The Fefferman-Stein-type inequality for the Kakeya maximal operator. II. (Q1862900)

Let \(f\) be a function on \(\mathbb R^d\), and consider the Kakeya maximal function \(K_\delta f(x)\), defined for \(x\) in \(\mathbb R^d\) and \(\delta > 0\) as \(K_\delta f(x) = \sup_{T \ni x} \int_T | f|\) where \(T\) ranges over all tubes containing \(x\) and having eccentricity \(\delta\). The Kakeya conjecture asserts that \(K_\delta\) is bounded from \(L^d(dx)\) to \(L^d(dx)\) with a norm of at most \(C_\alpha \delta^{-\alpha}\) for any \(\alpha\). A generalization of this conjecture would be that this operator is bounded from \(L^d(w dx)\) to \(L^d(K_\delta(w) dx)\) with a norm of at most \(C_\alpha \delta^{-\alpha}\) for any positive weight function \(w\). This was proven in two dimensions by \textit{D. Müller} and \textit{F. Soria} [J. Fourier Anal. Appl. Spec. Iss., 467--478 (1995; Zbl 0886.42016)] and in higher dimensions in the radial case by \textit{H. Tanaka} [Proc. Am. Math. Soc. 129, No. 8, 2373--2378 (2001; Zbl 0972.42013)]. Here the author demonstrates that the higher-dimensional conjecture is true in the range \(\alpha > (d-2)/2d\), improving on previous work of the author. The techniques involve a modification of Wolff's ``hairbrush'' argument as well as bilinear methods.