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 You can also see my Math 254B home page for a more leisurely-paced introduction, but it is getting a little out of date.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

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	arXiv:9807163 Slides: dvi + Figures 12 Summary: dvi see errata below
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 1234
A bilinear approach to cone multipliers I. Restriction estimates	Ana Vargas	GAFA 10 (2000), 185-215	dvi + Figures12345 ps Summary: dvi see errata below
A bilinear approach to cone multipliers II. Applications	Ana Vargas	GAFA10 (2000), 216-258	dvi + Figures12345 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	arXiv:9903166 Slides: dvi + Figures 1234567
Endpoint bilinear restriction theorems for the cone, and some sharp null form estimates		Math. Z. 238 (2001), 215-268	arXiv:9909066 Non-endpoint: dvi Informal: dvi Slides: dvi + Figures 1234567
Bounds on arithmetic projections, and applications to the Kakeya conjecture	Nets Katz	Math. Res. Letters 6 (1999), 625-630	arXiv:9906097 Slides: dvi + Figures 1234567
An x-ray transform estimate in R^n	Izabella Laba	Revista Mat. Iber. 17 (2001), 375-408	arXiv:9909032 Informal: dvi
Some connections between the Falconer and Furstenburg conjectures	Nets Katz	New York J. Math. 7 (2001), 148-187	arXiv:0101195 Finite field analogues
An improved bound for the Minkowski dimension of Besicovitch sets in medium dimension	Izabella Laba	GAFA 11 (2001), 773-806	arXiv: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	arXiv:0008098 addenda
Kakeya and restriction phenomena for finite fields	Gerd Mockenhaupt	Duke Math. J. 121 (2004), 35-74	arXiv:0204234
New bounds for Kakeya problems	Nets Katz	Journal d'Analyse de Jerusalem, 87 (2002), 231-263	arXiv:0102135 Slides + Figure1234567
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	arXiv: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	Comm. Anal. Geom. 12 (2004), 43-57	arXiv:0207225
A new bound for finite field Besicovitch sets in four dimensions		Pacific J. Math 222 (2005), 337-363	arXiv:0204251
A sharp bilinear restriction estimate for paraboloids		GAFA 13 (2003), 1359-1384	arXiv:0210084
Some recent progress on the Restriction conjecture		Fourier analysis and convexity, 217-243, Appl. Numer. Harmon. Anal., Birkhäuser Boston, Boston,MA, 2004.	arXiv:0303136 see errata below
Recent progress on the Restriction conjecture		to appear, Park City proceedings	arXiv:0311181 see errata below
On the Multilinear Restriction and Kakeya conjectures	Jon Bennett Tony Carbery	Acta Math. 196 (2006) 261—302	arXiv:0509262
The Kakeya set and maximal conjectures for algebraic varieties over finite fields	Jordan Ellenberg Richard Oberlin	Mathematika 56 (2010) 1-25	arXiv:0903.1879 discussion
Sharp bounds for multilinear curved Kakeya, restriction and oscillatory integral estimates away from the endpoint		Mathematika 66 (2020), 517--576	arXiv:1907.11342 discussion

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.

*errata*: There is a typo common to several papers above. Namely, in Lemma 4.5 of “A bilinear approach...” with Vargas and Vega, the condition $(n+1)/2 > alpha q$ should be $(n-1)/2 > alpha q$; and similarly the condition $\sigma + 1 > \alpha q$ should be $\sigma > \alpha q$ in Lemma 6.1 of “A bilinear approach to cone multipliers I” with Vargas, Theorem 5.1 of “Some recent progress on the restriction conjecture”, and Theorem 2.10 of “Recent progress on the restriction conjecture”. Thanks to Monica Visan for pointing out the error.

Short stories

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
Dvir’s proof of the finite field Kakeya conjecture	The Kakeya conjecture and the Ham Sandwich theorem
A remark on the Kakeya needle problem

Miscellaneous

Local smoothing / Bochner-Riesz / Kakeya survey + Figures 12 (1999)
Besicovitch set applet (1999)
Geometric maximal functions, oscillatory integrals, and non-linear evolution equations + Figures 123. 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)
Another description of my research (intended for incoming grad students, 2005; not really Kakeya-ish, though)

Back to my preprints page.