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However, we prove that if $\\psi$ is invariant under rotation by $\\frac{2\\pi}{m}$, for some $m\\geq 3$, and $\\Delta \\psi$ is locally bounded, then $$\\sup_{x\\in B_1(0)}\\frac{|\\nabla \\psi(x)|}{|x|}<\\infty.$$ This is sharp in that there are examples of functions $\\psi$ for which $\\Delta \\psi$ is locally bounded, which are invariant under rotation by $\\pi$ with $|\\psi(x)-\\psi(0)|\\approx |x|^2 |\\log|x||$ as $|x|\\rightarrow 0$. 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