Beurling-Helson型定理中的估计多维案例
Estimates in Beurling--Helson type theorems. Multidimensional case
摘要：
我们考虑$m$维环面$\mathbb T^m$上的函数$f$的空间$A_p(\mathbb T^m)$，使得傅立叶系数序列$\hat{f}=\{\hat{f}(k), ~k \in \mathbb Z^m\}$属于$l^p(\mathbb Z^m), ~1\leq p<2$ . $A_p(\mathbb T^m)$上的范数由$\|f\|_{A_p(\mathbb T^m)}=\|\hat{f}\|_{l^p(\mathbb Z^m)}$定义。我们研究了在$\mathbb T^m$上的$C^1$ -光滑实函数$φ$的范数$\|e^{iλφ}\|_{A_p(\mathbb T^m)}$作为$|λ|\rightarrow \infty, ~λ\in\mathbb R,$的增长率（作者在前面研究了一维情况）。我们得到的较低的估计与空间$A_p(\mathbb R^m)$有直接的类似。
英文摘要：
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 $\|e^{iλφ}\|_{A_p(\mathbb T^m)}$ as $|λ|\rightarrow \infty, ~λ\in\mathbb R,$ for $C^1$ -smooth real functions $φ$ 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 $A_p(\mathbb R^m)$.
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