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

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 $\beta$.