  We consider the counter images $\maclJ (\rr d)$ and $\maclJ _0(\rr d)$ of
entire functions with exponential and almost exponential bounds, respectively,
under the Bargmann transform, and we characterize them by estimates of powers
of the harmonic oscillator. We also consider the Pilipovi{\'c} spaces $\bsycalS
_s(\rr d)$ and $\bsySig _s(\rr d)$ when $0<s<1/2$ and deduce their images under
the Bargmann transform.
