  For $d\geq 1$ and $\alpha \in (0, 2)$, consider the family of pseudo
differential operators $\{\Delta+ b \Delta^{\alpha/2}; b\in [0, 1]\}$ on $\R^d$
that evolves continuously from $\Delta$ to $\Delta + \Delta^{\alpha/2}$. In
this paper, we establish a uniform boundary Harnack principle (BHP) with
explicit boundary decay rate for nonnegative functions which are harmonic with
respect to $\Delta +b \Delta^{\alpha/2}$ (or equivalently, the sum of a
Brownian motion and an independent symmetric $\alpha$-stable process with
constant multiple $b^{1/\alpha}$) in $C^{1, 1}$ open sets. Here a "uniform" BHP
means that the comparing constant in the BHP is independent of $b\in [0, 1]$.
Along the way, a uniform Carleson type estimate is established for nonnegative
functions which are harmonic with respect to $\Delta + b \Delta^{\alpha/2}$ in
Lipschitz open sets. Our method employs a combination of probabilistic and
analytic techniques.
