Order-zero Fourier multiplier

An order-zero Fourier multiplier here is $m(D)$ with symbol $m\in C^\infty(\mathbb R^n\setminus\{0\})$ such that

M_k(m):=\max_{|\gamma|\leq k}\sup_{\xi\ne0} |\xi|^{|\gamma|}|\partial^\gamma m(\xi)|<\infty \quad(k=0,1,2,\ldots).

Here $\gamma$ is a multi-index. In particular $m$ is bounded, so it defines an L2 multiplier.

The symbol has smooth derivatives on the punctured frequency space.

Real fields

The condition $m(-\xi)=\overline{m(\xi)}$ ensures that $m(D)$ preserves real functions. A scalar symbol acts componentwise on vector fields and preserves the divergence-free condition.