Under some regularity assumptions, we compute the modulus of continuity of the generalized Mazur map between unit balls of Orlicz spaces. Our estimate coincides with the known estimates in the setting of $L_p(\mu)$-spaces. We apply this estimate to approximate uniformly continuous mappings between balls of reflexive Orlicz spaces by $\alpha$-Hölder maps, with $\alpha$ as large as possible. We also relate this optimal value of $\alpha$ to the Boyd indices of the spaces and to the problem of isomorphic extension of Hölder maps.
"Modulus of continuity of the Mazur map between unit balls of Orlicz spaces and approximation by Hölder mappings." Illinois J. Math. 49 (1) 195 - 216, Spring 2005. https://doi.org/10.1215/ijm/1258138314