Distribution boundary value of analytic function and wave front sets

Question

Assume $f(z)$ is analytic in the tube domain $\mathbb R^n\oplus iC$, where $C\subset \mathbb R^n$ is a convex cone. Under the assumption $|f(x+iy)|\leq 1/|y|^k$, we know by a Theorem of Martineau (see also Hormander, volume 1, Theorem 3.1.15) that the limit $\lim_{y\to 0, y\in C} f(x+iy)$ exists as a tempered distribution $f(x)$ on $\mathbb R^n$, uniformly in proper cones $y\in C'\subset C$. The convergence is in the weak topology, and in fact in the strong topology on the space of tempered distributions of fixed order $k$.

Question: Let $\Gamma\subset T^*\mathbb R^n$ be the wave front set of $f(x)$. Is it true that $f(x+iy)\to f(x)$ also in the Hormander topology $C^{-\infty}_\Gamma$? If not true in general, can some conditions be given on $f$ that would ensure such convergence?

Answer 1

You need to check Condition (ii) in Definition 8.2.2 in the first volume of Hörmander's ALPDO. Let us note $f(x+i0)$ the limit-distribution of your question and let $\Gamma$ be its wave-front-set. Let $K\times V$ be a compact-conic subset of the phase space with empty intersection with $\Gamma$, let $\phi$ a smooth compactly supported function of $x$, supported in $K$. We must take a look at $$ \mathcal F\bigl(f(x+iy)\phi(x)-f(x+i0)\phi(x)\bigr)(\xi)\quad \text{for $\xi\in V$}. $$ With $T_y$ standing for the Fourier transform of $f(x+iy)-f(x+i0)$, we define for $\alpha\in \mathbb N^n$, $$ J_\alpha(\xi,y)= \xi^\alpha\int T_y(\eta) \hat \phi(\xi-\eta) d\eta= \int T_y(\eta)(\xi-\eta+\eta)^\alpha \hat \phi(\xi-\eta) d\eta. $$ We define for $\beta, \gamma\in \mathbb N^n$, $$ J_{\beta,\gamma}(\xi,y) =\int T_y(\eta)\eta^\beta(\xi-\eta)^\gamma \hat \phi(\xi-\eta) d\eta =\int T_y(\eta)\eta^\beta\ \widehat{D^\gamma \phi}(\xi-\eta) d\eta, $$ and we have for $1=\chi_0+\chi_1$, $\chi_1$ supported near $\Gamma$, $\chi_0$ supported in $V$, $$ J_{\beta,\gamma}(\xi,y)=\int\chi_{0}(\eta) T_y(\eta)\eta^\beta\ \hat{\phi}_\gamma(\xi-\eta) d\eta + \int \chi_1(\eta)T_y(\eta)\eta^\beta\ \hat{\phi}_\gamma(\xi-\eta) d\eta. $$ Let us verify the bounds and note that $\phi_\gamma$ is in the Schwartz space. The first integral ($\eta$ is there a fast-decreasing direction for $T_y(\eta)$ since we are away from the bad directions of the wave-front-set) is uniformly rapidly decreasing in $\xi$: we have $$ \vert\int\chi_{0}(\eta) T_y(\eta)\eta^\beta\ \hat{\phi}_\gamma(\xi-\eta) d\eta\vert \lesssim \int\chi_{0}(\eta) (1+\vert\eta\vert)^{-N} (1+\vert\xi-\eta\vert)^{-N} d\eta\lesssim (1+\vert\xi\vert)^{-N+n+1} $$ We check the second integral for $\xi \in V$: there we have from the empty-intersection above $$ 1+\vert\xi-\eta\vert\gtrsim 1+\vert \xi\vert +\vert \eta\vert $$ and this gives the uniform fast decay since $T_y(\eta)$ is bounded above by $(1+\vert\eta\vert)^{N_0}$: we have $$ \vert\int \chi_1(\eta)T_y(\eta)\eta^\beta\ \hat{\phi}_\gamma(\xi-\eta) d\eta\vert\lesssim \int (1+\vert\eta\vert)^{N_0+\vert \beta\vert}(1+\vert\xi-\eta\vert)^{-N} d\eta\lesssim \int (1+\vert\eta\vert)^{N_0+\vert \beta\vert}(1+\vert\xi\vert+\eta\vert)^{-N} d\eta \lesssim (1+\vert\xi\vert)^{-N+N_0+\vert \beta\vert+n+1}. $$ To complete the proof it is needed to check the convergence to 0 with $y$ of the above bounds.