  We consider a family of pseudo differential operators $\{\Delta+ a^\alpha
\Delta^{\alpha/2}; a\in [0, 1]\}$ on $\R^d$ that evolves continuously from
$\Delta$ to $\Delta + \Delta^{\alpha/2}$, where $d\geq 1$ and $\alpha \in (0,
2)$. It gives rise to a family of L\'evy processes \{$X^a, a\in [0, 1]\}$,
where $X^a$ is the sum of a Brownian motion and an independent symmetric
$\alpha$-stable process with weight $a$. Using a recently obtained uniform
boundary Harnack principle with explicit decay rate, we establish sharp bounds
for the Green function of the process $X^a$ killed upon exiting a bounded
$C^{1,1}$ open set $D\subset\R^d$. As a consequence, we identify the Martin
boundary of $D$ with respect to $X^a$ with its Euclidean boundary. Finally,
sharp Green function estimates are derived for certain L\'evy processes which
can be obtained as perturbations of $X^a$.
