\ell^p-\ell^q$ boundedness of a discrete wave equation.