Strongly singular Radon transforms on the Heisenberg group and folding singularities

Abstract

Abstract. We prove sharp L 2 regularity results for classes of strongly singular Radon transforms on the Heisenberg group by means of oscillatory integrals. We show that the problem in question can be effectively treated by establishing uniform estimates for certain oscillatory integrals whose canonical relations project with two-sided fold singularities; this new approach also allows us to treat operators which are not necessarily translation invariant.

Strongly singular integrals along curves

“…It follows from the proposition below that we may, with no loss in generality, assume that our curves γ(t) are of standard type; that is approximately homogeneous, taking the form (7) γ k (t) = t a k a k ! + higher order terms…”

Estimates for oscillatory strongly singular integral operators

“…Combining two estimates we can obtain a certain L 2 regularizing property of R. It would be interesting if one can obtain sharp L 2 improving estimates for R in mixed Sobolev spaces. Various L p estimates in various contexts for R can be found in [1,[5][6][7][8].…”