Sobolev Spaces for the Wave Equation

In spatial dimension 2 or above, solutions of the wave equation can lose regularity over time when measured in the scale of L^p norms, if p is not equal to 2. The number of derivatives one can lose over time is captured by a classical theorem of Peral and Miyachi, which was later extended by Seeger, Sogge and Stein to more general Fourier integral operators. Subsequently a more conceptual proof of such results emerged from the work of Smith, and Hassell, Portal and Rozendaal. Recently I joined the latter group in extending some of these results using ideas from Fourier decoupling. In this talk, I will explain some of the ideas involved, by looking at some simple examples. Joint work with Andrew Hassell, Pierre Portal and Jan Rozendaal.