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Let $T \in \mathscr D'(\mathbb R^n)$ be a distribution, such that its Fourier transform $\widehat T$ is a real analytic function on $\mathbb R^n$ but it can't be continued to an entire function on $\mathbb C^n$. As an example of such $\widehat T$ we can take $$ \widehat T(\xi_1,\xi_2) = \frac{B(1+i\xi_1,1+i\xi_2)}{B(2+2i\xi_1,2+2i\xi_2)}, \quad (\xi_1,\xi_2) \in \mathbb R^2, $$ where $B$ is the Beta-function.

Are there some theorems that allow us to say whether some point $x \in \mathbb R^n$ is included in the support of $T$ or not, given $\widehat T$? If the answer is positive are there some theorems that allow us to find the distance (or estimate it below) between $x$ and the support of $T$? The Schwartz-Paley-Wiener theorem can't be applied here because $\widehat T$ can't be continued to an entire function. Any references are very welcome.

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I think the FBI transform might be useful here. –  Johannes Hahn Jan 19 at 19:05
    
If $T$ is in $\mathscr{S}'$, you can obtain information on the analytic wavefront set of $T$ from $\widehat{T}$. Once you have that, the my best bet to answer your question is the microlocal Holmgren theorem due to Kashiwara and Kawai, which restricts the support of $T$ through knowledge of the analytic wavefront set of $T$. It's actually more natural to rephrase this problem in the context of hyperfunctions. Then, as suggested by Johannes Hahn, it's important to move from $\widehat{T}$ to the FBI transform of $T$, which is just a Gaussian convolution away from $\widehat{T}$. –  Pedro Lauridsen Ribeiro Jan 20 at 1:18

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