Kakeya and Restriction Preprints

Preprints in Kakeya and Restriction problems

There are other pages on the web on Kakeya and Restriction; Ben Green, Alex Iosevich, and Izabella Laba all maintain one. If you are new to this field and want to learn more, I can suggest starting with my Notices article surveying this field. If you then want to learn more about the Kakeya problem, you could try the El Escorial proceedings survey with Nets Katz, or the Edinburgh lecture notes on the Kakeya problem; if you want to learn more about the restriction problem, I can offer my Park city notes on the Restriction problem. You can also see my Math 254B home page for a more leisurely-paced introduction, but it is getting a little out of date. If you are more into the algebraic side of things, you can learn about the finite field analogues of these problems in this paper with Gerd Mockenhaupt. If you like the arithmetic combinatorial side of things, you can start with this short paper with Nets, or my Math 254A home page. If you are instead interested in the Bochner Riesz or local smoothing problems, you will have to go to my research papers, such as my second paper with Ana Vargas; I do not yet have a good survey of these problems (one should probably go look instead at the home pages of Izabella Laba or of the papers of Tom Wolff), although I mention these problems briefly in the Park city notes.

Math 254B home page - Restriction theorems, Bochner-Riesz, Kakeya, etc. (1999)

Papers, and projects close to completion

Title	With	Status	Download
Weak-type endpoint bounds for Riesz means		Proc. Amer. Math. Soc. 124 (1996), 2797-2805	dvi ps.Z
The Bochner-Riesz conjecture implies the restriction conjecture		Duke Math. J. 96 (1999), 363-376	dvi ps.Z Slides: dvi ps.Z
The weak-type endpoint Bochner-Riesz conjecture and related topics		Indiana U. Math. J. 47 (1998), 1097-1124	dvi Slides: dvi ps.Z
A bilinear approach to the restriction and Kakeya conjectures	Ana Vargas Luis Vega	J. Amer. Math. Soc. 11 (1998), 967-1000	math.CA/9807163 Slides: dvi + Figures 1 2 Summary: dvi
On the Maximal Bochner-Riesz conjecture in the plane, for p<2		Trans. Amer. Math. Soc. 354 (2002), 1947-1959	dvi + Figure 1 ps.Z Slides: dvi + Figures 1 2 3 4
A bilinear approach to cone multipliers I. Restriction estimates	Ana Vargas	GAFA 10 (2000), 185-215	dvi + Figures 1 2 3 4 5 ps Summary: dvi
A bilinear approach to cone multipliers II. Applications	Ana Vargas	GAFA10 (2000), 216-258	dvi + Figures 1 2 3 4 5 ps
An improved bound on the Minkowski dimension of Besicovitch sets in R^3	Nets Katz Izabella Laba	Annals of Math. 152 (2000), 383-446	math.CA/9903166 Slides: dvi + Figures 1 2 3 4 5 6 7
Endpoint bilinear restriction theorems for the cone, and some sharp null form estimates		Math. Z. 238 (2001), 215-268	math.CA/9909066 Non-endpoint: dvi Informal: dvi Slides: dvi + Figures 1 2 3 4 5 6 7
Bounds on arithmetic projections, and applications to the Kakeya conjecture	Nets Katz	Math. Res. Letters 6 (1999), 625-630	math.CO/9906097 Slides: dvi + Figures 1 2 3 4 5 6 7
An x-ray transform estimate in R^n	Izabella Laba	Revista Mat. Iber. 17 (2001), 375-408	math.CA/9909032 Informal: dvi
Some connections between the Falconer and Furstenburg conjectures	Nets Katz	New York J. Math. 7 (2001), 148-187	math.CA/0101195 Finite field analogues
An improved bound for the Minkowski dimension of Besicovitch sets in medium dimension	Izabella Laba	GAFA 11 (2001), 773-806	math.CA/0004015
From rotating needles to stability of waves: emerging connections between combinatorics, analysis, and PDE.		Notices Amer. Math. Soc. 48 (2001) No 3, 294-303	math.CA/0008098 addenda
Kakeya and restriction phenomena for finite fields	Gerd Mockenhaupt	Duke Math. J. 121 (2004), 35-74	math.CA/0204234
New bounds for Kakeya problems	Nets Katz	Journal d'Analyse de Jerusalem, 87 (2002), 231-263	math.CA/0102135 Slides + Figure 1 2 3 4 5 6 7
Recent progress on the Kakeya conjecture	Nets Katz	Publicacions Matematiques, Proceedings of the 6th International Conference on Harmonic Analysis and Partial Differential Equations, U. Barcelona 2002, 161-180	math.CA/0010069
Bochner-Riesz summability for analytic functions on the m-complex unit sphere and for cylindrically symmetric functions on R^{n-1} \times R	Adam Sikora	to appear, Comm. Anal. Geom.	math.CA/0207225
A new bound for finite field Besicovitch sets in four dimensions		to appear, Pacific J. Math	math.CA/0204251
A sharp bilinear restriction estimate for paraboloids		GAFA 13 (2003), 1359-1384	math.CA/0210084
Some recent progress on the Restriction conjecture		submitted, Proceedings, Fourier Analysis and Convexity Workshop	math.CA/0303136
Recent progress on the Restriction conjecture		submitted, Park City proceedings	math.CA/0311181

Some further papers related to Kakeya and restriction problems can be found on my PDE preprint page.
Some further papers dealing with more general aspects of harmonic analysis can be found here.

Short stories

These are generally very short, toy versions of real results due to other people, and are not publication-quality. Caveat emptor. All files other than figures are in dvi format. Unlike the preprints, these articles are fluid and subject to new developments. Please let me know if you have any comments, references, etc. on any of them.

Disclaimer: Many of the notes here are based on papers written by other people. My intention here is not to try to "beat" these authors' work in any way, but rather to isolate the main ingredients of the argument, which are often very beautiful, and try to present them in as simple and brief a context as possible (often sacrificing generality, rigour, and/or details in order to do this). Certainly I do not view these notes as worthy of publication in a refereed journal, and are definitely inferior to the original article in every single aspect, with the possible exception of brevity.

The Szemeredi-Trotter cell decomposition	Non-endpoint bilinear cone restriction theorems
Informal proof of bilinear cone restriction estimates	Informal proof of x-ray estimates
Sharpness of the Carleson-Sjolin theorem	An informal outline of Wolff's local smoothing estimate
Korner's Besicovitch set construction	Finite field analogues of the Erdos, Falconer, and Furstenburg problems

Miscellaneous

Local smoothing / Bochner-Riesz / Kakeya survey + Figures 1 2 (1999)
Besicovitch set applet (1999)
Geometric maximal functions, oscillatory integrals, and non-linear evolution equations + Figures 1 2 3. This was a talk for the Clay Millennium event. (2000)
The restriction and Kakeya conjectures. These are lecture notes for a two-hour mini-course at the IPAM OIDE conference. (2001)
Edinburgh lecture notes on the Kakeya problem. These are an expanded version of the Kakeya half of the IPAM notes, for the Edinburgh instructional conference. (2002)
A half-serious diagram of Bochner-Riesz and restriction conjecture inter-relationships. (note: you may need to set "EPS clip" on ghostview to see this properly).
A brief description of some analysis research at UCLA (intended for incoming grad students; 2000) [Figure 1] [Figure 2].
The Kakeya conjecture (intended for incoming grad students; 2001) [Figure 1] [Figure 2].
My research (intended for incoming graduate students; 2002)

Back to my preprints page.