On the Constants and Extremal Function and Sequence for Hardy Inequalities in $L_p$ and $l_p$

TL;DR

This paper investigates the optimal constants in Hardy inequalities within $L_p$ and $l_p$ spaces, analyzing their behavior, convergence rates, and near-extremal functions and sequences.

Contribution

It provides the exact convergence rates of the constants and identifies functions and sequences that nearly attain the bounds, advancing understanding of Hardy inequalities.

Findings

Determined the exact rate of convergence of the constants $d(a,b)$ and $d_n$.

Identified almost extremal functions and sequences for Hardy inequalities.

Established precise bounds and behaviors of the constants in $L_p$ and $l_p$ contexts.

Abstract

We study the behavior of the smallest possible constants $d(a,b)$ and $d_n$ in Hardy inequalities $$ \int_a^b\left(\frac{1}{x}\int_a^xf(t)dt\right)^p\,dx\leq d(a,b)\,\int_a^b [f(x)]^p dx $$ and $$ \sum_{k=1}^{n}\Big(\frac{1}{k}\sum_{j=1}^{k}a_j\Big)^p\leq d_n\,\sum_{k=1}^{n}a_k^p. $$ The exact rate of convergence of $d(a,b)$ and $d_n$ is established and the ``almost extremal'' function and sequence are found.