We consider the spaces Ap​(Tm) of functions f on the m
-dimensional torus Tm such that the sequence of the Fourier
coefficients f^​={f^​(k), k∈Zm} belongs to
lp(Zm), 1≤p<2. The norm on Ap​(Tm) is defined by
∥f∥Ap​(Tm)​=∥f^​∥lp(Zm)​. We study the rate of
growth of the norms ∥eiλφ∥Ap​(Tm)​ as
∣λ∣→∞, λ∈R, for C1 -smooth real
functions φ on Tm (the one-dimensional case was investigated
by the author earlier). The lower estimates that we obtain have direct
analogues for the spaces Ap​(Rm)