Weighted \(L^2\)-multipliers
The authors consider the action of multiplier operators, defined using the Fourier transform by \(Tf= (m\widehat f)\check{\phantom{f}}\), on \(L^p\) spaces with power weights of the form \(w(x)= |x|^{2k}\), \(k\in\mathbb{Z}\). Using ideas about the multiplier \(m\) that relate the multiplier to powers of \(|x|\) and induction, they provide a simple proof of results of \textit{B. Muckenhoupt}, \textit{R. L. Wheeden} and \textit{W.-S. Young} [Adv. Math. 49, 170-216 (1983; Zbl 0521.42012)] for the case of even power weights. Specifically, they use the Fourier transform result \(\widehat{xf}= \widehat f'\), the Hörmander condition \[ \sup_{R>0} R^{2\alpha-1} \int_{\{R<|x|\leq 2R\}}|D^\alpha m(x)|^2 dx<\infty \] and a condition for a two-weight version of Hardy's inequality \[ \sup_{r>0} \Biggl(\int^\infty_r |U(x)|^2 dx\Biggr)\Biggl(\int^r_0 |V(x)|^{-2} dx\Biggr)< \infty \] to view \(|m^{(\alpha)}|^2\) as a weight and to replace it by appropriate power of \(|x|\).
two-weight version of Hardy's inequality
scientific article; zbMATH DE number 878063