1. INTRODUCTION. The development of the famous Hardy inequality (in both its discrete and continuous forms) during the period 1906–1928 has its own history or, as we have called it, prehistory. Contributions of mathematicians other than G. are important here. In this article we describe some of those contributions. We also include and comment upon several… (More)
We present an equivalence theorem, which includes all known characterizations of the class B p , i.e., the weight class of Ariño and Muckenhoupt, and also some new equivalent characterizations. We also give equivalent characterizations for the classes B * p , B * ∞ and RB p , and prove and apply a " gluing lemma " of independent interest.
Let 1 < p ≤ q < ∞. Inspired by some results concerning characterization of weighted Hardy type inequalities, where the equivalence of four scales of integral conditions was proved, we use related ideas to find some new equivalent scales of integral conditions related to the Stieltjes transform. By applying our result to weighted inequalities for the… (More)
Let 1 < p ≤ q < ∞. Inspired by some recent results concerning Hardy type inequalities where the equivalence of four scales of integral conditions was proved, we use related ideas to find ten new equivalence scales of integral conditions. By applying our result to the original Hardy type inequality situation we obtain a new proof of a number of… (More)
We investigate the smoothing effect of the parabolic part of a quasilinear evolutionary equation on its solution as time evolves. More precisely, the following initial-boundary value problem with Dirichlet boundary conditions is considered: u(x, 0) = u0(x) for x ∈ Ω. Here, ∆p stands for the negative Dirichlet p-Laplacian defined by ∆pu ≡ div(|∇u| p−2 ∇u)… (More)