Suppose $D$ is a bounded domain of $\mathbb{R}^{m}$ ($m>1$). If $h$ is harmonic on $D$, do we have $$\int_{D}|h(x)|dx<\infty?$$
2026-03-25 21:47:30.1774475250
A question on integrability of harmonic functions
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Certainly not. In $\mathbb R^3$let $D=\{0<|x|<1\}.$ The function $h(x) = |x|^{-1}$ is harmonic in $D.$ Hence all derivatives of $h$ are harmonic in $D.$ But $$\frac{\partial^2 h}{\partial x_1^2}$$ fails the integrability condition.