Sharp Hardy's type inequality for Laguerre expansions

Abstract

A method of proving Hardy's type inequality for orthogonal expansions is presented in a rather general setting. Then, sharp multi-dimensional Hardy's inequality associated with the Laguerre functions of convolution type is proved for the type index 𝛼 ∈ [−1/2, ∞)^𝑑. The case of the standard Laguerre functions is also investigated. Moreover, the sharp analogues of Hardy's type inequality involving 𝐿¹ norms in place of 𝐻¹ norms are obtained in both settings.