  We extend the theorems of [G1] on $L^p$ to $L^p_s$ Sobolev improvement for
translation invariant Radon and fractional singular Radon transforms over
hypersurfaces, proving $L^p$ to $L^q_s$ boundedness results for such operators.
Here $q \geq p$ 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} $ such that one has $L^p$ to $L^q_{s}$ boundedness for $({1 \over p}, {1
\over q}, s)$ beneath $Z$, and in the case of Radon transforms one does not
have $L^p$ to $L^q_{s}$ boundedness for $({1 \over p}, {1 \over q}, s)$ above
the plane containing $Z$, thereby providing a Sobolev space improvement result
which is sharp up to endpoints for $({1 \over p}, {1 \over q})$ 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 $q$.
