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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)
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)
An interpolation inequality of Nirenberg, involving Le-besgue-space norms of functions and their derivatives, is modified, replacing one of the norms by a Hölder norm. In his paper [1], L. Nirenberg derived the inequality ∇ j u q C∇ m u a p u 1−a r (0.1) which holds for all functions u ∈ C ∞ 0 (R N) with a constant C > 0 independent of u. Here · s is the L(More)
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