Weighted Paley-Wiener theorem on the Hilbert transform (Q616727)

The aim of this paper is to prove the weighted analogues of the Paley-Wiener theorem for odd and even functions. In particular, it is shown that for a weight \(w(x)=|x|^\alpha\), the Hilbert transform is bounded in \(L(w)\), when \(-1<\alpha< 1\) provided that \(g\) is odd and monotone on \(\mathbb{R}_+\) or, when \(-2<\alpha< 0\) provided that \(g\) is even and monotone on \(\mathbb{R}_+\). Thus assuming monotonicity or general monotonicity of \(g\) allows to extend Flett's and Hardy-Littlewood results for \(p= 1\).