It is well known that the class of measurable functions which satisfy
$$
\sup_{|z|>0}\frac{\|f(\cdot+z)+f(\cdot-z)-2f(\cdot)\|_\infty}{|z|^\alpha}<\infty
$$ coincides with the class of Lipschitz functions for $0<\alpha<1$, with the
Zygmund class if $\alpha=1$ and they are the natural extension for
$1<\alpha<2.$ Consider $\mathcal{L}=-\Delta+V$ in $\mathbb{R}^n, \, n\ge 3,$
where $V$ is a nonnegative potential satisfying a reverse H\"older inequality
for $q>n/2$. We define as $C^{\alpha}_{\mathcal{L}},\, 0<\alpha <2,$ the class
of measurable functions such that $$ \|\rho(\cdot)^
\alpha}f(\cdot)\|_\infty<\infty \quad \, \, \text{and}\:\:
\quad
\sup_{|z|>0}\frac{\|f(\cdot+z)+f(\cdot-z)-2f(\cdot)\|_\infty}{|z|^\alpha}<\infty,
$$ where $\rho$ is the critical radius function associated to $\mathcal{L}$.
Let $W_y f = e^{-y\mathcal{L}}f$ be the heat semigroup of $\mathcal{L}$. Given
$\alpha >0,$ we denote by $\Lambda_{\alpha/2}^{{W}}$ the set of functions $f$
which satisfy \begin{equation*} \|\rho(\cdot)^{-\alpha}f(\cdot)\|_\infty<\infty
\hbox{ and } \Big\|\partial_y^k{W}_y f \Big\|_{L^\infty(\mathbb{R}^{n})}\leq
C_\alpha y^{-k+\alpha/2},\;\: \, {\rm with }\, k=[\alpha/2]+1, y>0.
\end{equation*} We prove that for $0<\alpha \le 2-n/q$,
$C^{\alpha}_{\mathcal{L} }= \Lambda_{\alpha/2}^{{W}},$ in the sense of normed
spaces with the obvious norms.
Parallel results are obtained for the classes defined through the Poisson
semigroup, $P_yf= e^{-y\sqrt{\mathcal{L}}}f.$ Regularity properties of
fractional powers (positive and negative) of the operator $\mathcal{L}$,
Schr\"odinger Riesz transforms, Bessel potentials and multipliers of Laplace
transforms type are also obtained. The proofs of these results need in an
essential way the language of semigroups.