Smoothing theorems for Radon transforms over hypersurfaces and related operators

We extend the theorems of [G1] on $L^p$ to $L_s^p$ Sobolev improvement for translation invariant Radon and fractional singular Radon transforms over hypersurfaces, proving $L^p$ to $L_s^q$ boundedness results for such operators. Here $q\ge p$ but $s$ can be positive, negative, or zero. For many such operators we will have a triangle $Z \subset (0,1) \times (0,1) \times {\mathbb R} $suchthatonehas$L^p$to$L^q_{s}$boundednessfor$({1 \over p}, {1 \over q}, s)$beneath$Z$, and in the case of Radon transforms one does not have $L^p$ to $L_s^q$ boundedness for $(\frac{1}{p},\frac{1}{q},s)$ above the plane containing $Z$, thereby providing a Sobolev space improvement result which is sharp up to endpoints for $(\frac{1}{p},\frac{1}{q})$ below $Z$. This triangle $Z$ intersects the plane $\{(x_1,x_2,x_3):x_3=0\}$, and therefore we also have an $L^p$ to $L^q$ improvement result that is also sharp up to endpoints for certain ranges of $p$ and $q$.