Hardy’s Inequalities with Remainders and Lamb-Type Equations

Abstract

© 2020, Pleiades Publishing, Ltd. We study Hardy-type integral inequalities with remainder terms for smoothcompactly-supported functions in convex domains of finite inner radius.New $ L_{1} $- and $ L_{p} $-inequalities are obtained with constants depending on the Lamb constant which is the first positive solutionto the special equation for the Bessel function.In some particular cases the constants are sharp.We obtain one-dimensional inequalities and their multidimensional analogs.The weight functions in the spatial inequalities contain powersof the distance to the boundary of the domain.We also prove that some function depending onthe Bessel function is monotone decreasing.This property is essentially used in the proof of the one-dimensional inequalities.The new inequalities extend those by Avkhadiev and Wirths for $ p=2 $to the case of every $ p\geq 1 $.