We know that if a function $u\in C（\Omega）$ satisfies mean value property（i.e. harmonic），it holds that $$ u*\phi_{\varepsilon}=u $$ where $\phi_{\varepsilon}$ is standard mollifier. My question is that when the formula holds, can we deduce that $u$ must be a harmonic function or other properties"
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1What exactly is $\phi_\epsilon$? This is important for your question, and "standard mollifier" does not have a universally accepted definition. – Giuseppe Negro Sep 28 '22 at 13:55