The $L^p$ CR Hartogs separate analyticity theorem for strictly convex domains

For a large class of convex domains in ${\mathbf{C}}^{\mathbf{n}}$, it is shown that an $L^p$ function on the boundary is CR if there are holomorphic extensions on almost all slices of D by complex lines parallel to the coordinate axes. As an application, a new operator limit for the Szeg\H{o} projection is obtained.