Estimates in Beurling--Helson type theorems. Multidimensional case

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 $l^p({\mathbb{Z}}^m),1\le p<2$. The norm on $A_p({\mathbb{T}}^m)$ is defined by $\Vert f{\Vert }_{A_p({\mathbb{T}}^m)}=\Vert \hat{f}{\Vert }_{l^p({\mathbb{Z}}^m)}$. We study the rate of growth of the norms $\Vert e^{i\lambda \phi }{\Vert }_{A_p({\mathbb{T}}^m)}$ as $|\lambda |\to \infty ,\lambda \in \mathbb{R},$ for $C^1$ -smooth real functions $\phi$ 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 $A_p({\mathbb{R}}^m)$.

Cited by in corpus (4)

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• Homeomorphic Changes of Variable and Fourier Multipliers
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