Estimates in Beurling--Helson type theorems. Multidimensional case

TL;DR

This paper investigates how the norms of exponential functions grow in certain Fourier spaces on multidimensional tori as the frequency parameter increases, extending previous one-dimensional results to higher dimensions.

Contribution

It extends Beurling--Helson type theorems to the multidimensional case, providing lower estimates for the growth of Fourier norms of exponential functions with smooth phases.

Findings

Derived lower bounds for the growth of norms ^{i} in A_p spaces as || ightarrow .

Results have direct analogues for spaces A_p(^m).

Generalized previous one-dimensional results to the multidimensional setting.

Abstract

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 $A_p(\mathbb R^m)$.