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Propagation of singularities and maximal functions in the plane. (English) Zbl 0754.35004
(Author’s summary.) 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 propagation 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.

##### MSC:
 35A20 Analyticity in context of PDEs 35A27 Microlocal methods and methods of sheaf theory and homological algebra applied to PDEs 58J47 Propagation of singularities; initial value problems on manifolds 47G20 Integro-differential operators
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