Let $(X,\mu)$ be a measure space and $T_t : L_2(\mu) \to L_2(\mu)$ for $t \geq 0$ a symmetric Markovian semigroup. Local ultracontractivity estimates of the form:

$$ \| T_t : L_p(\mu) \to L_q(\mu)\| \leq \frac{C_{p,q}}{t^{\frac{d}{2} ( \frac{1}{p} - \frac{1}{q} )}} \text{, } t \leq 1. $$

had been studied in connection with local Sobolev inequalities, see for example theorem II.4.2 of [1]. In particular they are all equivalent for different $p < q$. Abusing slightly the terminology we will say that the bounds on the norm $\| T_t : L_p \to L_q \|$ are polynomial if they are of the form $t^{- \alpha}$.

**Question 1** Are there concrete examples of symmetric Markovian semigroups for which the optimal bound of $\| T_t : L_p(\mu) \to L_q(\mu)\|$ with $p < q$ is not polynomial. More generally, given a decreasing function $\Phi : (0,1] \to [0,\infty)$ are there any known techniques to construct $(X,\mu,(T_t)_t)$ such that:

$$ C_1 \Phi(t) \leq \| T_t : L_2(\mu) \to L_\infty(\mu)\| \leq C_2 \Phi(t) \text{, } t \leq 1. $$

**Question 2** If $(G,\mu)$ is a LCH unimodular group with its Haar measure and $(T_t)_t$ is a right (resp. left) invariant symmetric Markovian semigroup, can local ultracontractivity estimates be non polynomial? For which pairs $(G, (T_t)_t )$ we are forced to have polynomial bounds?

For the second question there seems to be some partial results known in the literature. For instance, if $G$ is an unimodular Lie group and $T_t = e^{-t L}$ where the infinitesimal generator $L$ can be expressed as:

$$ L = \sum_{i=1}^{k}{X^{\ast}_i X_i} = - \sum_{i=1}^{k}{X_i X_i} $$

for some $\{X_i\}_i$ generating the whole Lie algebra. Then, using the argument of 5.6.1 in [2], one can prove an scale invariant Poincaré inequality. As small balls grow polynomially for the subriemannian metric $d_L$ associated to $L$ we have that $(G,d_L,\mu)$ is a doubling metric measure space for small balls. Following 5.5.1 in [2] a scale invariant Poincaré inequality together with a doubling condition for small balls implies a Parabolic Harnack principle for small regions. Using the parabolic Harnack principle bilateral Gaussian bounds can be found for the heat kernel, obtaining:

$$ \frac{C_1}{\mu(B_e(\sqrt{t}))} e^{\beta_1 \frac{d(x,e)^2}{t}} \leq h_t(x) \leq \frac{C_2}{\mu(B_e(\sqrt{t}))} e^{\beta_2 \frac{d(x,e)^2}{t}} \text{, } t \leq 1. $$

Since the volume of balls for small $t$ is comparable to $t^{d_0}$, where $d_0$ is the homogeneous dimension, the optimal ultracontractivity estimates are polynomial.

Can this construction be extended to more general infinitesimal generators in Lie groups or more general LCH groups?

My intuition is that, since measure spaces with a Markovian semigroup are a very big class, the first question will have a positive answer. For the second question the extra rigidity of working with invariant semigroups will reduce the possible ways of decay and maybe the answer is negative.

[1] Varopoulos, Saloff-Coste & Coulhon: The Analysis and Geometry of Groups.

[2] Saloff-Coste , "Aspect of Sobolev Type Inequalities".