Some \(l^p\)-Improving Bounds For Radon-Like Transforms

We prove (L^p-L^q) boundedness for a wide class of Radon-like transforms. The technique of proof leverages the existing one-dimensional theory to produce a non-trivial bounds in any dimension. For certain combinatorially simple transforms, this range is sharp up to endpoints. Additionally, we make observations connecting the (L^p)-improving properties of a Radon-like transform to the zero set of certain homogeneous polynomials.