Nagel Stein Wainger estimates for balls associated with the Bismut condition (Q2702394)

Let \(X_i\), \(i\in \{0,1,\dots, m\}\), be real vector fields on \(\mathbb{R}^d\) satisfying the so-called \(r\)-weak Hörmander condition, \(\Phi_\varepsilon\) be the solution (as a flow on the Cameron-Martin space \({\mathcal H}\)) of the control PDE corresponding to \(X_i\). Define the pseudo-metrics \(d_\varepsilon(x, y)=: \inf\{\|h\|_{{\mathcal H}}: \Phi_\varepsilon(h)= y\}\) and \(d_{\varepsilon, R}(x,y)=: \inf\{\|h\|_{{\mathcal H}}:\Phi_\varepsilon\) is a submersion in \(h\) and \(\Phi_\varepsilon(h)= y\}\), the corresponding open balls by \(B_\varepsilon(x,\delta)\), \(B_{\varepsilon, R}(x,\delta)\), respectively.NEWLINENEWLINENEWLINEUnder the Bismut condition, the authors prove upper and lower uniform estimates for the volumes of \(B_\varepsilon(x\varepsilon)\), \(B_{\varepsilon,R}(x,\varepsilon)\) by \(\text{const}\cdot\sum_{I\in\Delta}|\lambda_I(x)|\varepsilon^{\|I\|}\) where \(|\Delta|\leq \left(\begin{smallmatrix} r(m+1)-1\\ d\end{smallmatrix}\right)\), \(I= \{[\alpha_1],\dots, [\alpha_d]\}\) is a set of multi-indices, \(X_{[\alpha_i]}\) is the vector field generated by the Lie bracket operation, \(\lambda_I(x)=:\text{det}(X_{[\alpha_1]},\dots, X_{[\alpha_d]})(x)\) and \(\|I\|\) is the weight of \(I\).NEWLINENEWLINEFor the entire collection see [Zbl 0953.00049].