This question is related to a prior question i asked, see Fourier Transform ; half space elliptic baby problem.
Essentially I am asking the same question now but taking a lot more care.
So lets examine the solutions of $ \Delta u(x,y) =0$ in $ (x,y) \in R \times (0,\infty)$ with $ u(x,0)=0$ and suppose we impose $|u(x,y)| \le C y$ some constant $C$. So of course we know the only solutions are $ u(x,y)=Cy$.
The goal of this question is to use a Fourier transforms approach to this Liouville type theorem and not lose any solutions. Now since $u(x,y)=Cy$ is a solution then we expect the Fourier transform in $x$; $ \hat{u}(\xi,y)= C_0 \delta_0(\xi) y$. So our goal is that when we perform the Fourier transform we don't lose this solution. (so the real goal is to prove a Liouville theorem for a more involved equation using a Fourier transform approach and hence I can't lose any solutions).
So if we take Fourier transform in $x$ we arrive at $ \partial_{yy} \hat{u}(\xi,y) - \xi^2 \hat{u}(\xi,y)=0$ and with $ \hat{u}(\xi,0)=0$. Solving this ode and taking into account the boundary condition we should get $$ \hat{u}(\xi,y)=A(\xi) \left( e^{\xi y} - e^{-\xi y} \right).$$
Now let $ \psi=\psi(\xi)$ denote some compactly supported smooth function and we write out $$ \int_{R} \hat{u}(\xi,y) \psi(\xi) d \xi = \int_{R} u(x,y) \hat{\psi}(x) dx$$ and substitute the formula for $ \hat{u}$ to arrive at $$\int_{R} A(\xi) \left( e^{\xi y} - e^{- \xi y} \right) \psi(\xi) d \xi = \int_{R} u(x,y) \hat{\psi}(x) dx.$$ Now fix $ \xi_0 >0$ and assume $ \psi$ is localized around $ \xi_0$ and break apart the left hand side and send $ y \rightarrow \infty$ and use the bound for $u$. This seems to suggest that $A(\xi_0)=0$ (or at least a.e. $ \xi$ near $ \xi_0$). Now do the same for $ \xi_0<0$. So this appears to suggest $ A(\xi)=0$ for all $ \xi \neq 0$. So this would say that $ \xi \mapsto \hat{u}(\xi,y)$ is supported on $ \{ \xi=0\}$ which seems to be allowing for the possibility of the Dirac mass.
So I would like to try and make the above rigourous. Note a weak formulation for the ODE satisfied by $ y \mapsto \hat{u}(\xi,y)$ is $$ 0 = \int_0^\infty \hat{u}(\xi,y) \left( \gamma''(y) - \xi^2 \gamma(y) \right) dy \qquad (*)$$ for all $ \gamma \in C_c^\infty([0,\infty) )$ with $\gamma(0)=0$. Note that since $\gamma(0)=0$ but not identically zero near $y=0$ this should encode the boundary condition $ \hat{u}(\xi,0)=0$. So our goal is to try and prove this weak formulation. Since $ u$ satisfies an equation we can use weak formulation to that to see $$0= \int_0^\infty \int_{R} u(x,y) \Delta ( \gamma(y) \hat{\psi}(x)) dx dy$$. If you expand this and use some Fourier transform result it appears you can arrive at $$0= \int_{R} \psi(\xi) \left( \int_0^\infty \hat{u}(\xi,y) \left( \gamma''(y) - \xi^2 \gamma(y) \right) dy \right) dx$$ and since $ \psi$ arbitrary it appears we have (*) at least for a.e. $ \xi$.
So now maybe we can use some ODE results to say that for a.e. $ \xi$ this solution is smooth and agrees with the formula we got before.
QUESTION. Is there any hope of arguing like this to arrive at the desired Liouville type theorem or is this a hopeless approach? any comments are greatly appreciated.
