Let $\alpha \in \mathbb{C}$. I want to prove that $$ (e^{i2\theta}\xi_1^2 + \xi_2^2 + \dots + \xi_n^2)^{-\alpha} \longrightarrow (Q(\xi)-i0)^{-\alpha}, $$
in $D’(\mathbb{R}^n\setminus \left\{0\right\})$ when $\theta \to - \frac{\pi}{2}$, where $Q(\xi) := -\xi_1^2 + \xi_2^2 + \dots + \xi_n^2$. The distribution $(Q(\xi)-i0)^{-\alpha} $ is well-defined as the pullback of $(t-i0)^{-\alpha}$ by the quadratic form $Q$ (provided by a condition verified by the wavefront set of $(t-i0)^{-\alpha} \in D’(\mathbb{R})$). I could give more details about this but I do not think it is necessary for the limit I want to find.
For that my first idea is to apply (or at least follow the proof of) Theorem 3.1.15 in The Analysis of Linear Partial Differential Operators I written by Hörmander, which I recall :
Theorem 3.1.15. Let $X$ be an open set in $\mathbb{R}^n$, $\Gamma$ an open convex cone in $\mathbb{R}^n$, and set for some $\gamma > 0$
$$ Z = \left\{ z \in \mathbb{C}^n ; \mathrm{Re}(z) \in X, \mathrm{Im}(z) \in \Gamma, \vert \mathrm{Im}(z) \vert \leq \gamma \right\}. $$ If $f$ is an analytic function in $Z$ such that
$$ \vert f(z) \vert \leq C \vert \mathrm{Im}(z) \vert^{-N}, \; \; \; z \in Z, $$
then $f(. + iy)$ has a limit $f_0 \in D’^{N+1}(X)$ as $y \to 0$, that is,
$$ \lim_{y \to 0} \int_X f(x+iy)\,\phi(x) \, dx = \langle f_0, \phi \rangle, \; \; \; \phi \in C_0^{N+1}(X).$$
But I have hard time finding the right sets $X$ and $\Gamma$ to describe the situation I am interested in. Do you think that could do it ? Or is there another way to prove this ? Any help would be appreciated. Thanks.
