We consider the spaces $A_p(\mathbb T^m)$ of functions $f$ on the $m$ -dimensional torus $\mathbb T^m$ such that the sequence of the Fourier coefficients $\hat{ f}=\{ \hat{ f}(k), ~k \in \mathbb Z^m\}$ belongs to $l^p(\mathbb Z^m), ~1\leq p<2$. The norm on $A_p(\mathbb T^m)$ is defined by $\|f\|_{ A_p(\mathbb T^m)}=\|\hat{ f}\|_{ l^p(\mathbb Z^m)}$. We study the rate of growth of the norms $\|e^{ i\lambda\varphi}\|_{ A_p(\mathbb T^m)}$ as $|\lambda|\rightarrow \infty, ~\lambda\in\mathbb R,$ for $C^1$ -smooth real functions $\varphi$ on $\mathbb T^m$ (the one-dimensional case was investigated by the author earlier). The lower estimates that we obtain have direct analogues for the spaces $A_p(\mathbb R^m)$.
Categories: math.CA