  We prove $L^p\times L^q\rightarrow L^r$ bounds for certain lacunary bilinear
maximal averaging operators with parameters satisfying the H\"older relation
$1/p+1/q=1/r$. The boundedness region that we get contains at least the
interior of the H\"older boundedness region of the associated single scale
bilinear averaging operator. In the case of the lacunary bilinear spherical
maximal function in $d\geq 2$, we prove boundedness for any $p,q\in
(1,\infty]^2$, which is sharp up to boundary; we then show how to extend this
result to a more degenerate family of surfaces where some curvatures are
allowed to vanish. For the lacunary triangle averaging maximal operator, we
have results in $d\geq 7$, and the description of the sharp region will depend
on a sharp description of the H\"older bounds for the single scale triangle
averaging operator, which is still open.
