Quantitative aspects of the Beurling--Helson theorem: Phase functions of a special form

Abstract

We consider the space $A({\mathbb{T}}^d)$ of absolutely convergent Fourier series on the torus ${\mathbb{T}}^d$. The norm on $A({\mathbb{T}}^d)$ is naturally defined by $\Vert f{\Vert }_A=\Vert \hat{f}{\Vert }_{l^1}$, where $\hat{f}$ is the Fourier transform of a function $f$. For real functions $\phi$ of a certain special form on ${\mathbb{T}}^d,d\ge 2,$ we obtain lower bounds for the norms $\Vert e^{i\lambda \phi }{\Vert }_A$ as $\lambda \to \infty$. In particular, we show that if $\phi (x,y)=a(x)|y|$ for $|y|\le \pi$, where $a\in A(\mathbb{T})$ is an arbitrary nonconstant real function, then $\Vert e^{i\lambda \phi }{\Vert }_{A({\mathbb{T}}^2)}\gtrsim |\lambda |$.

The introduction and Remarks modified to improve clarity

References in corpus (2)

• Estimates in Beurling--Helson type theorems. Multidimensional case
• Absolutely convergent Fourier series. An improvement of the Beurling--Helson theorem