Multipliers in the scale of periodic Bessel potential spaces with smoothness indices of different signs

We prove a general type description result for the multipliers acting between two periodic Bessel potential spaces, defined on the $n$–dimensional torus, in a case when their smoothness indices are of different signs. This is done through the detailed examination of a periodic analogue of the linear operator $J_s$, which is employed in the definition of the scale of the Bessel potential space defined on the whole space ${\mathbb{R}}^n$. Our method of defining this periodic analogue of $J_s$ uses the results about an asymptotic behaviour of the generalized Fourier coefficients and existence of a natural homeomorphism between the spaces ${\mathcal{D}}^{\mathrm{\prime }}({\mathbb{T}}^n)$ and $S_{2\cdot \pi }^{\mathrm{\prime }}({\mathbb{R}}^n)$, where the latter consists of all $2\cdot \pi$–periodic distributions from the dual Schwartz space ${\mathcal{S}}^{\mathrm{\prime }}({\mathbb{R}}^n)$.