Timeline for Distribution boundary value of analytic function and wave front sets
Current License: CC BY-SA 4.0
8 events
| when toggle format | what | by | license | comment | |
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| May 17, 2020 at 17:21 | comment | added | Dima | Yes, that's true. But in higher dimension it can happen that $\Gamma$ is smaller than $C^o$, and then it's not clear how to proceed. | |
| May 15, 2020 at 19:15 | comment | added | Bazin | Then it is true in one dimension, since you have only two directions for the variable $\xi$ in the wave-front-set and one direction is ruled out by the inclusion of Theorem 8.1.6. | |
| May 15, 2020 at 15:06 | comment | added | Dima | One can use the argument in Hormander's theorem 8.1.6 to show there is convergence in $C^{-\infty}_{C^o}$, where $C^o$ is the dual cone - that's what happens in the example you consider. However it could happen that $\Gamma$ is strictly smaller than $C^o$, and then I don't see any way to approach the convergence question in the general case. | |
| May 15, 2020 at 9:01 | comment | added | Bazin | Yes, I think so, at least in the framework of Theorem 3.1.11 in the Hörmander's ALPDO. I guess that it is indeed possible to have an explicit expression for the Fourier transform using Formula (3.1.13) there; also the homogeneous example $(x+i0)^a$ has a Fourier transform $\hat T$ supported in $\mathbb R_+$ and the Fourier transform of $(x+iy)^a$ is $e^{-y\xi} \hat T(\xi)$, so it vanishes as well on $\xi<0$. To go back to that Formula (3.1.13), it is quite likely that there is an analogous device in several dimensions. | |
| May 14, 2020 at 19:32 | comment | added | Dima | Do you think Hormander convergence $f(x+iy)\to f(x+i0)$ holds in general? Computing explicitly is only possible in very few cases. | |
| May 13, 2020 at 18:39 | comment | added | Bazin | @Dima I should have written this as a claim: for the first integral, since $\eta\in V$, thus away from the bad directions for $f(x+i0)$ you have fast decay. For instance, if you take a look at the one-dimensional homogeneous $(x+i0)^a$ with wave-front-set $\{0\}\times\mathbb R_+^*$, then I think that a direct calculation of the Fourier transform of $(x+iy)^a$ shows a uniform fast decay for $y\ge 0$ for $\xi$ negative. I believe indeed that this is the place where the assumption is used. | |
| May 13, 2020 at 16:29 | comment | added | Dima | Thanks for the answer! I don't quite understand how to think about the first summand, the one with $\chi_0$. We have the Schwartz distribution $T_y(\eta)$, which is paired with the Schwartz function $\chi_0(\eta)\hat{\phi_\gamma}(\xi-\eta)$. I don't see how to use the knowledge $\eta\in V$ to be able to write $T_y(\eta)\leq C(1+|\eta|)^{-N}$, let alone have $C$ independent of $y$. In fact, everything you write applies to any weakly converging sequence, which need not also be in Hormander topology; surely somehow the particular case of holomorphic boundary value should be used? | |
| May 13, 2020 at 12:06 | history | answered | Bazin | CC BY-SA 4.0 |