  In this paper we study the domain of stable processes, stable-like processes
and more general pseudo- and integro-differential operators which naturally
arise both in analysis and as infinitesimal generators of L\'evy- and
L\'evy-type (Feller) processes. In particular we obtain conditions on the
symbol of the operator ensuring that certain (variable order) H\"{o}lder and
H\"{o}lder-Zygmund spaces are in the domain. We use tools from probability
theory to investigate the small-time asymptotics of the generalized moments of
a L\'evy or L\'evy-type process $(X_t)_{t \geq 0}$, \begin{equation*}
  \lim_{t \to 0} \frac 1t\left(\mathbb{E}^x f(X_t)-f(x)\right), \quad
x\in\mathbb{R}^d, \end{equation*} for functions $f$ which are not necessarily
bounded or differentiable. The pointwise limit exists for fixed $x \in
\mathbb{R}^d$ if $f$ satisfies a H\"{o}lder condition at $x$. Moreover, we give
sufficient conditions which ensure that the limit exists uniformly in the space
of continuous functions vanishing at infinity. As an application we prove that
the domain of the generator of $(X_t)_{t \geq 0}$ contains certain H\"{o}lder
spaces of variable order. Our results apply, in particular, to stable-like
processes, relativistic stable-like processes, solutions of L\'evy-driven SDEs
and L\'evy processes.
