Maybe this is rather obvious, but I'm stuck. Let's consider the Poisson equation in the upper half plane with boundary condition $g$, $i.e.$
$$
\Delta u(x,y)=0, u(x,0)=g(x).
$$
Then the solution is given by the Poisson integral, 
$$
u(x,y)=P_y*g.
$$
Then I know some pointwise bounds if $g$ is good enough. The question is:

If $g\in L^p(\mathbb{R}),$ $1<p<\infty$, can I say $u\in L^p(\mathbb{R}\times\mathbb{R}^+)$?