The \(\infty \)-Poincaré inequality on metric measure spaces
A metric measure space \((X,d,\mu)\) is said to support a weak \(\infty\)-Poincaré inequality if there exist constants \(C>0\) and \(\lambda \geq1\) such that, for every Borel measurable function \(u:X\rightarrow \mathbb R \cup \{\infty\}\) and every upper gradient \(g:X\rightarrow [0,\infty]\) of \(u\), the pair \((u,g)\) satisfies the inequality \[ \frac {1}{\mu(B(x,r))} \int_{B(x,r)}|u-u_{B(x,r)}|\, d\mu\leq Cr\|g\|_{L^{\infty}(B(x,\lambda r))} \] for each ball \(B(x,r)\subset X\). The \(\infty\)-Poincaré inequality is a weaker condition than any \(p\)-Poincaré inequality with finite \(p\geq 1\), but it still gives reasonable information on the geometry of the space. The main result of this paper states that a connected complete doubling metric measure space supports a weak \(\infty\)-Poincaré inequality if and only if it is thick quasiconvex, i.e., every pair of points can be connected by a ``thick'' family of quasiconvex curves in the sense that the \(\infty\)-modulus of this family is positive. The authors also prove that supporting an \(\infty\)-Poincaré inequality is also equivalent to the condition that \(LIP^\infty(X)=N^{1,\infty}(X)\).
scientific article; zbMATH DE number 6141588