Multipliers from Sobolev space $H_p^{\text{al}}$ into $H_p^{-\text{al}}$

A function $q(x)$ is said to be a multiplier from the Sobolev space $H_p^{\text{al}}(R^n)$ into $H_p^{-\text{al}}(R^n)$ if the operator $Lf(x)=q(x)f(x)$ is a bounded operator from the first space into the second one. Let $M_p^{\text{al}}$ the the space of such multipliers. In this paper we give the description of the spaces $M_p^{\text{al}}$ provided the condition $\text{al}>min(n\mathrm{/}p,n\mathrm{/}{p}^{\mathrm{\prime }})$. In the case $\text{al}\le min(n\mathrm{/}p,n\mathrm{/}{p}^{\mathrm{\prime }})$ we obtain some embedding theorems for Sobolev spaces with negative smoothness indices into $M_p^{\text{al}}$.