$\omega \left( x, y \right)$ is an analytic function, we can find a function $\lambda \left( x, y \right)$ satisfying $$ \partial _{xx} \lambda -\partial _{yy} \lambda =\omega $$ (it's a wave equation with sources in physics).
Take Fourier transform for $x$ and $y$, we get $$ \tilde{\omega} \left( k_x, k_y \right) = -k_x^2 \tilde{\lambda} \left( k_x, k_y \right) + k_y^2 \tilde{\lambda} \left( k_x, k_y \right). $$
Now we have $$ \tilde{\omega} \left( k, k \right) = -k^2 \tilde{\lambda} \left( k, k \right) + k^2 \tilde{\lambda} \left( k, k \right) = 0. $$
But it's obvious that $\tilde{\omega} \left( k, k \right) = 0$ is not true in the most case. Where is the problem?