Derivative estimate proof of one sided harmonic liouville

The ball mean property and the continuous mean-value theorem make $u$ smooth. Its partial derivatives are therefore harmonic by the smooth derivative lemma, whose smoothness hypothesis is now satisfied.

On every sphere of radius $0<s<R$, the trace of $u$ is smooth. Harmonic replacement and uniqueness represent $u$ inside the smaller ball by this trace. Differentiate the kernel at its center and write $y=a+s\theta$. It gives $\mathrm{\nabla }u(a)=\frac{n}{s}\frac{1}{\mathrm{\mid }\mathrm{\partial }{B}_s\mathrm{\mid }}{\int }_{\mathrm{\partial }{B}_s(a)}u(y)\theta dS(y)$, while the undifferentiated center formula gives $u(a)=\mathrm{\mid }\mathrm{\partial }{B}_s{\mathrm{\mid }}^{-1}{\int }_{\mathrm{\partial }{B}_s(a)}u$. Derivative passage is justified by the same compact-sphere bounds as in replacement.

Taking vector norms and using $\mathrm{\mid }\theta \mathrm{\mid }=1$ bounds the gradient by $(n\mathrm{/}s)$ times the average of $\mathrm{\mid }u\mathrm{\mid }$. This is at most $nS\mathrm{/}s$ in the bounded signed case, and exactly $nu(a)\mathrm{/}s$ as an upper bound in the nonnegative case. Let $s\uparrow R$ to obtain both estimates.

For a nonnegative entire harmonic function, the positive estimate holds at each fixed $a$ for every $R>0$. Let $R\to \mathrm{\infty }$ to obtain $\mathrm{\nabla }u(a)=0$. This is also a second route to one-sided Liouville after shifting and, if necessary, negating the function.