Weakly Lipschitz mappings in higher dimensions

We introduce and investigate the main features of weakly Lipschitz mappings whose pointwise Lipschitz constant is controlled from above by the derivative of a set function. The main results give conditions under which such mappings belong to $W^{1,n}_{\rm loc},$ and show that the class contains metrically quasiconformal mappings as a proper subclass. In addition, under a Rado--Reichelderfer type oscillation assumption, we derive diameter control, image-measure bounds, sufficient conditions for Lusin's $(N)$-property, and an area formula.