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friends.

My question is the following:

So I need to find a Green function (an inverse operator kernel) for an operator, which is 'kind of similar' to a transport operator.

$$ L = \frac{\partial}{\partial x_1} + \frac{\partial}{\partial x_2} - i $$

writing an equation, I get:

$$ Lg(x) = \delta(x) $$

Using Fourier transform, the equation is turned into an algebraic one:

$$ (-iy_1 - iy_2 - i)h(y) = 1 $$

I cannot solve it further.

Namely, there are three problems:

0) $ 1/(-iy_1 - iy_2 - i) $ doesn't belong to the Schwarz space, which means that I need to fins some regularization of it. How? If the function only had a point support -- I could have done something like a PV integral, but here I am puzzled.

1)Since $ 1/(-iy_1 - iy_2 - i) $ doesn't belong to Schwarz space, the homogenious equation has non-trivial roots.

$ (-iy_1 - iy_2 - i) h(y) = 0 $

Again, I don't know how to find them. They could have been just deltas at various points of the domain, where the inverse doesn't exist, but here there is a whole line.

2) I don't know how to find an inverse Fourier of $ 1/ (-iy_1 - iy_2 - i) $, since it has a singularity on the real line and thus I don't know how to apply residue theory to them.

Any suggestions?

Actually, I have already asked for help, and received the following hint:

http://www.codecogs.com/gif.latex?\hat{F}^{-1}_{k_1}\left[%20\frac{1}{-ik_1%20-%20ik_2%20-i}\right]%20=%20e^{-i(1+k_2)x_1}\left(\frac{1}{2}-\theta(x_1)\right)

But to be honest, I don't know how to use it :-( I guess, it should help me take the inverse Fourier, but I am yet far from there.

Any suggestions (or links to book chapters) would be greatly appreciated.

Actually, I know a solution for the case when we have 1 instead of $i$. Then the operator is exactly a transport equation, and It is solved by plain dividing by $1/(−iy1−iy2−1)$ , but here it is not the case.

Also, I kind of know something about the homogeneous part.... just by plugging in $e^{ix_1 + ix_2}$

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    A change of variables gives you an operator $L = \partial_t - const $ where variables are $s=x_1-x_2$ and $t=x_1+x_2$. I'd start by looking at distributions of the form $1(\xi)\otimes PV(1/\eta)$ to solve the problem after Fourier transform. – 2017-02-16
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    Mmmm, the operator becomes simpler, true, but the right part becomes worse... I mean, it probably works for the homogeneous case and explains why the last term is exponential. But for the inhomogeneous part, I't have to make a substitution the the right hand side as well, under the delta, which would complicate things a bit. – 2017-02-17
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    I think the hint was about "Instead of applying the inverse FT with respect to both variables, apply inverse FT first with respect to one variable, and then with respect to another". – 2017-02-17

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