The question is related to a technical point in Bakry-Emery calculus (page 130 of Bakry, Gentil, and Ledoux's book "Analysis and geometry of Markov diffusion operators"): given a Markov semigroup $(P_t)_{t \geq 0}$ acting on $L^2(\mu)$, we sometimes want to differentiate an expression of the form $\Psi(s) = P_s((P_{t-s}f)^2)$ for $s \in [0, t]$ where $t > 0$ is some fixed time.
In order for this expression to make sense we first need, I think, $f$ to belong to a space for which $(P_{t-s}f)^2 \in L^2(\mu)$; for instance if $(P_t)_{t \geq 0}$ is the Euclidean heat kernel on $\mathbb{R}^n$ say then we can take $f \in \mathcal{S}$ where $\mathcal{S}$ is the Schwartz space of functions with rapid decay; then $P_{t-s}f \in \mathcal{S}$ and $(P_{t-s}f)^2 \in \mathcal{S} \subset L^2(\mathbb{R}^n)$.
Another example is the Ornstein-Uhlenbeck semigroup $(P_t)_{t \geq 0}$ generated by $L = \Delta - \nabla |x|^2 \cdot \nabla$; we can also check, because we have an explicit expression for the density (the Mehler kernel), that $P_t$ preserves $\mathcal{S}$. Once we know $(P_{t-s}f)^2 \in L^2(\mu)$ where $\mu$ is a Gaussian measure, we can use the fact $\partial_sP_sf = LP_sf = P_sLf$ and then get some results.
My understanding is that one reason why Bakry-Emery calculus is powerful is the fact we do not need to know anything about the invariant measure in order to perform pointwise calculations, but it seems like we need to know (at least for these two examples) something about the heat kernels in order to proceed. The book mentions that for most applications it suffices to work on the algebra of functions $\mathcal{A}$ of smooth functions (or maybe $\mathcal{A}_0$ the space of compactly supported smooth functions), but how do we know for a given example of Markov semigroup what class of functions actually works?
This brings me to the question: what regularity results are available for second order diffusion type operators, e.g. $$ Lf = \sum_{i,j=1}^n a_{ij}(x)\partial_i\partial_jf + \sum_{i=1}^n b_i(x)\partial_if? $$ I guess the regularity should be mainly controlled by the second order part, so I expect if $b_i$ is smooth and say the second order part preserves $\mathcal{S}$ then $L$ should too. I am also interested in the case where $b_i(x)$ has some singularity, e.g. $b_i(x) = x_i/|x|$. Are there any references available? Or must we check case by case?
EDIT: Here is one example of a result I can find in the literature: if $L = \Delta + \sum_{i=1}^n b_i\partial_i$ and $\mathcal{A}$ is the space of smooth functions with all derivatives growing at most polynomially at infinity, then $\mathcal{A}$ is preserved under $L$, $P_tf$, and $\partial_tP_tf = LP_tf = P_tLf$. Are there more results of this type?