  We prove $L^p-$estimates for lacunary maximal functions associated with
uncentered spherical averages and Nikodym averages associated with spheres. In
the process, we require to prove $L^p-$improving properties for single scale
averaging operators in this context. In the second half of the paper, we define
bilinear analogues of uncentered spherical maximal function and the Nikodym
maximal function and study their $L^p-$boundedness properties. Interestingly,
in the bilinear setting, one-dimensional case is also meaningful. We obtain
$L^p-$estimates for lacunary uncentered bilinear maximal functions in all
dimensions.
