Sharp Lp–Lq estimates for generalized k-plane transforms

In this paper, optimal Lp–Lq estimates are obtained for operators which average functions over polynomial submanifolds, generalizing the k-plane transform. An important advance over previous work (e.g., [P. Gressman, Lp-improving properties of X-ray like transforms, Math. Res. Lett. 13 (5) (2006) 787–803]) is that full Lp–Lq estimates are obtained by methods which have traditionally yielded only restricted weak-type estimates. In the process, one is led to make coercivity estimates for certain functionals on Lp for p<1.