Estimates in Beurling–Helson Type Theorems: Multidimensional Case

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 Fourier coefficients $\widehat{f}=\{\widehat{f}(k),\,k\in\mathbb Z^m\}$ belongs to $l^p(\mathbb Z^m)$, $1\le p<2$. The norm on $A_p(\mathbb T^m)$ is defined by $\|f\|_{A_p(\mathbb T^m)}=\|\widehat{f}\|_{l^p(\mathbb Z^m)}$. We study the rate of growth of the norms $\|e^{i\lambda\varphi}\|_{A_p(\mathbb T^m)}$ as $|\lambda|\to\infty$, $\lambda\in\mathbb R$, 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 analogs for the spaces $A_p(\mathbb R^m)$.