  We prove $L^p({\mathbb R}^3)$ to $L^p_s({\mathbb R}^3)$ Sobolev improvement
theorems for local averaging operators over real analytic surfaces in ${\mathbb
R}^3$. For most such operators, in a sense made precise in the paper, the set
of $(p,s)$ for which we prove $L^p({\mathbb R}^3)$ to $L^p_s({\mathbb R}^3)$
boundedness is optimal up to endpoints. Using an interpolation argument in
conjunction with these $L^p({\mathbb R}^3)$ to $L^p_s({\mathbb R}^3)$ results
we obtain an $L^p({\mathbb R}^3)$ to $L^q({\mathbb R}^3)$ improvement theorem,
and the set of exponents $(p,q)$ obtained will also usually be optimal up to
endpoints. The advantage the methods of this paper have over those of the
author's earlier papers is that the oscillatory integral methods of the earlier
papers, closely tied to the Van der Corput lemma, allow one to only prove 1/2
of a derivative of surface measure Fourier transform decay, while the methods
of this paper, when combined with appropriate resolution of singularities
methods, allow one to go up to the maximum possible 1 derivative. This allows
us to prove the stronger sharp up to endpoints results.
