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.