Symmetrization and norm of the Hardy-Littlewood maximal operator on Lorentz and Marcinkiewicz spaces

We prove that when a function on the real line is symmetrically rearranged the distribution function of its uncentered Hardy-Littlewood maximal function increases pointwise, while it remains unchanged only when the function is already symmetric. Using these results, we then compute the exact norms of the maximal operator acting on Lorenz and Marcinkiewicz spaces and we determine extremal functions that realize these norms. The best constants on L^p are obtained as a special case of our results.