An Analogue to a Theorem of Fefferman and Phong for Averaging Operators Along Curves with Singular Fractional Integral Kernel

Thus Tf(x) can be viewed as averaging the function f over the curve t → γ(x, t) with respect to the fractional integral kernel k(t). The condition γ(x, 0) = x is a way of saying the curve at x is ”centered at x”, and the condition that ∂γ ∂t (x, t) 6= 0 ensures that the averaging is smooth. In fact, the arguments of this paper will go through with k(t) replaced by k(x, t) satisfying appropriate x-derivative conditions, as will be described at the end of this paper. However, to simplify the exposition we asssume (1.3). It should be pointed out that recent work of Seeger and Wainger [SW] has also dealt with Radon transforms with fractional integral kernel, proving L to L estimates under rather different hypotheses.