Adapting analysis/synthesis pairs to pseudodifferential operators

Many problems in harmonic analysis are resolved by producing an analysis/synthesis of function spaces. For example the Fourier or wavelet decompositions. In this talk I will discuss how to use Fourier integral operators to adapt analysis/synthesis pairs (developed for the constant coefficient PDE case) to the pseudodifferential setting. I will demonstrate how adapting a wavelet decomposition can be used to prove $L^p$ bounds for joint eigenfunctions.