extendability and that $$\begin{aligned}
||U_pa(Q,P)U_p^{-1}||_{B(L^q(\lambda))} &\leq e^\epsilon\frac{1}{c^d}\int_{\mathbb{R}^d}G_a\left(\frac{x}{c}\right)dx \\
&= e^\epsilon\int_{\mathbb{R}^d}G_a\left(x\right)dx.\end{aligned}$$ Taking infimum over all $G_a$ dominating $g_a$ in the sense of Definition \[defnD\] gives $||U_pa(Q,P)U_p^{-1}||_{B(L^q(\lambda))} \leq e^\epsilon||a||_{HS_0(M)}$. Removing the isometries on $L^p$ we obtain our desired result.

\[remAffine\] If $p=2$, the function taking value $0$, and $\epsilon=0$ is a valid growth pair for any $\phi \in C^2(\mathbb{R}^d)$. If $\phi(x) = xN(x) + lx + \epsilon(x)$, where $N$ is a linear map $\mathbb{R}^d \to \mathbb{R}^d$, $l \in \mathbb{R}^d$, and $\epsilon$ is a bounded $C^2$ function, then $M$ can be taken to be an affine function of $x$ with real-symmetric linear part, depending only on $c, p, N$ and $l$. We will not make such an assumption until Section \[secMoreHSM\], and so we will keep $M$ as a general measurable function from $\mathbb{R}^d$ to itself unless otherwise specified.

Note that for $\phi$ of the form $x \mapsto xN(x) + lx + \epsilon(x)$ with $\epsilon(x)$ a bounded $C^2$ function, an operator will be bounded on $L^p(\mu)$ if and only if it is bounded on $L^p(\tilde{\mu})$, where $\tilde{\mu}$ is associated with $\tilde{\phi}(x) = xN(x) + lx$, as the two measures are equivalent and the Radon-Nikodym derivative of one with respect to the other is 