Sharp Hardy's Inequalities in Hilbert Spaces

TL;DR

This paper investigates the optimal constants in Hardy's inequalities within Hilbert spaces, determining exact values, convergence rates, and extremal functions, thus advancing the theoretical understanding of these inequalities.

Contribution

It provides the exact value of the constant $d(a,b)$, the convergence rate of $d_n$, and identifies extremal functions and sequences, which were previously unknown.

Findings

Exact constant $d(a,b)$ established.

Convergence rate of $d_n$ determined.

Extremal functions and sequences identified.

Abstract

We study the behavior of the smallest possible constants $d(a,b)$ and $d_n$ in Hardy's inequalities $$ \int_a^b\left(\frac{1}{x}\int_a^xf(t)dt\right)^2\,dx\leq d(a,b)\,\int_a^b [f(x)]^2 dx $$ and $$ \sum_{k=1}^{n}\Big(\frac{1}{k}\sum_{j=1}^{k}a_j\Big)^2\leq d_n\,\sum_{k=1}^{n}a_k^2. $$ The exact constant $d(a,b)$ and the precise rate of convergence of $d_n$ are established and the extremal function and the ``almost extremal'' sequence are found.