  Hardy's inequality on $H^p$ spaces, $p\in(0,1]$, in the context of orthogonal
expansions is investigated for general basis on a subset of $\mathbb{R}^d$ with
Lebesgue measure. The obtained result is applied to various Hermite, Laguerre,
and Jacobi expansions. For that purpose some delicate estimates of the higher
order derivatives for the underlying functions and of the associated heat
kernels are proved. Moreover, sharpness of studied Hardy's inequalities is
justified by a construction of an explicit counterexample, which is adjusted to
all considered settings.
