Computing best constants and extremals for Maximal Operators and Fourier Multipliers. Recent results

A sharp rearrangement inequality for the uncentered Hardy-Littlewood
maximal operator allows us to prove many sharp norm inequalities, on $L^p$ and
beyond, for this sub-linear operator. We do characterize the extremals of our
norm inequalities in those cases where the best constants are attained. It is
conceivable that a suitable rearrangement approach could work also in the case
of the Hilbert Transform, possibly leading to a real-variable proof of the
results of Cole-Pichorides, possibly leading to a solution of the 80-year
old problem of understanding the exact $l^p$ norm behaviour of some discrete
Hilbert Transforms.