On the Hilbert transform (Q1093126)

The Hilbert transform of f is given by the formula \(Hf(x)=(1/\pi)\int^{\infty}_{-\infty}(x-t)^{-1}f(t)dt,\) where the integral is taken in the principal value sense. Let \(L^*\) be the collection of all function f such that \((1+| t|)^{-1}f(t)\) is integrable on (-\(\infty,\infty)\), and let \(L^ p_{\alpha}(R)\) be the class of function f for which \(\| f\|_{p,\alpha}=(\int_{R}| f(t)|^ p| t|^{\alpha}dt)^{1/p}<\infty,\) where \(p>0\), \(R=(-\infty,\infty)\) and \(\alpha\in R\). The author obtains some necessary and sufficient conditions for the function \(f\in L^ p_{\alpha}(R)\cap L^*\) to have its Hilbert transform Hf in \(L^ p_{\alpha}(R)\).