L Improving Bounds for Averages along Curves

Let n ≥ 2, and let M1 and M2 be two smooth n − 1-dimensional manifolds, each containing a preferred origin 0M1 and 0M2 . We shall abuse notation and write 0 for both 0M1 and 0M2 . For the purposes of integration we shall place a smooth Riemannian metric on M1 and M2, although the exact choice of this metric will not be relevant. All our considerations shall be local to the origin 0. We are interested in the local L improving properties of averaging operators on curves. Before we give the rigorous description of these operators, let us first give an informal discussion. Informally, we assume that we have a smooth assignment x2 7→ γx2 taking points inM2 to curves inM1, with a corresponding dual assignment x1 → γ∗ x1 taking points in M1 to curves in M2, such that x1 ∈ γx2 ⇐⇒ x2 ∈ γ∗ x1 . We then form the operator R taking functions on M1 to functions on M2, defined by