We present sufficient conditions for the transience and the existence of local times of a Feller process, and the ultracontractivity of the associated Feller semigroup; these conditions are sharp for Lévy processes. The proof uses a local symmetrization technique and a uniform upper bound for the characteristic function of a Feller process. As a by-product, we obtain for stable-like processes (in the sense of R. Bass) on
R
d
\mathbb {R}^d
with smooth variable index
α
(
x
)
∈
(
0
,
2
)
\alpha (x)\in (0,2)
a transience criterion in terms of the exponent
α
(
x
)
\alpha (x)
; if
d
=
1
d=1
and
inf
x
∈
R
α
(
x
)
∈
(
1
,
2
)
\inf _{x\in \mathbb {R}} \alpha (x)\in (1,2)
, then the stable-like process has local times.