Consider solving the Laplace equation $\nabla^2 u = 0$ on the upper half plane in $\mathbb{R}^2$, with boundary conditions that $u(x,0)=f(x)$ and $|u(x,y)|\leq M$.
If we Fourier transform the equation in the $x$ variable, we get
$$\mathfrak{F}_x[\partial^2_x u+\partial^2_yu]=-k^2 \mathfrak{F}_x[u]+\partial^2_y\mathfrak{F}_x[u]=0.$$
This is an ODE for each $k\in \mathbb{R}$ fixed that can be immediately solved to give
$$\mathfrak{F}_x[u](k,y)=A(k)e^{ky}+B(k)e^{-ky}.$$
Now the boundary condition $u(x,0)$ translates, to the Fourier transform, as $\mathfrak{F}_x[u](k,0)=\mathfrak{F}[f](k)$.
On the other hand we have the boundary condition $|u(x,y)|\leq M$. This "function must be bounded" boundary condition I'm in doubt on how to translate to the Fourier transform.
I mean, I have one very imprecise notion that: well, in the upper-half plane we may take $y\to \infty$ and then $e^{ky}\to \infty$ if $k>0$ so that we better get rid of that term so we may set $A(k)=0$ for $k>0$, the same for $e^{-ky}\to \infty$ when $k<0$ so that we may set $B(k)=0$ for $k < 0$.
This is horribly non-rigorous. So my question is how can we approach this rigorously? How can we impose the condition $|u(x,y)|\leq M$ when solving this problem via Fourier transform?