The transformations of functions acting on sublevel sets that satisfy a Pólya-Szegő inequality are characterized as those being induced by transformations of sets that do not increase the associated capacity.
Van Schaftingen, J. (2014). Equivalence between Pólya–Szegő and relative capacity inequalities under rearrangement. Archiv der Mathematik, 103(4), 367−379. https://doi.org/10.1007/s00013-014-0695-4 (Original work published 2014)