Quasiextremals for a Radon-like Transform

where for each x in some ambient manifold, each set Mx is a smooth submanifold of a second ambient manifold,Mx varies smoothly with x, and σx is a smooth multiple of the induced surface measure on Mx. A transversality hypothesis is also imposed, guaranteeing that the transpose of T is a generalized Radon transform in the same sense. As is well known, the particular operator T defined by (1.1) maps L(d+1)/d(Rd) boundedly to Ld+1(Rd), but does not map Lp boundedly to Lq for any other exponents p, q. The localized operator