$\|\mathcal R\chi_E\|_3\lesssim\|\chi_E\|_{3/2}$

for the maximal Radon transform $\mathcal R$ (maximal averages over 2-

planes) in $\mathbb R^3$? (no Fourier transform, no complex

interpolation)

A suitable variant of Bourgain's "bush" argument doesn't give the

sharp bound and Cordoba's classical argument gives just an $L^2

\rightarrow L^2$ bound.