# Hörmander-Mikhlin theorem on the torus

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 $$(\ast)$$?

There are two ways of proceeding. You can impose $$m$$ to be the restriction to $$\mathbb{Z}^n \subset \mathbb{R}^n$$ of a function satisfying the condition $$(\ast)$$ or you can formulate an analogue of condition $$(\ast)$$ with discrete derivatives.