## 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 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 12 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 1234 A bilinear approach to cone multipliers I. Restriction estimates Ana Vargas GAFA 10 (2000), 185-215 dvi + Figures 12345 ps Summary: dvi A bilinear approach to cone multipliers II. Applications Ana Vargas GAFA10 (2000), 216-258 dvi + Figures 12345 ps An improved bound on the Minkowski dimension of Besicovitch sets in R^3 Annals of Math. 152 (2000),  383-446 math.CA/9903166 Slides: dvi + Figures 1234567 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 1234567 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 1234567 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 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 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 1234567 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.

### Miscellaneous

Back to my preprints page.