Bilinear restriction estimates for surfaces of codimension bigger than 1

Abstract

Abstract. In connection with the restriction problem in R n for hypersurfaces including the sphere and paraboloid, the bilinear (adjoint) restriction estimates have been extensively studied. However, not much is known about such estimates for surfaces with codimension (and dimension) larger than one. In this paper we show sharp bilinear L 2 × L 2 → L q restriction estimates for general surfaces of higher codimension. In some special cases, we can apply these results to obtain the corresponding linear estimates.

Restriction estimates to complex hypersurfaces

“…See Remark 4.7. However, for n = 3, the L p − L q estimates for p, q satisfying 1/p + 2/q < 1 and q > 10/3 were obtained in [6]. Our result doesn't recover this and it is a manifestation that multilinear strategy has certain inefficiency in capturing the curvature property of the underlying surface.…”

The almost optimal multilinear restriction estimate for hypersurfaces with curvature: the case of $n-1$ hypersurfaces in R^n

“…The bilinear case was intensely studied, see [10,42,36,37,34,28,29,4,40,25,39,33,11,1,38] and references therein. We should highlight the works of Wolff [42] and Tao [37] where the conjectured was established, up to the endpoint, for subsets of the cone, respectively paraboloids.…”

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.…”