On Kakeya's maximal function (Q1091558)

Let s be the family of rectangles in \({\mathbb{R}}^ d\) with dimensions \(a\times a\times...\times a\times aN\) but with arbitrary direction. Set \(Mf=\sup_{x\in A\in S}| A|^{-1}\int_{A}| f| dx.\) For \(d\geq 3\) the author proves the inequality \(\| Mf\|_{L^ d({\mathbb{R}}^ d)}\leq C(\log N)^{3/2}\| f\|_{L^ d({\mathbb{R}}^ d)}\) with an absolute constant C. In the case \(d=2\) the same is true with the exponent \(1/2\) instead of 3/2. [Cf. \textit{A. Cordoba}, The Kakeya maximal function and the spherical summation multipliers, Am. J. Math. 99, 1-22 (1977; Zbl 0384.42008)].