Maximal averages and packing of one dimensional sets (Q1126755)

The author summarizes some recent work in real analysis related to Kakeya type maximal functions, considered in the classical situations of lines and circles. One defines a Kakeya set to be a compact set \(E\subset \mathbb R^n\) which contains a unit line segment in each direction, \(\forall e\in \mathbb P^{n-1} \exists x\in \mathbb R^n: x+te\in E\) for all \(t\in [-1/2, 1/2]\) where one regards \(\mathbb P^{n-1}\) as being the unit sphere with antipodal points identified. If \(\delta\) is a small positive number and \(f:\mathbb R^n\to \mathbb R\) then one defines the Kakeya maximal function of \(f\), \(f_\delta^\ast:\mathbb P^{n-1}\to \mathbb R\), via \(f_\delta^\ast(e)= \sup_a | T_e^\delta(a)| ^{-1}\int_{f_\delta^\ast(a)}| f(x)| dx\) where \(f_\delta^\ast(a)\) is the cylinder centered at \(a\) with length 1, cross section radius \(\delta\) and axis in the \(e\) direction. Also one defines the \(\delta\)-entropy \(\mathcal N_\delta(E)\) to be the maximum possible cardinality for a \(\delta\)-separated subset. Then the following three questions are open if \(n\geq 3\). 1. Is it true that if \(E\) is a Kakeya set in \(\mathbb R^n\) then \(\limsup_{\delta\to 0} \log \mathcal N_\delta(E)/\log \delta^{-1} = n\)? 2. Is it true that a Kakeya set in \(\mathbb R^n\) must have Hausdorff dimension \(n\)? 3. Is the following estimate true \(\forall \varepsilon >0\;\exists C_\varepsilon: \| f_\delta^\ast \| _{L^n(\mathbb P^{n-1})}\leq C_\varepsilon\delta^{-\varepsilon} \| f\| _{L^n(\mathbb P^{n-1})}\)? The author discusses the partial results that have been proved on this problem and some work on a class of related problems involving circles in the plane, quoting some recent works of several authors (Bourgain, Kolasa, Mistis, Schlag, Wolff, etc).