\mapsto a(x, \xi +itM(x))$ will be a real-analytic function on $\mathbb{R}^d \times (-1,1)$ and will satisfy some modified form of the Cauchy-Riemann equations.

That functions $a \in HS_0(M)$ possess for each fixed $x\in \mathbb{R}^d$ such a “pseudo-holomorphic" extension to the strip $\{\xi \in \mathbb{R}^d, \exists t \in [-1,1] \text{ s.t. } \eta = tM(x)\}$ is where the name Holomorphic Strip originated.

We should compare the symbol class $HS_0(M)$ to standard symbol classes giving rise to bounded operators through the Weyl calculus. If we take $\phi(x) = 0$ our space of functions is $L^p(\lambda)$ and our generalised Weyl pair is the standard one, in which there are many known symbol classes giving rise to bounded operators (see for instance [@Stein], Chapter 6). These classes typically assume boundedness and decay in the $\xi$ variable of sufficiently many derivatives of $a(x,\xi)$, and allow for some singular integral operators. If $\phi(x)=0$ we can take $M=0$, in which case $HS_0(0)$ will be the space of functions whose Fourier transform in the second variable is dominated by an integrable function, thus not including singular integral operators. This implies boundedness, continuity in $\xi$ and decay of $a(x,\xi)$ (by the Riemann-Lebesgue Lemma), but does not give a rate 