A note on the Kakeya maximal function
The author presents a proof of an estimate for the Kakeya maximal function in two dimensions which is slightly more complicated then the original proof of Cordoba but which gives a sharp bound and has the possibilities of extension to higher dimensions. The Kakeya maximal function is defined as follows. A rectangle R in \(R^ n\) has dimension \(\xi\) \(\times N\delta\) if there is a motion of \(R^ n\) that maps R onto \([0,\delta]^{n-1}\times [0,N\delta]\). Denote the collection of \(\xi\) \(\times \delta N\) rectangles by \({\mathcal B}_{\delta,N}\), and the union of all such over \(\delta >0\) by \({\mathcal B}_ N\). The Kakeya maximal function is \[ M_ Nf(x)=\sup_{x\in R\in {\mathcal B}_ N}\frac{1}{| R|}\int_{R}| f(y)| dy, \] and it is usually studied by means of \[ M_{\delta,N}f(x)=\sup_{x\in R\in {\mathcal B}_{\delta,N}}\frac{1}{| R|}\int_{R}| f(y)| dy. \] \textit{A. Cordoba} [Am. J. Math. 99, 1-22 (1977; Zbl 0384.42008)] proved that \(M_ N\) is bounded on \(L^ 2\) with a bound that grows like (log N)\({}^ 2\) by showing that \(\| M_{\delta,N}f\|_ 2\leq C(\log 2N)^{1/2}\| f\|_ 2.\) The author gives an alternate proof of this fact and notes that \(\| M_{\delta,N}f\|_ n\geq c(\log N)^{(n-1)/n}\| f\|_ n\) for suitable f.
scientific article; zbMATH DE number 4073683