Let $u$ be a harmonic function in $\mathbb{R}^n$ with $\int_\mathbb{R} |u|^p < \infty$. Then $u \equiv 0$.

Question

This post Let $u$ harmonic. Then $\int_{\mathbb R^d}|u|^2<\infty \implies u=0$ answers this question in the case when $p=2$ using the Cauchy-Schwarz Inequality and an application of the Mean-Value Property. Since Cauchy-Schwarz only works for $p=2$, I was wondering if there is a generalization for $p \geq 2$. I tried it using Holder's Inequality, but I was not able to make that work.

Answer 1

Hint: Apply Hölder for $u$ and $1$ over $B(x,r)$.