Restriction estimates to complex hypersurfaces

Abstract

The restriction problem is relatively well understood for hypersurfaces and recent progresses have been made by means of bilinear and multilinear approaches and, more recently, by means of polynomial partitioning combined with those approaches. However, for surfaces with codimension bigger than 1, bilinear and multilinear generalizations of restriction estimates are more involved and effectiveness of these multilinear estimates is not so well understood yet. Regarding the restriction problem for surfaces with codimensions bigger than 1, the current state of the art is still at the level of the TT⁎ method which is known to be useful for obtaining Lq–L2 estimates. In this paper, we consider a special type of codimension 2 surfaces, given by the graphs of complex analytic functions, and attempt to make progress beyond the L2 restriction estimates.