On the restriction of the Fourier transform and Fourier series to circles of lacunary radii (Q1100671)

Given in \({\mathbb{R}}^ 2 \)a family \(\{\Gamma_ n\}_{n\in {\mathbb{Z}}}\) of circles centered at the origin of lacunary radii, that is, such that \(R_{n+1}/R_ n\leq p>1\) for every \(n\in {\mathbb{Z}}\) if \(R_ n\) is the radius of the circle \(R_ n\), we obtain the restriction for the Fourier transform \[ (\int_{\Gamma}| \hat f(\xi)| \quad qd\sigma (\xi))^{1/q}\leq C_ p\| f\|_{L\quad p(R^ 2)} \] where \(\Gamma =\cup_{n}\Gamma_ n\), \(\beta +1/q=2/p'\), \(p'>2(1- \beta)/(1+\beta)\), \(-1<\beta \leq 1/3\), \(1/p+1/p'=1\) and \(d\sigma\) is the measure on \(\Gamma_ n\), \(d\sigma =R_ n^{\beta -1}d\mu\), \(d\mu\) being the induced Lebesgue measure on \(\Gamma_ n\). In the second part of this paper we obtain a similar restriction Theorem for double Fourier series in the plane.