  In this paper, optimal $L^p-L^q$ estimates are obtained for operators which
average functions over polynomial submanifolds, generalizing the $k$-plane
transform. An important advance over previous work is that full $L^p-L^q$
estimates are obtained by methods which have traditionally yielded only
restricted weak-type estimates. In the process, one is lead to make coercivity
estimates for certain functionals on $L^p$ for $p < 1$.
