Analysis of degenerate elliptic operators of Grushin type

We analyze degenerate, second-order, elliptic operators $H$ in divergence form on $L_2({\bf R}^{n}\times{\bf R}^{m})$. We assume the coefficients are real symmetric and $a_1H_\delta\geq H\geq a_2H_\delta$ for some $a_1,a_2>0$ where $$H_\delta=-\nabla_{x_1} c_{\delta_1, \delta'_1}(x_1) \nabla_{x_1}-c_{\delta_2, \delta'_2}(x_1) \nabla_{x_2}^2 .$$ Here $x_1\in{\bf R}^n$, $x_2\in{\bf R}^m$ and $c_{\delta_i, \delta'_i}$ are positive measurable functions such that $c_{\delta_i, \delta'_i}(x)$ behaves like $|x|^{\delta_i}$ as $x\to0$ and $|x|^{\delta_i'}$ as $x\to\infty$ with $\delta_1,\delta_1'\in[0,1>$ and $\delta_2,\delta_2'\geq0$. Our principal results state that the submarkovian semigroup $S_t=e^{-tH}$ is conservative and its kernel $K_t$ satisfies bounds $$0\leq K_t(x ;y)\leq a (|B(x ;t^{1/2})| |B(y ;t^{1/2})|)^{-1/2}$$ where $|B(x ;r)|$ denotes the volume of the ball $B(x ;r)$ centred at $x$ with radius $r$ measured with respect to the Riemannian distance associated with $H$. The proofs depend on detailed subelliptic estimations on $H$, a precise characterization of the Riemannian distance and the corresponding volumes and wave equation techniques which exploit the finite speed of propagation. We discuss further implications of these bounds and give explicit examples that show the kernel is not necessarily strictly positive, nor continuous.