Let me first recall a particular case of the classical Hörmander-Mikhlin multiplier theorem: Let $m$ be a bounded function on $\mathbb {R} ^{n}$ which is smooth except possibly at the origin, and such that the function $$\text{ $|\xi|^{k}|\nabla ^{k}m(\xi)|$ is bounded for all integers $0\leq k\leq \frac n2+1$}, \tag{$\ast$}$$ then the Fourier multiplier $m(D)$ is a bounded endomorphism of $L^p(\mathbb R^n)$ for all $1 < p < ∞$ and is bounded from $L^1(\mathbb R^n)$ into $L^{1,\infty}(\mathbb R^n)$.

There is no doubt in my mind that an analogous statement holds true on the torus, but precisely, what means analogous? A Fourier multiplier on the torus $\mathbb T^n$ is simply a function $m$ defined on $\mathbb Z^n$ and we have $$ (m(D) u)(x)=\sum_{k\in \mathbb Z^n} m(k) \hat u(k)e^{2π i x\cdot k}. $$ Assuming $m$ bounded is of course enough for $L^2(\mathbb T^n)$ boundedness (and also necessary), but is certainly not sufficient for $L^p(\mathbb T^n)$ boundedness or Marcinkiewicz' $L^1\rightarrow L^{1,\infty}$ continuity.

Question: What is the discrete relevant assumption on $m$ which could replace on the lattice $\mathbb Z^n$ the kind of homogeneity assumption given by $(\ast)$?