Weighted Hardy Type Inequalities and Parametric Lamb Equation

Abstract

This paper is devoted to weighted Hardy type inequalities. Using the Bessel functions, we prove one-dimensional inequalities and give some remarks on extensions of the one-dimensional inequalities to $$n$$ -dimensional convex domains with finite inner radius. Constants in those inequalities depend on the roots of parametric Lamb equation for the Bessel function and turn out to be sharp in some particular cases. We establish the inequalities in $$L^{p}$$ spaces, with $$p\in[1,\infty)$$ . Also new $$L_{2}$$ - inequality with sharp constants is obtained.