Quasiconformal, Lipschitz, and BV mappings in metric spaces

Consider a mapping $f\text{\hspace{0.17em}⁣}:X\to Y$ between two metric measure spaces. We study generalized versions of the local Lipschitz number $\mathrm{L}\mathrm{i}\mathrm{p}f$, as well as of the distortion number $H_f$ that is used to define quasiconformal mappings. Using these, we give sufficient conditions for $f$ being a BV mapping $f\in B{V}_{\mathrm{l}\mathrm{o}\mathrm{c}}(X;Y)$ or a Newton-Sobolev mapping $f\in N_{$\mathrm{l}\mathrm{o}\mathrm{c}$}^{1,p}(X;Y)$,with$1\le p<\infty$.