Mixed Norm Estimates for Certain Generalized Radon Transforms

In this paper we investigate the mapping properties in Lebesgue-type spaces of certain generalized Radon transforms defined by integration over curves. Let X and Y be open subsets of R, d ≥ 2, and let Z be a smooth submanifold of X×Y ⊂ R2d of dimension d+1. Assume that the projections π1 : Z → X and π2 : Z → Y are submersions at each point of Z. For each y ∈ Y , let γy = {x ∈ X : (x, y) ∈ Z} = π1π 2 (y). In this case, γy are smooth curves in X which vary smoothly with y ∈ Y . For every y ∈ Y , choose a smooth, non-negative measure σy on γy which varies smoothly with y in the natural sense. A generalized Radon transform T (see e.g. [2, 10, 12]) is defined as an operator taking functions on X to functions on Y via