Let $G$ a connected unimodular Lie group, endowed with Haar measure $X={X_1,\cdots,X_k}$ a Hörmander system of left-invariant vector fields. The sublaplacian $\Delta = - \sum_{i=1}^k X_i^2$ generates a symmetric Markovian semigroup $H_t$ and $H_t$ admits a right convolution kernel $h_t$ such that $$H_tf(x) = \int_G h_t(y^{-1}x) f(y)dy. $$
For Lie groups G with exponential growth, i.e., there exists $D \in \mathbb{N}$ such that $V (t) \approx t^D$, $t> 1$ ( $V(t)$ is the Haar measure of ball of radius $t$), is shown in Lemma VIII 2.5 of Analysis and Geometry on Groups
Lemma VIII 2.5. For every $m \in \mathbb{N}$ and $p \geq 1$, there exists $C_m$ such that $$ \| \left( \dfrac{\partial}{ \partial t} \right)^m h_t(\cdot) \exp(\alpha \rho(\cdot)) \|_p \leq C_mt^{-m} V(\sqrt{t})^{-1/p} e^{C_m \alpha^2 t}, \quad \alpha>0, t>0. $$
The following corollary that results from this Lemma, gives the analyticity of the semigroup in $L^1:$
VIII.2.6 Corollary The semigroup $(H_t)_{t>0}$ is bounded analytic on $L^1$ when G has polynomial growth.
The proof of Corollary follows from Lemma VIII 2.5. by noting that: $\| \dfrac{\partial h_t}{ \partial t} \|_1 \leq C t^{-1}. $
I have two questions, first, I'm not sure what criterion the author used to say that the semigroup is analytic based on the calculation of the norm of the derivative of the semigroup in $L^1$, it seems to me that he used Theorem 5.2 (d) from section 2.5 from Pazy's book, a condition like $\|A H_t \| \leq Ct^{-1}$, where $A$ is the infinitesimal generator of $H_t$, but you need to know more about the semigroup, that is, that it is of class $C_0$, differentiable and $0 \in spec(A)$ .
Second, and more interesting, would the semigroup be analytic also in $L^p$, $p>1$? What would be the problem in $L^p$ to guarantee analyticity? It is not possible to try to do a calculation of the same type as the Lemma VIII 2.5.?
