The Poincaré inequality for \(C^{1,\alpha}\)-smooth vector fields (Q483430)

Let \(M\) be a connected Riemannian \(C^\infty\)-manifold \(M\). Assume \(g\in M\), and \(r_0>0\). The author establishes the Poincaré inequality for John domains in the form \[ \| f- f_\Omega\|_{p,\Omega}\leq C\Biggl({b\over a}\Biggr)^\nu\text{\,diam}(\Omega)\|\nabla_H f\|_{p,\Omega}, \] with \(1\leq p<\infty\), where \(\Omega\subset B(g,r_0)\subset M\) is a John domain with inner diameter \(a\) and outer diameter \(b\), while \(f\in C^\infty(\Omega)\), and \(\nu\) is the Hausdorff dimension of \(M\). Moreorer, the constant \(C\) is independent of \(\Omega\) and \(f\), \(\nabla_H\) is the horizontal gradient. Equivalently, in the paper under review, the Poincaré inequality is obtained for the equiregular Carnot-Carathéodory spaces spanned by vector fields with Hölder class derivatives.