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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)
Abstract. Pointwise interpolation inequalities, in particular, |∇ku(x)| c (Mu(x))1−k/m (M∇mu(x)) , k < m, and |Izf(x)| c(MIζf(x)) z/Re ζ(Mf(x))1−Re z/Re ζ , 0 < Re z < Re ζ < n, where ∇k is the gradient of order k, M is the Hardy-Littlewood maximal operator, and Iz is the Riesz potential of order z, are proved. Applications to the theory of multipliers in(More)
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)
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)
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