The Fourier restriction conjecture and the Bochner-Riesz conjecture ask for Lebesgue space mapping properties of certain oscillatory integral operators. They both are central in harmonic analysis, are open in dimensions $\geq 3$, and notably have the same conjectured exponents. In the 1970s, H\"{o}rmander asked if a more general class of operators all satisfy the same $L^p$-boundedness as in the above two conjectures. A positive answer to H\"{o}rmander's question would imply the above two conjectures and would have more applications such as in the manifold setting. Unfortunately H\"{o}rmander's question is known to fail in all dimensions $\geq 3$, even if one adds a ``positive curvature'' assumption. Bourgain showed that in dimension $3$ one always has the failure if a derivative condition is not satisfied everywhere. Joint with Shaoming Guo and Hong Wang, we generalize this condition to arbitrary dimension and call it ``Bourgain's condition''. We unify Fourier restriction and Bochner-Riesz by conjecturing that every H\"{o}rmander type operator satisfying Bourgain's condition should have the same $L^p$-boundedness as in those two conjectures. As evidences, we prove that the failure of Bourgain's condition immediately implies the failure of such an $L^p$-boundedness in every dimension. We also prove that current techniques on the two conjectures apply equally well in our conjecture and make some progress on our conjecture that consequently improves the two conjectures in higher dimensions.

I will talk about history and results, leaving comments on proof techniques mainly to my IAS talk.