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