A geometric inequality with applications to the Kakeya problem in three dimensions
The Besicovitch sets [see, for instance, \textit{K. Falconer}: ``The geometry of fractal sets'' (1985; Zbl 0587.28004)] are involved in deep problems arising in Harmonic Analysis, such as for \(L^p\)-boundedness of the ball multiplier and the Bochner-Riesz spherical summation operators and for the Fourier transform restriction problems as well. Estimates of the Hausdorff dimension \(d\) of these Besicovitch sets have drawn the attentions of many authors. For example, a celebrated result of \textit{J. Bourgain} [Geom. Funct. Anal. 1, No. 2, 147-187 (1991; Zbl 0756.42014)] states that \(d\geq {7\over 3}\) in \({\mathbb R}^3\). The main purpose of this paper under review is to introduce and prove some geometric inequality (see Proposition 2.8) which can be viewed as a sharp bound on the measure of the set of directions of all lines that come \(\delta\)-close (\(0<\delta<1\)) to three given lines. Next this geometric inequality is used by the author (in Propositions 3.1 and 3.2) to derive the boundedness of the Kakeya maximal function \({\mathcal M}_{\delta}\) from \(L^p({\mathbb R}^3)\) into \(L^q({\mathbf S}^2)\) for \(p={7\over 3}\) and \(q=2p'={7\over 2}\) and with the bound essentially given by \(\delta^{-{3\over p}+1-\varepsilon}\) for any \(\varepsilon >0\). Here \({\mathbf S}^2\) is the unit sphere in \({\mathbb R}^3\) and for each direction \(e\in{\mathbf S}^2\), \(({\mathcal M}_{\delta}f)(e)\) is defined by taking the supremum of all averages of \(f\) over tubes of unit length with thickness \(\delta\) and oriented in the direction of \(e\). Therefore, by making use of Lemma 2.15 in J. Bourgain's paper quoted above, the author's work leads to a different geometric approach of this celebrated estimate: \(d\geq p={7\over 3}\).
scientific article; zbMATH DE number 1197119