  We show that a smooth function $f$ on $\mathbf R^d$ belongs to the
Pilipovi{\'c} space $\mathcal H_{\flat _\sigma}(\mathbf R^d)$ or the
Pilipovi{\'c} space $\mathcal H_{0,\flat _\sigma}(\mathbf R^d)$, if and only if
the $L^p$ norm of $H_d^Nf$ for $N\ge 0$, satisfy certain types of estimates.
Here $H_d=|x|^2-\Delta _x$ is the harmonic oscillator.
