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Recent work connected with the Kakeya problem. (English) Zbl 0934.42014

Rossi, Hugo (ed.), Prospects in mathematics. Invited talks on the occasion of the 250th anniversary of Princeton University. Papers from the conference, Princeton, NJ, USA, March 17-21, 1996. Providence, RI: American Mathematical Society. 129-162 (1999).
The author gives a thorough survey of the Kakeya set conjecture and many related problems. This is a fast-moving field, and some exciting new discoveries have been made since the appearance of this survey, but nevertheless this is an excellent reference and contains the first published appearance of some interesting conjectures and folklore. The survey begins with the Kakeya set problem - does a compact subset of \(\mathbb R^n\) which contains a unit line segment in every direction necessarily have full Hausdorff dimension? (It is known that these sets can have measure zero). The conjecture is solved for \(n=2\) but for \(n > 2\) only partial progress is known. One can ask the more quantitative question of estimating the Kakeya maximal function \[ f^*(\omega) = \sup_l \int_l f \] where \(f\) is a function on \(\mathbb R^n\), \(\omega \in S^{n-1}\), and \(l\) ranges over all lines in the direction \(\omega\). These problems have discrete analogues such as the Zarankiewicz problem in extremal graph theory, and the problem of counting incidences between points and lines, as in the Szemeredi-Trotter theorem. The author studies the two-dimensional case in depth. In this case most problems related to Kakeya are solved, although there is an interesting variant related to some work of Furstenberg (replace unit line segments by \(\beta\)-dimensional subsets of the segment) which remains open, and is closely related to the original Kakeya problem. Finite field analogues are also considered; for various reasons, the finite field case is in fact far less technical to deal with than the Euclidean case, although not all Euclidean results have analogues for finite fields. The author also considers the variants in which line segments are replaced by other curves, and specifically by circles, in which case the problem becomes that of bounding the circular maximal function. Here one must use some sophisticated combinatorial tools together with geometric facts such as quantitative versions of the theorem of Apollonius. The author also discusses the connection between Kakeya and oscillatory integral problems such as the restriction and Bochner-Riesz conjectures, as well as the connection to certain Dirichlet sum problems, and specifically to Montgomery’s conjecture. Finally, the author gives an application of Kakeya sets to show the non-existence of a certain type of local smoothing. Specifically, one cannot locally estimate the \(L^p\) norm of solution to the wave equation in terms of \(L^\infty\) bounds on the initial data for \(p > 2\) without losing at least an epsilon of derivatives, even if one averages in time.
For the entire collection see [Zbl 0902.00027].

MSC:

42B25 Maximal functions, Littlewood-Paley theory
42-02 Research exposition (monographs, survey articles) pertaining to harmonic analysis on Euclidean spaces
42B20 Singular and oscillatory integrals (Calderón-Zygmund, etc.)
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