I found some assertion in this article (bottom of third page) of François Trèves which in a way states that taking a bounded simply connected domain $D$, $H$ the open upper half plane, a holomorphic function $f$ in $D\cap H$, such that, given an integer $m$, any compact $K$, $\mid f(z)\mid<\frac{1}{\mid\Im(z)\mid^{m}}$ for $z\in K\cap D\cap H$. Then it is possible to find a holomorphic function $g$, continuous in $\overline{D\cap H}$ such that
$$f(x+it) = \left(\frac{\partial}{\partial t}\right)^{m+1}g(x+it)$$
Anyone knows how to prove that ? (I am not concerned with the polynomial growth condition, because even for abounded function, $m=0$, I don't know how to do that)
2 remarks :
This is used to prove the following well-know complex analysis fact : given an integer $m$, any compact $K$, and a holomorphic function on a domain $D = U+i\Gamma$, where $U$ is an open set in the real space, $\Gamma$ a cone not containing $0$, $\mid f(z)\mid<\frac{1}{\mid\Im(z)\mid^{m}}$ for $z\in K\cap D$, the following limit exists
$$\lim_{\varepsilon\rightarrow 0}\int_{D}f(x+i\varepsilon\gamma)\phi(x)dx$$
where $\gamma\in\Gamma$
- I precise that in his article, this in stated in a several complex variables context, using holomorphic functions on wedges with real edges