Hamilton type gradient estimate for the sub-elliptic operators (Q2256559)

Let \(M \) be a smooth, connected, finite-dimensional manifold endowed with a smooth measure and a symmetric, second-order, locally sub-elliptic and negative diffusion operator \(L \) satisfying \(L 1=0 \) as in [\textit{F. Baudoin} et al., Math. Ann. 358, No. 3--4, 833--860 (2014; Zbl 1287.53025)]. Furthermore, one assumes that \( M \) is equipped with another symmetric, positive and first-order differential bilinear form \(\Gamma^Z : C^\infty (M) \times C^\infty (M) \to C^\infty(M)\) satisfying \[ \Gamma^Z ( f g, h) = f \Gamma^Z (g, h) + g \Gamma^Z ( f, h). \] Introduce now the second-order differential forms \[ \Gamma_2 ( f, g) =\frac{1}{2}( L\Gamma( f, g) -\Gamma( f, Lg) -\Gamma(g, L f ) ) \] and \[ \Gamma^Z ( f, g) =\frac{1}{2} ( L\Gamma^Z ( f, g) - \Gamma^Z ( f, Lg) - \Gamma^Z (g, L f ) ) \] on \(M \). As in the above mentioned paper, the operator \(L \) is said to satisfy the generalized curvature-dimension inequality \(\mathrm{CD}(\rho_1,\rho_2,k,d) \) if there exist constants \(\rho_1\in\mathbb R\), \(\rho_2\geq 0\), \(k\geq 0\), \(d\in [2,\infty]\), such that the inequality \[ \Gamma_2(f,f)+\nu \Gamma_2^Z(f,f)\geq \frac{1}{d}(L(f)^2)+\left(\rho_1-\frac{k}{\nu}\right)\Gamma(f,f)+\rho_2 \Gamma^Z (f,f) \] holds for every \(f \in C^\infty (M) \) and every \(\nu > 0 \). Examples of operators \(L\) satisfying these conditions can be found in [\textit{F. Baudoin} and \textit{N. Garofalo}, ``Curvature-dimension inequalities and Ricci lower bounds for sub-Riemannian manifolds with transverse symmetries'', Preprint, \url{arXiv:1101.3590}]. Under these conditions on \(L\), the author proves a Hamilton gradient-type estimate for smooth bounded functions on \(M \) and as a corollary a pointwise estimate for bounded \(L \)-harmonic functions and also (if \(\rho_1\geq 0) \) the following gradient estimate for the heat kernel \(p(x,y,t) \) of \(L \): there exists a constant \(C=C(\rho_2,k,d) \) such that \[ \Gamma_y(\log p(x,y,t))+\rho_2 t \Gamma^Z_y(\log p(x,y,t))\leq \frac{C}{t}\left(1+\frac{d^2(x,y)}{t}\right) \] for all \( t>0, x,y\in M \), where \(d \) denotes the canonical distance on \(M \) determined by \(L \).