The Fefferman-Stein type inequality for the Kakeya maximal operator

Let K δ K_\delta , 0 > δ >> 1 0>\delta >>1 , be the Kakeya maximal operator defined as the supremum of averages over tubes of the eccentricity δ \delta . We shall prove the so-called Fefferman-Stein type inequality for K δ K_\delta , \[ ‖ K δ f ‖ L p ( R d , w ) ≤ C d , p ( 1 δ ) d / p − 1 ( log ⁡ ( 1 δ ) ) α ( d ) ‖ f ‖ L p ( R d , K δ w ) , \|K_\delta f\|_{L^p(\mathbf R^d,w)} \le C_{d,p} (\frac {1}{\delta })^{d/p-1} (\log (\frac {1}{\delta }))^{\alpha (d)} \|f\|_{L^p(\mathbf R^d,K_\delta w)}, \] in the range ( 1 > p ≤ ( d 2 − 2 ) / ( 2 d − 3 ) (1>p\le (d^2-2)/(2d-3) , d ≥ 3 d\ge 3 , with some constants C d , p C_{d,p} and α ( d ) \alpha (d) independent of f f and the weight w w .