2023/11/24 by Harris, Terence L. J.
Mathematics · #28A78 #28A80 #Advanced Harmonic Analysis Research #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Limits and Structures in Graph Theory #Mathematical Analysis and Transform Methods
paper · pdf · doi:10.48550/arxiv.2311.14667
openalex publication_date 2023/11/24 · openalex created_date 2023/11/29 · openalex updated_date 2026/07/28
It is shown that SL2 Besicovitch sets of measure zero exist in ℝ3. The proof is constructive and uses point-line duality analogously to Kahane's construction of measure zero Besicovitch sets in the plane. A corollary is that the SL2 Kakeya maximal inequality cannot hold with uniform constant. A counterexample is given to show that the SL2 Kakeya maximal inequality cannot hold for p> 3/2; even in the model case where the δ-tubes have δ-separated directions and the cardinality of the tube family is ∼ δ-2. It is then shown that, with Cε δ-ε loss, the SL2 Kakeya maximal inequality does hold if p ≤ 4/3, whenever the tubes satisfy a 2-dimensional ball condition (equivalent to the Wolff axioms in the SL2 case). The proof is via an L4/3 inequality for restricted families of projections onto planes. For both inequalities, the range 4/3 < p ≤ 3/2 remains an open problem. A related L6/5 inequality is derived for restricted projections onto lines. Finally, an application is given to generic intersections of sets in ℝ3 with "light rays" and "light planes".