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 $u\in L^p(\mathbb{R}\times\mathbb{R}^+)$?

Maybe this is rather obvious, but I'm stuck. Let's consider the Laplace 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 $u\in L^p(\mathbb{R}\times\mathbb{R}^+)$?