In this paper, a necessary and sufficient condition is obtained for the scale
invariant boundary Harnack inequality (BHP in abbreviation) for a large class
of Hunt processes on metric measure spaces that are in weak duality with
another Hunt process. We next consider a discontinuous subordinate Brownian
motion with Gaussian component Xtβ=WStββ in Rd for which the
L\'evy density of the subordinator S satisfies some mild comparability
condition. We show that the scale invariant BHP holds for the subordinate
Brownian motion X in any Lipschitz domain satisfying the interior cone
condition with common angle ΞΈβ(cosβ1(1/dβ),Ο), but fails
in any truncated circular cone with angle ΞΈβ€cosβ1(1/dβ), a
Lipschitz domain whose Lipschitz constant is larger than or equal to
$1/\sqrt{d-1}.