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New Kakeya estimates using Gromov’s algebraic lemma. (English) Zbl 1459.42032

Summary: This paper presents several new results related to the Kakeya problem. First, we establish a geometric inequality which says that collections of direction-separated tubes (thin neighborhoods of line segments that point in different directions) cannot cluster inside thin neighborhoods of low degree algebraic varieties. We use this geometric inequality to obtain a new family of multilinear Kakeya estimates for direction-separated tubes. Using the linear / multilinear theory of J. Bourgain and L. Guth [Geom. Funct. Anal. 21, No. 6, 1239–1295 (2011; Zbl 1237.42010)], these multilinear Kakeya estimates are converted into Kakeya maximal function estimates. Specifically, we obtain a Kakeya maximal function estimate in \(\mathbb{R}^n\) at dimension \(d(n) = (2 - \sqrt{2}) n + c(n)\) for some \(c(n) > 0\). Our bounds are new in all dimensions except \(n = 2, 3, 4\), and 6.

MSC:

42B25 Maximal functions, Littlewood-Paley theory
42B20 Singular and oscillatory integrals (Calderón-Zygmund, etc.)
42B10 Fourier and Fourier-Stieltjes transforms and other transforms of Fourier type
14P10 Semialgebraic sets and related spaces

Citations:

Zbl 1237.42010
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References:

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