  We first prove De Giorgi type level estimates for functions in
$W^{1,t}(\Omega)$, $\Omega\subset\mathbb{R}^N$, with $t>N\geq 2$. This
augmented integrability enables us to establish a new Harnack type inequality
for functions which do not necessarily belong to De Giorgi's classes as
obtained in [Di Benedetto--Trudinger, AIHP (1984)] for functions in $W^{1,2}$.
As a consequence, we prove the validity of the strong maximum principle for
uniformly elliptic operators of any even order, in fairly general domains in
dimension two and three, provided second order derivatives are taken into
account.
