Lemma: One complex variable with smooth parameters

Let $F(z,y)$ be holomorphic for $\mathrm{Im}z>0$, smooth in real parameters $y$, and locally bounded by a power of $(\mathrm{Im}z{)}^{-1}$, together with each fixed finite number of $y$-derivatives. Then it has a distributional boundary value on compact real coordinate patches, and

For a lower-half-plane boundary value, $\lambda \le 0$.

The assertion is local and remains valid after a real-analytic change of the complex transverse coordinate with the corresponding half-plane orientation.

Proof. This is the distributional boundary-value theorem with parameters HormanderI. We recall the two ingredients needed here. Sufficiently many holomorphic primitives in $z$ have controlled boundary limits; differentiating those limits defines the original distributional boundary value. The estimates with parameters make this construction continuous on joint tests.

For the wavefront assertion, localize and Fourier transform. In a cone where $\lambda <-c|(\lambda ,\nu )|$, shift the $z$ contour upward. An almost-analytic extension of the real cutoff makes its $\bar{\partial }$-error vanish to arbitrarily high order at the boundary. That order can be chosen larger than the fixed polynomial growth order of $F$. The shifted term decays exponentially in the negative $\lambda$-cone; the cutoff errors decay to every inverse power. This proves (eq:BVorientation). Reflection proves the lower-half-plane case. The same contour argument in an analytic transverse coordinate proves the last assertion.