An embedding theorem for functions whose Fourier transforms are weighted square summable (Q1332668)

Summary: Our main result is, by elementary means only, an embedding theorem for functions \(f\) whose Fourier transforms \(\widetilde{f}\) are weight square summable, i.e. \(\int_{\mathbb{R}^ n} | \widetilde{f} (\lambda)|^ 2 w(\lambda) d\lambda<\infty\), where the weight function \(w: \mathbb{R}^ n\to (0,\infty)\) satisfies the Kolmogorov condition \(\int_{\mathbb{R}^ n} w(\lambda)^{-1} d\lambda <\infty\). The necessary and sufficient condition on a non-negative Borelian measure on \(\mathbb{R}^ n\) is given, for the inequality \(\int_{\mathbb{R}^ n} | f(t)|^ 2 \mu(dt)\leq A\int_{\mathbb{R}^ n} |\widetilde{f} (\lambda)|^ 2 w(\lambda)d\lambda\) to hold for every such function \(f\), with a constant \(A<\infty\) not depending on \(f\).