Fourier transform of Laplace Equation

if given :

has unique bounded solution:

**Use the above information to solve:**

Ok, now it's the first time i've seen the use of an exponent in a boundary condition. This question is worth 5 marks, which makes me think that my lecturer feels that this is easy,and not a lot of work. But i'm not sure of anything here.

Where should i start ? If the condition was simply then i could easily solve for in the given form of the solution and work from there, but now with that pesky exponent w, i'm not sure.

Any ideas welcome !