Let $(M,g)$ be a complete Riemannian manifold, $\Delta$ its Laplace-Beltrami operator and $T_t = (e^{t \Delta})_{t \geq 0}$ the associated heat semigroup. We can define the subordinated Poisson semigroup $(P_s)_{s \geq 0}$ as the semigroup generated by the fractional Laplacian $A = (-\Delta)^\frac12$. Denote also by $p_s(x,y)$ the integral kernel of the Poisson semigroup and by $h_t(x,y)$ the integral kernel of the heat semigroup, ie the functions satisfying that \begin{eqnarray} (P_s f)(x) & = & \int_M p_s(x,y) \, f(y) \, d \mbox{vol}(y),\\ (T_t f)(x) & = & \int_M h_t(x,y) \, f(y) \, d \mbox{vol}(y). \end{eqnarray}
In the case of $\mathbf{R}^n$ with the usual Laplacian, the Poisson kernel satisfy the following off-diagonal estimates $$ p_s(x,y) = c_n \, \frac{1}{t^n} \frac1{\Big(1 + \frac{|x - y|^2}{t}\Big)^{\frac{n+1}{2}}} $$ (off-diagonal here is just a jargon term meaning that the kernel tends to $0$ as the distance $d(x,y) \to \infty$).
Question: Is there any reference for the existence of off-diagonal estimates for the Poisson kernel over manifolds, perhaps under certain extra hypotheses like having positive Ricci curvature.
Motivation: It is known, after the works Saloff-Coste and Grigor'yan, that if $(M,g)$ satisfy the following Poincare inequality for every $0 < r$ $$ \int_{B_x(r)} |f - f_B|^2 \, d \mbox{vol} \, \lesssim \, r^2 \int_{B_x(2 \, r)} |\nabla f|^2 \, d \mbox{vol} $$ and the measure is doubling, meaning that $\mbox{vol}(B_x(2 r)) \leq D \, \mbox{vol}(B_x(r))$, then we have (bilateral) Gaussian bounds of the form $$ \frac1{\mbox{vol}(B_x(\sqrt{t}))} e^{-c \, \frac{d(x,y)^2}{t}} \lesssim h_t(x,y) \lesssim \frac1{\mbox{vol}(B_x(\sqrt{t}))} e^{-C \, \frac{d(x,y)^2}{t}}, \label{Gaussian} \tag{G} $$ see, [SC,Theorem 5.4.12]. In principle, one can use \eqref{Gaussian} to get off-diagonal estimates of the Poisson kernel just by expressing $p_t$ in terms of $h_t$ via Stein's subordination formula and apply the bounds above. But this seems like an overkill since (intuitively) the polynomial type bounds of the Poisson kernel should be much less restrictive than the Gaussian bounds above.
[SC] Saloff-Coste, Laurent, Aspects of Sobolev-type inequalities, London Mathematical Society Lecture Note Series. 289. Cambridge: Cambridge University Press. x, 190 p. (2002). ZBL0991.35002.
