Convolution of L^p function with a Poisson Kernel

Question

I am trying to work this problem from Papa Rudin. Any hints, suggestions on where to begin are very much appreciated. I'm not sure if I should try to show the Laplacian is zero/Mean value property holds or if there is a slicker way to do this.

Answer 1

First notice that $$\Delta(f*h_{\lambda})=f*\Delta h_{\lambda} $$ Now take the Fourier transform on the $x$ axis: $$\widehat{(\Delta h_{\lambda})}(x)=\left(-x^2+\frac{\partial^2}{\partial\lambda^2}\right)e^{-\lambda |x|}=0$$ By invertibility of the Fourier transform, and since $f*0=0$, we are done.