extended to a holomorphic function for $\eta$ in $B(0, |M(x)|)$, and such that there exists a constant $K>0$ such that for any multiindex $\alpha$ with $|\alpha| \leq (d+1)$ we have $$\sup_{u \in \mathbb{R}^d, \eta \in B(0, |M(u)|)} \int_{\mathbb{R}^{d}}\left|\partial_\xi^{(\alpha)}a\left(u, \xi+i\eta\right)\right| d\xi \leq K.$$ Then $a \in HS_0(M)$. However, this integrability condition is much stronger than what we actually require for admission into $HS_0(M)$. Although, as we shall see, some “pseudo-holomorphic" nature is always apparent for symbols in $HS_0(M)$.

\[remHolomorphicExtension\]

It is well-known that if a function $f \in L^\infty(\mathbb{R}^d)$ has Fourier transform of the form $$(\mathcal{F}f)(\xi) = (2\pi)^{\frac{d}{2}}\exp \left(-a|\xi|\right) g(\xi)$$ for some $a>0$, $g \in L^1(\mathbb{R}^d)$, then $f$ almost everywhere agrees with the restriction to $\mathbb{R}^d$ of a function holomorphic on the cylinder $\{\xi + i \eta; |\eta| < a\}$ (this can be verified via a change of contour in the integral expression for the inverse Fourier transform). Furthermore, this holomorphic extension has a continuous extension to the closure of the cylinder, and the supremum norm of said continuous extension is bounded by $||g||_{L^1(\mathbb{R}^d)}$.

There is an analogous statement for elements of $HS_0(M)$. Fix $a \in HS_0(M)$, $x, \xi \in \mathbb{R}^d$ and $t \in [-1, 1]$. Then we find $$\begin{aligned}
\frac{1}{(2\pi)^\frac{d}{2}} 