Subelliptic Poincaré inequalities: the case \(p<1\)
The paper is devoted to the Poincaré-type inequality \[ \biggl(\frac 1{| B| }\int_B| f(x)-f_B| ^q\,dx\biggr)^{1/q}\leq Cr\Biggl( \frac 1{| B| }\int_B\biggl(\sum_j| \langle X_j,\nabla f(x)\rangle| ^2\biggr)^{p/2}\Biggr)^{1/p}, \] where \(0<p<1\), \(q>p\) is tied to \(p\) by a natural balance condition, \(\{X_j\}\) is a finite collection of smooth vector fields satisfying the Hörmander condition, and the subelliptic gradient of \(f\) satisfies certain weak reverse Hölder condition. The point here is that \(p<1\); in this case general Poincaré-type results are almost impossible to get, but it turns out that under the weak reverse Hölder condition imposed on \(\nabla f\) the above inequality holds. Weighted cases are treated, too.
scientific article; zbMATH DE number 889852