Restriction estimates to complex hypersurfaces

Abstract

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 T T * 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.

Decoupling inequalities for quadratic forms

“…Beyond decoupling theory, problems associated with quadratic d-surfaces (d ě 2) of co-dimension bigger than one have also attracted much attention, in particular in Fourier restriction theory and related areas. We refer to Christ [Chr85], [Chr82], Mockenhaupt [Moc96], Bak and Lee [BL04], Bak, Lee and Lee [BLL17], Lee and Lee [LL19], Guo and Oh [GO20] for the restriction problems associated with manifolds of co-dimension two and higher, Bourgain [Bou91], Rogers [Rog06] and Oberlin [Obe07] for the planar variant of the Kakeya problem, and Oberlin [Obe04] for sharp L p Ñ L q improving estimates for a quadratic 3-surface in R 5 . Recently, Gressman [Gre19a], [Gre19b] and [Gre20] has made significant progress in proving sharp L p -improving estimates for Radon transforms of intermediate dimensions.…”

Fourier restriction estimates for surfaces of co-dimension two in $\mathbb{R}^5$

“…Previously we have mentioned Christ [Chr82,Chr85] and Mockenhaupt [Moc96]. Other than these, Oberlin [Obe04] studied Fourier restriction estimates and L p → L q improving estimates for the surface (ξ 1 , ξ 2 , ξ 3 , ξ 2 1 + ξ 2 2 , ξ 2 2 + ξ 2 3 ), which is a surface that satisfies the (CM) condition; Banner [Ban02] studied restriction estimates for certain surfaces not satisfying the (CM) condition; Bak and Lee [BL04], and Oberlin [Obe02] obtained sharp Fourier restriction estimates for certain quadratic moment surfaces of high co-dimensions; Bak, Lee and Lee [BLL17] studied bilinear Fourier restriction estimates for surfaces of co-dimension bigger than one and deduce a linear restriction estimate for the complex paraboloid in C 3 , which can be identified with (ξ 1 , ξ 2 , ξ 3 , ξ 4 , ξ 2 1 − ξ 2 2 + ξ 2 3 − ξ 2 4 , ξ 1 ξ 2 + ξ 3 ξ 4 ); Lee and Lee [LL19] studied restriction estimates for complex hyper-surfaces, which can also be viewed as special cases of real surfaces of co-dimension two, in some high dimensions.…”