  This paper is concerned with establishing uniform weighted $L^p$-$L^q$
estimates for a class of operators generalizing both Radon-like operators and
sublevel set operators. Such estimates are shown to hold under general
circumstances whenever a scalar inequality holds for certain associated
measures (the inequality is of the sort studied by Oberlin, relating measures
of parallelepipeds to powers of their Euclidean volumes). These ideas lead to
previously unknown, weighted affine-invariant estimates for Radon-like
operators as well as new $L^p$-improving estimates for degenerate Radon-like
operators with folding canonical relations which satisfy an additional
curvature condition of Greenleaf and Seeger for FIOs (building on the ideas of
Sogge and Mockenhaupt, Seeger, and Sogge); these new estimates fall outside the
range of estimates which are known to hold in the generality of the FIO
context.
