The Mazur map establishes that the unit balls of \(L_{p}(\mu )\) spaces are mutually Hölder homeomorphic with explicit Hölder estimates for \(0<p<\infty \) . In this paper, we present a sharp analysis of Hölder estimates for the Mazur map on \(L_{p}(\mu )\) for \(0<p\le {1}\) , along with an application to coarse embeddings. Moreover, we also employ the notion of almost Lipschitz homeomorphism to investigate the homeomorphisms between unit balls of \(L_{p}(\mu )\) for \(0<p<1\) . In particular, we prove that the unit balls of \(L_{p}[0,1]\) and \(\ell _{q}\) are not \((\alpha ,\beta )\) -almost Lipschitz homeomorphic for \(0<p,q<{1}\) , where \(\alpha ,\beta >0, \alpha +\beta <{1}\) .
