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 $\alpha \in [-1/2, \infty)^{d}$. The case of the standard Laguerre functions is also investigated. Moreover, the sharp analogues of Hardy's type inequality involving $L^{1}$ norms in place of $H^{1}$ norms are obtained in both settings.