# Sharp Hardy's inequality for Jacobi and symmetrized Jacobi trigonometric expansions

## TL;DR

This paper establishes sharp Hardy's inequalities for Jacobi and symmetrized Jacobi trigonometric systems across broad parameter ranges, including $L^1$-analogues, expanding the understanding of boundedness and inequalities in these settings.

## Contribution

It proves the sharp Hardy's inequalities for Jacobi systems in the widest parameter ranges and derives their $L^1$-analogues, extending previous results.

## Key findings

- Sharp Hardy's inequality for polynomial systems with $eta,eta
otin(-1, ext{infinity})$
- Sharp Hardy's inequality for function systems with $eta,eta
otin[-1/2, ext{infinity})$
- Established $L^1$-analogues of Hardy's inequality with the same parameter restrictions.

## Abstract

Four Jacobi settings are considered in the context of Hardy's inequality: the trigonometric polynomials and functions, and the corresponding symmetrized systems. In the polynomial cases sharp Hardy's inequality is proved for the type parameters $\alpha,\beta\in(-1,\infty)^d$, whereas in the function systems for $\alpha,\beta\in[-1/2,\infty)^d$. The ranges of these parameters are the widest in which the corresponding orthonormal bases are composed of bounded functions. Moreover, the sharp $L^1$-analogues of Hardy's inequality are obtained with the same restrictions on the parameters $\alpha$ and $\beta$.