Abstract.
Along the way, a uniform Carleson type estimate is established for nonnegative functions which are harmonic with respect to ∆ + ∆^{α/2} in Lipschitz open sets. Our method employs a combination of probabilistic and analytic techniques.
For $dgeq 1$ and $alpha in (0, 2)$, consider the family of pseudo
differential operators ${Delta+ b Delta^{alpha/2}; bin [0,
1]}$ 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
$bin [0, 1]$. 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.