the form $g \exp\left(\left(\frac{1}{2}-\frac{1}{p}\right)\phi \right)$ for $g \in C_c^\infty(\mathbb{R}^d)$, which are all bounded since $\exp\left(\left(\frac{1}{2}-\frac{1}{p}\right)\phi \right)$ is continuous and hence bounded on the compact set $\text{supp}(g)$. Let $X \subset L^p(\lambda)$ be those functions which have compact support. Note that $X$ is dense in $L^p(\lambda)$ and $U_p\mathcal{C}_\phi$ is contained in $X$, so if we can show $U_p\mathcal{C}_\phi$ is dense in $X$ then we are done. To this effect, fix some $f \in X$. Since $f$ is compactly supported, $\exp\left(-\left|\frac{1}{2}-\frac{1}{p}\right)\phi\right)$ attains a maximum on $\text{supp}(f)$, and so $F(\cdot) = f(\cdot)\exp\left(-\left(\frac{1}{2}-\frac{1}{p}\right)\phi(\cdot)\right) \in L^p(\lambda)$. Since $C_c^\infty(\mathbb{R}^d)$ is dense in $L^p(\lambda)$, we can choose a sequence $\{g_n\} \subset C_c^\infty(\mathbb{R}^d)$ (furthermore, each with support $\text{supp}(g_n) \subset 2\text{supp}(f)$, say) such that $||F-g_n||_{L^p(\lambda)}\to 0$. Thus we have: $$\begin{aligned}
||f(\cdot)-g_n(\cdot)\exp\left(\left(\frac{1}{2}-\frac{1}{p}\right)\phi(\cdot)\right)||_{L^p(\lambda)}^p &= \int_{2\text{supp}(f)}\left|f(x)-g_n(x)\exp\left(\left(\frac{1}{2}-\frac{1}{p}\right)\phi(x)\right)\right|^p dx \\
&= \int_{2\text{supp}(f)}\exp\left(\left(\frac{p}{2}-1\right)\phi(x)\right)|F(x)-g_n(x)|^p dx \\
&\leq C \int_{2\text{supp}(f)}|F(x)-g_n(x)|^p dx \\
&\leq C ||F - g_n||_{L^p(\lambda)}^p \\
&\to 0.\end{aligned}$$ So we are done.

The formula in Theorem \[thmKernel\] and the formula for the semigroup generated by $\frac{1}{2}(Q^2+P^2-d)$ (Theorem \[TheoremSGEasy\]) is what allowed van Neerven and Portal to deduce such significant results for the classical Ornstein-Uhlenbeck semigroup via the Weyl calculus in [@vNPWeylPairs]. The explicit formula for kernels of $a(Q,P)$ will allow us to work on boundedness on 