Equation $\Delta u=0$ is called the Laplace equation, btw. The answer to your question is yes. The standard reference is Garnett, Bounded analytic functions, Ch II, section 1. This is also true for $p=\infty$ (maximum principle).

Equation $\Delta u=0$ is called the Laplace equation, btw.

Edit. The answer to your question is no. Consider $f(z)=1/(z+i)$. On the real line it belongs to $L^p$ with any $p>1$. In the upper half-plane it does not belong to any $L^p$ if $p\leq 2$. If you want a real function, take a real or imaginary part of this.

Equation $\Delta u=0$ is called the Laplace equation, btw. The answer to your question is yes. The standard reference is Garnett, Bounded analytic functions, Ch II, section 1.

Equation $\Delta u=0$ is called the Laplace equation, btw. The answer to your question is yes. The standard reference is Garnett, Bounded analytic functions, Ch II, section 1. This is also true for $p=\infty$ (maximum principle).