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

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 α,β∈(−1,∞)d, whereas in the function systems for α,β∈[−1∕2,∞)d. The ranges of these parameters are the widest in which the corresponding orthonormal bases are composed of bounded functions. Moreover, the sharp L1-analogues of Hardy’s inequality are obtained with the same restrictions on the parameters α and β.