Marcinkiewicz inequalities based on Stieltjes zeros (Q1298556)

The authors consider the generalized Ditzian-Totik (GDT) weights of the form \[ U(x)= \prod^M_{k=0}|x-t_k|^{\Gamma_k}\widetilde\omega_k(|x-t_k|^{\delta_k}), \] where \(\Gamma_k\in\mathbb{R}\), \(-1= t_0< t_1<\cdots< t_{M- 1}< t_M= 1\), \(\delta_k= {1\over 2}\) if \(k\in\{0, M\}\) and \(\delta_k=1\) otherwise. The function \(\widetilde\omega_k\) is either equal to 1 or is a concave modulus of continuity of the first order, i.e. \(\widetilde\omega_k\) is semi-additive, nonnegative, continuous and nondecreasing on \([0,1]\), \(\widetilde\omega_k(0)= 0\) and \(2\widetilde\omega_k\left({(a+ b)\over 2}\right)\geq \widetilde\omega_k(a)+ \widetilde\omega_k(b)\) for all \(a,b\in [0,1]\), and for every \(\varepsilon> 0\), \(\widetilde\omega_k(x)/x^\varepsilon\) is a nonincreasing function on \((0,1)\) with \(\lim_{x\to 0+}\widetilde\omega_k(x)/x^\varepsilon= \infty\). Their main aim is to find necessary and sufficient conditions for GDT weighted Marcinkiewicz inequalities based at Stieltjes zeros.