$a(Q,P)$, defined as in Definition \[defnWeylHS0M\], extends to a bounded operator on $L^p(\mu)$ and $$||a(Q,P)||_{B(L^p(\mu))} \leq e^\epsilon||a||_{HS_0(M)}.$$ That is, the generalised Weyl calculus extends to a bounded linear map $HS_0(M) \to B(L^p(\mu))$.

Let $U_p:L^p(\mu) \to L^p(\lambda)$ be the isometry from Definition \[defnUp\]. Then $a(Q,P)$ has a bounded extension to $L^p(\mu)$ if and only if $U_pa(Q,P)U_p^{-1}$ has a bounded extension to $L^p(\lambda)$, in which case $||a(Q,P)||_{B(L^p(\mu))} = ||U_pa(Q,P)U_p^{-1}||_{B(L^p(\lambda))}$. We will use a Young’s convolution inequality argument to show that under our conditions $U_pa(Q,P)U_p^{-1}$ extends boundedly to $L^q(\lambda)$ for all $q \in [1, \infty]$ with norm at most $e^\epsilon ||a||_{HS_0(M)}$. We then remove the isometries on $L^p$ to obtain the desired result.

From Definition \[defnWeylHS0M\], $U_pa(Q,P)U_p^{-1}$ can be expressed as an integral operator on $L^q(\lambda)$ with kernel $k:\mathbb{R}^{2d} \to \mathbb{C}$ where for $x, y \in \mathbb{R}^d$, $$k(y,x) = \frac{1}{(2\pi)^{d/2} c^d} \mathcal{F}_2a\left(\frac{x+y}{2c}, \frac{x-y}{c}\right) \exp\left(\left(\frac{1}{2}-\frac{1}{p}\right)(\phi(y)-\phi(x))\right).$$ Using the definition of $\mathcal{F}_2 a$ and $HS_0(M)$, we find $$\begin{aligned}
k(y,x) &= \frac{1}{(2\pi)^{d/2} c^d} \mathcal{F}_2a\left(\frac{x+y}{2c}, \frac{x-y}{c}\right)  \exp\left(\left(\frac{1}{2}-\frac{1}{p}\right)(\phi(y)-\phi(x))\right)\\
&= \frac{1}{c^d}g_a\left(\frac{x+y}{2c},\frac{x-y}{c}\right)\exp\left(\left(\frac{1}{2}-\frac{1}{p}\right)(\phi(y)-\phi(x))-\left|\left(\frac{x-y}{c}\right) M\left(\frac{x+y}{2c}\right)\right|\right),\\\end{aligned}$$ for a unique $g_a \in \mathcal{D}_{\infty,1}$. Our assumption implies $\left(\frac{1}{2} -\frac{1}{p}\right)(\phi(x)-\phi(y)) - \left|\left(\frac{x-y}{c}\right) M\left(\frac{x+y}{2c}\right)\right| \leq \epsilon$, which we incorporate to find $$\begin{aligned}
|k(y,x)| &\leq   e^\epsilon \frac{1}{c^d}\left|g_a\left(\frac{x+y}{2c},\frac{x-y}{c}\right)\right|\\
&\leq e^\epsilon \frac{1}{c^d}G_a\left(\frac{x-y}{c}\right),\end{aligned}$$ for some $G_a \in L^1(\mathbb{R}^d)$. Young’s convolution inequality implies 