Quantitative estimates in Beurling-Helson type theorems

We consider the spaces $A_p(\mathbb T)$ of functions $f$ on the circle $\mathbb T$ such that the sequence of Fourier coefficients $\widehat f=\{\widehat f(k),\,k\in\mathbb Z\}$ belongs to $l^p$, $1\le p<2$. The norm in $A_p(\mathbb T)$ is defined by $\|f\|_{A_p}=\|\widehat f\|_{l^p}$. We study the rate of growth of the norms $\|e^{i\lambda\varphi}\|_{A_p}$ as $|\lambda|\to\infty$, $\lambda\in\mathbb R$, for $C^1$-smooth real functions $\varphi$ on $\mathbb T$. The results have natural applications to the problem of changes of variable in the spaces $A_p(\mathbb T)$.
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