  In this paper we study the large time asymptotic behaviour of the heat
equation with Hardy inverse-square potential on corner spaces
$\mathbb{R}^{N-k}\times (0,\infty)^k$, $k\geq 0$. We first show a new improved
Hardy-Poincar\'{e} inequality for the quantum harmonic oscillator with Hardy
potential. In view of that, we construct the appropriate functional setting in
order to pose the Cauchy problem. Then we obtain optimal polynomial large time
decay rates and subsequently the first term in the asymptotic expansion of the
solutions in $L^2(\mathbb{R}^{N-k}\times (0,\infty)^k)$. Particularly, we
extend and improve the results obtained by V\'{a}zquez and Zuazua (J. Funct.
Anal. 2000), which correspond to the case $k=0$, to any $k\geq 0$. We emphasize
that the higher the value of $k$ the better time decay rates are. We employ a
different and simplified approach than V\'{a}zquez and Zuazua, managing to
remove the usage of spherical harmonics decomposition in our analysis.
