On some sufficient conditions for Fourier multipliers and absolute convergence of the Fourier transform

We present new sufficient conditions for Fourier multipliers. These conditions are given in terms of simultaneous
behavior of (quasi-)norms of a function in different Lebesgue and Besov spaces. We also provide some sufficient
conditions for the representation of a function as an absolute convergence Fourier integrals in terms of belonging
of a function simultaneously to several spaces of smooth functions.