\left|\int_{\mathbb{R}^d} \mathcal{F}_2a(x, k)\exp(i k (\xi + i t M(x)) ) dk\right| &\leq \frac{1}{(2\pi)^\frac{d}{2}}\int_{\mathbb{R}^d} |\mathcal{F}_2a(x, k)|\exp(- t k M(x)) ) dk \\
&= \int_{\mathbb{R}^d} \exp\left(- |k M(x)| - t k M(x) \right)|g_a(x,k)|dk \\
&\leq \int_{\mathbb{R}^d} G_a(k) dk \\
&\leq ||g_a||_{\mathcal{D}_{\infty,1}} \\
&= ||a||_{HS_0(M)},\end{aligned}$$ where the infimum is over all $G_a$ dominating $g_a$ as in Definition \[defnD\]. By the Fourier inversion formula, the above agrees $\xi$-almost everywhere with $a(x,\xi)$ when $t=0$. We thus take the above as a definition of an extension of $a$ to the set $$D_M = \{ (x, \xi+i\eta) \in \mathbb{R}^d \times \mathbb{C}^d; \xi \in \mathbb{R}^d, \exists t \in [-1,1] \text{ s.t. } \eta = tM(x).$$ We denote this extension at $x, \xi, \eta$ as $a(x, \xi+ i\eta)$. For each fixed $x$, continuity as a function of $\xi$ and $t$ follows by the dominated convergence theorem. If $d=1$, for fixed $x$ this extension is holomorphic as a function of $\xi + i\eta \in \mathbb{C}$, which also follows from the DCT.

If $d \geq 2$ and fixed $x\in \mathbb{R}^d$, it does not make sense to speak of holomorphy of $a(x, \cdot)$ due to its domain not being an open subset of $\mathbb{C}^d$. However, for fixed $x \in \mathbb{R}^d$, $(\xi, t) 