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. Equivalently, if M is the maximal operator and S the symmetrization, then SMf(x)≤MSf(x) for every x, and equality holds for all x if and only if, up to translations, f(x) = S f(x) almost everywhere. Using these results, we then compute the exact norms of the maximal operator acting on Lorentz and Marcinkiewicz spaces, and we determine extremal functions that realize these norms.