The restriction and Kakeya conjectures

[摘要] We survey the progress made on the restriction problem since it was first conjectured in the 1960s by E. M. Stein, in particular the oscillatory-integral approach which culminated in the Tomas-Stein theorem of 1975. We also examine the connections between the restriction and Kakeya problems, the latter evolving from a problem posed by S. Kakeya in 1917. In particular we devise a correspondence between the restriction and Kakeya set conjectures which is able to compare progress on the two problems in a quantitative way. Finally we discuss the latest developments which rely on bilinear, and their natural extension, multilinear, estimates and which have been found to provide the best known results for their linear counterparts (i.e. on the original problems).

[发布机构] University:University of Birmingham;Department:School of Mathematics

[关键词] Q Science;QA Mathematics