Quantitative decompositions of Lipschitz mappings into metric spaces

We study the quantitative properties of Lipschitz mappings from Euclidean spaces into metric spaces. prove that it is always possible to decompose domain such a mapping pieces on which “behaves like projection mapping” along with “garbage set” arbitrarily small in an appropriate sense. Moreover, our control quantitative, i.e., independent both particular and space maps into. This improves theorem Azzam-Schul paper “Hard Sard”, answers question left open paper. The proof uses ideas differentiation, as well detailed how supplement by additional coordinates form bi-Lipschitz mappings.