Multiplicative sufficient conditions for Fourier multipliers

We get new sufficient conditions for Fourier multipliers in Hardy spaces $H_p(\mathbb R^n)$, $0<p\le 1$, and $L_p(\mathbb R^n)$, $1\le p\le\infty$. Being of a multiplicative character, these conditions are stated in terms of the joint behaviour of ‘norms’ of functions in $L_q(\mathbb R^n)$ and Besov spaces $B_{r,\infty}^s(\mathbb R^n)$.