The Fefferman-Stein type inequality for the Kakeya maximal operator (Q2719012)

Let \(n > 1\). For each \(0 < \delta \ll 1\) let \(K_\delta f(x)\) be the maximal operator NEWLINE\[NEWLINE \sup_T |T|^{-1} \int_T |f|NEWLINE\]NEWLINE where \(T\) ranges over all tubes of eccentricity \(\delta\) containing \(x\). The Kakeya conjecture asserts that NEWLINE\[NEWLINE \|K_\delta f \|_p \lesssim \delta^{-n/p+1-} \|f\|_pNEWLINE\]NEWLINE for all \(1 \leq p \leq n\). For dimensions \(n > 2\) this has only been proven for \(p \leq (n+2)/2\) (with a recent improvement in large dimension to \(p \leq (4n+3)/7\)). NEWLINENEWLINENEWLINEThe natural weighted generalization of this conjecture is that NEWLINE\[NEWLINE \|K_\delta f \|_{L^p(w)} \lesssim \delta^{-n/p+1-} \|f\|_{L^p(K_\delta w)}.NEWLINE\]NEWLINE This conjecture was already verified in two dimensions by Muller and Soria, and for \(p \leq (n+1)/2\) by Vargas. NEWLINENEWLINENEWLINEBy using the ``hairbrush'' arguments of Wolff and some previous estimates of the author, the author extends the higher dimensional result to \(p \leq (n^2-2)/(2n-3)\), which is a little shy of the \((n+2)/2\) exponent obtained using this argument for the unweighted problem. It seems natural to wonder whether the techniques in this paper can be improved to match the \((n+2)/2\) exponent.