Remarks on Strichartz estimates for null forms (Q1850216)

The author uses ideas related to the div-curl lemma (and in particular use of Littlewood-Paley decomposition) to derive some new null form estimates. In particular, estimates on the \(L^{q/2}_t L^{r/2}_x\) norm for expressions such as \(|\nabla|^{-\sigma} Q_{ij}(\phi, \psi)\) are obtained, where \(\phi\), \(\psi\) solve the wave equation and \(Q_{ij}(\phi,\psi) = \partial_i \phi \partial_j \psi - \partial_j \phi \partial_i \psi\) is a ``null form''; this expression arises for instance in the study of the Yang-Mills and Maxwell-Klein-Gordon equations. Further refinements are then obtained using the ``improved Strichartz estimate'' of Klainerman and Tataru, which deals with waves whose Fourier transform is supported on a narrow angular band. Similar estimates are obtained for the related null form \(\partial_i \phi R_j \psi - R_j \phi \partial_i \psi\), which arises in the wave map system in the Coulomb gauge.