  Let $\mathcal{L}= \partial_t- \Delta_x+|x|^2$. Consider its Poisson semigroup
$e^{-y\sqrt{\mathcal{L}}}$. For $\alpha >0$ define the Parabolic
Hermite-Zygmund spaces
  $$
  \Lambda^\alpha_{\mathcal{L}}=\left\{f: \:f\in L^\infty(\mathbb{R}^{n+1})\:\;
{\rm and} \:\; \left\|\partial_y^k e^{-y\sqrt{\mathcal{L}}} f
\right\|_{L^\infty(\mathbb{R}^{n+1})}\leq C_k y^{-k+\alpha},\;\: {\rm with }\,
k=[\alpha]+1, y>0. \right\},
  $$
  with the obvious norm. It is shown that these spaces have a pointwise
description of H\"older type.
  The fractional powers $\mathcal{L}^{\pm \beta}$ are well defined in these
spaces and the following regularity properties are proved:
  \begin{eqnarray*}
  \alpha, \beta >0, \quad \|\mathcal{L}^{-\beta} f\|_{
\Lambda^{\alpha+2\beta}_{\mathcal{L}}}\le C \|f\|_{
\Lambda^\alpha_{\mathcal{L}}}.
  \end{eqnarray*}
  \begin{eqnarray*}
  0< 2\beta < \alpha, \quad \|\mathcal{L}^\beta
f\|_{\Lambda_{\mathcal{L}}^{\alpha-2\beta}}\le C
\|f\|_{\Lambda^\alpha_{\mathcal{L}}}.
  \end{eqnarray*}
  Parallel results are obtained for the Hermite operator $- \Delta +|x|^2.$ The
proofs use in a fundamental way the semigroup definition of the operators
$\mathcal{L}^{\pm \beta}$ and $(-\Delta+|x|^2)^{\pm \beta}$. The
non-convolution structure of the operators produce an extra difficulty of the
arguments.
