On Ahlfors-David regular weighted bounds for the extension operator associated to the circle

This paper addresses the sharpness of a weighted $L^2$-estimate for the Fourier extension operator associated to the circle, obtained by J. Bennett, A. Carbery, F. Soria and A. Vargas in 2006. A point left open in their paper was the necessity of a certain $\log R$-factor in the bound. Here, I show that the factor is necessary for all $1\mathrm{/}2$-Ahlfors-David regular weights on the circle, but it can be removed for $s$-Ahlfors-David regular weights with $s\mathrm{\ne }1\mathrm{/}2$.