Weighted restriction theorems for Bessel potentials (Q1109271)

Consider \(R^ k\) as a subset of \(R^ n\) in the usual way. Fix \(p\in (1,\infty)\) and \(\alpha >(n-k)/p\), \(\alpha\neq n\). Suppose w is a nonnegative Lebesgue-measurable function on \(R^ k\) with \(0<w<\infty\) a.e.; suppose, further, \(w^ p\) is focally-integrable. Extend w to \(R^ n\) by \(w(x_ 1,...,x_ k,...,x_ n)\equiv w(x_ 1,...,x_ k).\) Define the Bessel potential \(J_{\alpha}f\) of Lebesgue-measurable f on \(R^ n\) by \((J_{\alpha}f)(x)=\int_{R^ n}G_{\alpha}(x-y)f(y)dy,\) where \(\hat G_{\alpha}(\xi)=(2\pi)^{-n/2}(1:| \xi |^ 2)^{-\alpha /2}.\) We obtain a necessary condition and a sufficient condition in order that \(C>0\) exist with \[ \int_{R^ k}| (G_{\alpha}f)(x_ 1,...,x_ k,0,...,0)w(x_ 1,...,x_ k)|^ p dx_ 1...dx_ k\leq \] \[ \leq C\int_{R^ n}| f(x_ 1,...,x_ n)w(x_ 1,...,x_ n)|^ p dx_ 1...dx_ n. \]