Some remarks about metric spaces, spherical mappings, functions and their derivatives (Q677843)

Given a metric \(d\) in \(\mathbb{R}^{n}\) and \(p\in \mathbb{R}^{n}\), the author investigates the existence of a function \(\theta _{p}:\mathbb{R}^{n}\backslash \{p\}\rightarrow S^{n-1}\) satisfying a ``Lipschitz'' condition \[ \left| \theta _{p}(x)-\theta _{p}(y)\right| \leq C\;\frac{d(x,y)}{\min (d(x,p),d(y,p))} \] and approximating the radial projection \(\pi _{p}:\mathbb{R} ^{n}\backslash \{p\}\rightarrow S^{n-1}\), \(\pi _{p}(x)=\frac{x-p}{\left| x-p\right| }.\) The existence of such mappings is used to derive Sobolev-type inequalities. The author constructs \(\theta _{p}\) when \(d\) satisfies the condition \[ d(x,y)\leq td(x,z)\text{ implies }\left| x-y\right| \leq \eta (t)\left| x-z\right| ,\text{ }(x,y\in \mathbb{R}^{n},t>0) \] where \(\eta :[0,\infty)\rightarrow [0,\infty)\) is a homeomorphism, and obtains an estimate for the values of a \(C^{1}\) function in terms of its gradient when \(d\) is a quasimetric equivalent to a metric , \(d\) being generated by a ``doubling measure'' \(\mu \) on \(\mathbb{R}^{n}\) (i.e. \(\mu (B(x,2r))\leq C\mu (B(x,r))\))