Weighted spherical restriction theorems for the Fourier transform (Q1186673)

Let \(\Sigma_{n-1}\) be the surface of the unit ball in \(\mathbb{R}^ n\) with the usual surface measure \(d\sigma\), and let \(v\) be a weight function, i.e., a positive measurable function defined on \(\mathbb{R}^ n\). The authors consider weighted versions of restriction theorems for the Fourier transform of the form: \[ \|\hat f\mid_{\Sigma_{n-1}}\|_ q\leq C\| f\|_{L^ p(v,\mathbb{R}^ n)} \] under suitable conditions on \(p\), \(q\) and \(v\). In particular, they study in detail the case when \(v\) is of the form \[ v(x)=\begin{cases} | x|^ \alpha &\text{if }| x|\leq 1\\ | x|^ \beta &\text{if } | x|>1.\end{cases} \] {}.