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

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.