PROPAGATION OF SINGULARITIES AND MAXIMAL FUNCTIONS IN THE PLANE

In this work we mainly generalize Bourgain's circular maximal function to include variable coefficient averages. Our techniques involve a combination of Bourgain's basic ideas plus microlocal analysis. In particular, to see the role of curvature, we rely heavily on methods used in studying propogation of singularities for hyperbolic differential equations. We also show that, for p > 2, there is local smoothing in L(p) for solutions to the wave equation.