A weighted inequality for the Kakeya maximal operator with a special base (Q5930103)

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, with partial results in higher dimensions. NEWLINENEWLINENEWLINEIn this paper the author shows the conjecture is true if the tubes are restricted to intersect a single line (in the spirit of the ``hairbrush'' argument of Wolff). For the unweighted estimate this was shown earlier by Igari.