82 research outputs found
Restriction and decay for flat hypersurfaces
In the first part we consider restriction theorems for hypersurfaces [Gamma] in Rn, with the affine curvature [fòrmula] introduced as a mitigating factor. Sjolin, [19], showed that there is a universal restriction theorem for all convex curves in R2. We show that in dimensions greater than two there is no analogous universal restriction theorem for hypersurfaces with non-negative curvature. In the second part we discuss decay estimates for the Fourier transform of the density [fòrmula] supported on the surface and investigate the relationship between restriction and decay in this setting. It is well-known that restriction theorems follow from appropriate decay estimates; one would like to know whether restriction and decay are, in fact, equivalent. We show that this is not the case in two dimensions. We also go some way towards a classification of those curves/surfaces for which decay holds by giving some sufficient conditions and some necessary conditions for decay
Counting joints in vector spaces over arbitrary fields
We give a proof of the "folklore" theorem that the
Kaplan--Sharir--Shustin/Quilodr\'an result on counting joints associated to a
family of lines holds in vector spaces over arbitrary fields, not just the
reals. We also discuss a distributional estimate on the multiplicities of the
joints in the case that the family of lines is sufficiently generic.Comment: Not intended for publication. References added and other minor edits
in this versio
On the Multilinear Restriction and Kakeya conjectures
We prove -linear analogues of the classical restriction and Kakeya
conjectures in . Our approach involves obtaining monotonicity formulae
pertaining to a certain evolution of families of gaussians, closely related to
heat flow. We conclude by giving some applications to the corresponding
variable-coefficient problems and the so-called "joints" problem, as well as
presenting some -linear analogues for .Comment: 38 pages, no figures, submitte
The endpoint multilinear Kakeya theorem via the Borsuk--Ulam theorem
We give an essentially self-contained proof of Guth's recent endpoint
multilinear Kakeya theorem which avoids the use of somewhat sophisticated
algebraic topology, and which instead appeals to the Borsuk-Ulam theorem
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